numerical solution of boundary-value and initial-boundary-value problems using...

Ghulam Ishaq Khan Institute of

Engineering Sciences and Technology

NUMERICAL SOLUTION OF

BOUNDARY-VALUE

AND

INITIAL-BOUNDARY-VALUE PROBLEMS

USING SPLINE FUNCTIONS

By

Fazal-i-Haq

Supervised by

Professor Dr. Syed Ikram Abbas Tirmizi

Co-Supervisor

Dr. Siraj-ul-Islam

This thesis is submitted in partial fulfillment of

The requirement of Degree of Doctor of Philosophy (PhD)

in Engineering Sciences

May 2009

Faculty of Engineering Sciences

GIK Institute of Engineering Sciences and Technology,

Topi, Pakistan.

ii

Dedicated to

My Late Mother,

Wife and Children

iv

Acknowledgements

I would like to bow my head before Allah Almighty, the Most Gracious and the Most

Merciful, whose benediction bestowed upon me talented teachers, provided me

sufficient opportunity and enabled me to undertake and execute this research work.

Throughout the completion of this work, I have been supported and guided by

several people. I would like to take this opportunity to express my gratitude to all those

people. My deepest and sincere gratitude and appreciation goes to my supervisor

Professor Dr. S. I. A. Tirmizi for his guidance at each stage of this work. His patience,

encouragement and support have been very valuable in the completion of this thesis.

His advices were always stimulating and helpful when I was facing difficulties in my

research. His mission of producing high-quality work will always help me grow and

expand my thinking. Secondly, I wish to express a sincere acknowledgement to my co-

supervisor Dr. Siraj-ul-Islam for all the guidance, kind comments and constructive

criticism which have made the long hours of labor worthwhile. From the very

beginning, his dedication to the work, his concern for my well being and his confidence

in me and my work gave me the energy and inspiration to work through the difficulties.

I am also thankful to all my teachers in the Faculty of Engineering Sciences for their

encouragement during this research work.

I wish to record my deepest obligations to friends, parents, brothers and sisters for

their unfailing support and many sacrifices at every stage during these years. I would

like to thank my late mother, wife and children for their love, care, understanding,

encouragement and enduring support. I am also grateful to entire staff and research

students of faculty of Engineering Sciences, GIK Institute of Engineering Sciences and

Technology, Topi, for their help and support in one way or another during the course of

my research.

I would also like to gratefully acknowledge NWFP Agricultural University

Peshawar for study leave and to Higher Education Commission (HEC) Islamabad for

financial assistance during my Ph.D.

Fazal-i-Haq

May, 2009

v

Published and submitted papers

Certain aspects of this dissertation are based on the following published/submitted

papers:

Published Journal papers

[1] S. Islam, S. I. A. Tirmizi, F. Haq (2007): Quartic non-polynomial splines

approach to the solution of a system of second-order boundary-value problems.

Int J. High Performance Computing Applications 21 (1), 42-49.

[2] S. Islam, S. I. A. Tirmizi, F. Haq, M. A. Khan (2008): Non-Polynomial splines

approach to the solution of sixth-order boundary-value problems. Appl. Math.

Comput. 195, 270-284.

[3] S. Islam, S. I. A. Tirmizi, F. Haq, S. K. Taseer (2008): Family of numerical

methods based on non-polynomial splines for solution of contact problems.

Comm. Nonlinear Science and Numer. Simul. 13, 1448-1460.

[4] S. I. A. Tirmizi, F. Haq, S. Islam (2008): Non-polynomial spline solution of

singularly perturbed boundary-value problems. Appl. Math. Comput. 196, 6-16.

Submitted Journal papers

[5] F. Haq, S. I. A. Tirmizi, S. Islam (2009): A numerical solution of Kuramo-

Sivashinsky equation using quartic and quintic B-splines. J. Comput. Math.

[6] F. Haq, S. Islam, S. I. A. Tirmizi (2009): Numerical solution of MRLW

equation using quintic B-spline functions. Appl.Math. Comput.

[7] S. Islam, S. I. A. Tirmizi, F. Haq (2009): Collocation method using quartic B-

Splines for numerical solution of the modified equal width equation. J. Comput.

Appl. Math.

[8] S. I. A. Tirmizi, F. Haq, S. Islam (2009): A numerical technique for solution of

the MRLW equation using quartic B-splines. Appl. Math. Modell.

Fazal-i-Haq

vi

Abstract

The following two types of problems in differential equations are investigated:

(i) Second and sixth-order linear and nonlinear boundary-value problems in

ordinary differential equations using non-polynomial spline functions.

(ii) One dimensional nonlinear Initial-boundary-value problems in partial

differential equations using B-spline collocation method.

Polynomial splines, non-polynomial splines and B-splines are introduced. Some

well known results and preliminary discussion about convergence analysis of

boundary-value problems and stability theory are described.

Quartic non-polynomial spline functions are used to develop numerical methods for

computing approximations to the solution of linear, nonlinear and system of second-

order boundary-value problems and singularly perturbed boundary-value problems.

Convergence analysis of the method is discussed.

Numerical methods for computing approximations to the solution of linear and

nonlinear sixth-order boundary-value problems with two-point boundary conditions are

developed using septic non-polynomial splines. Second-, Fourth- and Sixth-order

convergence is obtained.

Numerical method based on collocation method using quartic B-spline functions for

the numerical solution of one-dimensional modified equal width (MEW) wave equation

is developed. The scheme is shown to be unconditionally stable using Von-Neumann

approach. Propagation of a single wave, interaction of two waves and Maxwellian

initial condition are discussed.

Algorithms based on quartic and Quintic B-spline collocation methods are designed

for the numerical solution of the modified regularized long wave (MRLW) equation.

Stability analysis is performed. Propagation of a solitary wave, interaction of multiple

solitary waves, and generation of train of solitary waves are also investigated.

Quartic and quintic B-spline functions have been used to develop collocation

methods for the numerical solution of Kuramoto-Sivashinsky (KS) equation. Also,

using splitting technique, the equation is reduced to a problem of second order in space.

Using error norms L2 and L∞ and conservative properties of mass, momentum and

energy, accuracy and efficiency of the suggested methods is established through

comparison with the existing numerical techniques. Performance of the algorithms is

tested through application of the methods on benchmark problems.

vii

CONTENTS

Acknowledgement iv

Published and submitted papers v

Abstract vi

Contents vii

CHAPTER 1 Introduction

1.1 Historical note and literature survey 1

1.2 Existence and uniqueness of two-point boundary-value problems 3

1.3 Polynomial and non-polynomial spline functions 5

1.4 B-Splines 7

1.5 Von Neumann stability analysis 8

1.6 Convergence of two-point boundary-value problems 9

1.7 Newton’s method 13

1.8 Summary 14

CHAPTER 2 Non-Polynomial Splines Methods for the Solution of

Second-Order Boundary-Value Problems

2.1 Introduction 17

2.2 Numerical Methods 20

2.2.1 Second-order convergence 23

2.2.2 Fourth-order convergence 23

2.2.3 Sixth-order convergence 24

2.3 Convergence Analysis 24

2.4 Numerical results and discussion 27

2.4.1 Linear problems 28

viii

2.4.2 Non-linear problem 29

2.4.3 System of differential equations 29

2.4.4 Singular perturbation problems 30

2.5 Conclusion 51

CHAPTER 3 Non-Polynomial Splines Approach to the Solution of

Sixth-Order Boundary-Value Problems

3.1 Introduction 52

3.2 Numerical Methods 54

3.3 Numerical methods of different orders 60

3.3.1 Second-order convergence 60

3.3.2 Fourth-order convergence 60

3.3.3 Sixth-order convergence 61

3.3.4 Eight-order convergence 61

3.4 Properties of the coefficient matrix A0 62

3.5 Convergence 66

3.6 Numerical results and discussion 68

3.6.1 Non-linear problems 68

3.6.2 Linear problems 68

3.7 Conclusion 76

CHAPTER 4 Collocation Method using Quartic B-Spline for

Numerical Solution of the Modified Equal Width

Equation

4.1 Introduction 77

4.2 Quartic B-spline solution 78

4.3 Stability analysis 81

4.4 Test problems and discussion 82

4.4.1 A single solitary wave 83

3.3.2 Interaction of two solitary waves 84

3.3.3 The Maxwellian initial condition 85

4.5 Conclusion 95

ix

CHAPTER 5 Solitary Wave Solutions of the Modified Regularized

Long Wave Equation

5.1 Introduction 96

5.2 The B-spline collocation methods 97

5.2.1 Quartic B-spline collocation method 97

5.2.2 Quintic B-spline collocation method 99

5.3 Stability of the proposed scheme 101

5.3.1 Stability of scheme based on quartic B-spline

collocation method

101

5.3.2 Stability of scheme based on quintic B-spline

collocation method

102

5.4 Numerical Tests and Results 103

5.4.1 Single solitary wave 103

5.4.2 Two solitary waves 105

5.4.3 Three solitary waves 106

5.4.4 The Maxwellian initial condition 106

5.5 Conclusion 122

CHAPTER 6 Collocation Method using B-Splines for Numerical

Solution of Kuramoto-Sivashinsky Equation

6.1 Introduction 123

6.2 The B-spline collocation methods 124

6.2.1 Quintic B-Spline Collocation Method 1 124

6.2.2 Quintic B-Spline Collocation Method 1I 126

6.2.3 Quartic B-Spline Collocation Method 129

6.3 Numerical validation 131

6.4 Conclusion 137

Future Work 138

References 140

Chapter 1

Introduction

1.1 Historical note and literature survey

The rapid development of spline functions is primarily due to their great usefulness in

applications. Classes of spline functions possess many nice structural properties as well

as excellent approximation powers. Splines have many applications in the numerical

solution of a variety of problems in applied mathematics and engineering. Some of

them are, data fitting, function approximation, Integro-differential equations, optimal

control problems, Computer-Aided Geometric Design (CAGD), Wavelets and so on.

Programs based on spline functions have found their way in most of computer

applications.

It is commonly accepted that the first mathematical reference to splines was made in

the year 1946 in an interesting paper by Schoenberg (Schoenberg [121]) , which is

probably the first place that the word "spline" is used in connection with smooth,

piecewise polynomial approximation. However, the ideas have their roots in the aircraft

and shipbuilding industries. Splines are types of curves, originally developed for ship-

building in the days before computer modeling. Naval architects needed a way to draw

a smooth curve through a set of points. The solution was to place metal weights

(called knots) at the control points, and bend a thin metal or wooden beam (called

a spline) through the weights. The physics of the bending spline meant that the

influence of each weight was greatest at the point of contact and decreased smoothly

further along the spline. To get more control over a certain region of the spline, the

draftsman simply added more weights. This scheme had obvious problems with data

exchange. There was a need for mathematical way to describe the shape of the

curve. Cubic Polynomials splines are the mathematical equivalent of the draftsman's

wooden beam. Through the advent of computers, splines have gained more importance.

They were first used as a replacement for polynomials in interpolation and then as a

tool to construct smooth and flexible shapes in computer graphics.

Chapter 1 Introduction

2

As late as 1960, there were no more than a handful of papers mentioning spline

functions by name. Some of the papers which have made great contributions in the

development of splines include (Ahlberg and Nilson [6], Brikhoff and Garabedian [25],

Loscalzo and Talbot [85], Maclaren [86], Rubin and Khosla [115], Sastry [120],

Schoenberg [122]). Univariate splines were studied intensely in the 60s, and by the

mid-70s they were sufficiently well understood to permit a fairly comprehensive

treatment in books form. Some of the books which discuss splines thoroughly include

(Ahlberg et al. [7], deBoor [35], Prenter [102], Schumaker [123], Shikin and Plis [124],

Spath [136]). Some of the earliest papers using spline functions for smooth approximate

solution of ordinary and partial differential equations (PDEs) include (Albasiny and

Hoskins [11], Bickely [21], Crank and Gupta [32], Jain and Aziz [64], Jain and Aziz

[65], Rubin and Khosla [115], Sastry [120], Usmani [152], Usmani and Sakai [153],

Usmani and Warsi [154]) . These papers demonstrate the approximate methods of

solving second-, third- and fourth-order linear boundary-value problems (BVPs) and

solution of elliptic and parabolic equations by spline functions of various degrees.

Today, there are hundreds of research papers on this subject, and it remains an active

research area. Recently, non-polynomial spline method has turned out to be an effective

tool for solving ordinary and partial differential equations. In many papers various

techniques using quadratic, cubic, quartic, quintic, sextic, septic and higher degree non-

polynomial splines have been discussed for the numerical solution of linear and non-

linear BVPs. Non-polynomial splines were used for numerical solution of system of

second-order BVPs in (Islam et al. [55], Islam and Tirmizi [56], Islam et al. [59]) and

numerical solution of second-order singularly perturbed BVPs has been obtained in

(Rashidinia et al. [108], Rashidinia and Mahmoodi [112], Tirmizi et al. [143]) . Third-

order BVPs have been treated in (Islam et al. [54], Islam and Tirmizi [57]) and fourth-

order equations are discussed in (Islam et al. [58], Islam et al. [61], Ramadan et al.

[105], Rashidinia and Jalilian [109], Siddiqi and Akram [127], Siddiqi and Akram

[129], Van-Daele et al. [155]) . Fifth-order BVPs have been considered in (Islam et al.

[62], Siddiqi and Akram [126], siddiqi et al. [131]) and sixth-order problems are

discussed in (Akram and Siddiqi [8], Islam et al. [60], Ramadan et al. [106]). A survey

on recent spline techniques for solving boundary value problems in ordinary differential

equations using cubic, quintic and sextic polynomial and non-polynomial splines is

given in (Kumar and Srivastava [82]) . Techniques for the solution of eight-, tenth- and

twelfth- order BVPs using non-polynomial spline are applied in (Rashidinia et al.


3

[110], Siddiqi and Akram [128], Siddiqi and Akram [130]). The use of non-polynomial

splines for the numerical solution of parabolic equations was proposed in (Ramadan et

al. [104], Rashidinia and Mahmoodi [111]).

1.2 Existence and uniqueness of two-point BVPs

In chapters 2 and 3 numerical methods for the solution of boundary value problems of

second and sixth orders are considered. Since existence and uniqueness theory plays an

important role in analyzing numerical methods for solving BVPs, therefore, some of its

aspects are studied here. Consider the second-order linear differential equation

0 1 2 , , ,p x y x p x y x p x y x r x x a b (1.1)

where the functions 0 1 2, ,p x p x p x and r x are continuous in ,a b . Consider

the general boundary conditions of the form:

1 0 1 0 1

2 0 1 0 1

,

,

l y a y a a y a b y b b y b A

l y c y a c y a d y b d y b B

(1.2)

where , , , , 1, 2i i i ia b c d i and ,A B are given constants. The particular types of the

boundary conditions (1.2) are:

i. Dirchlet boundary conditions,

, ;y a A y b B (1.3)

ii. Neuman’s boundary conditions,

, ,

or , ;

y a A y b B

y a A y b B

(1.4)

iii. Mixed boundary conditions,

2 20 1 0 1

2 20 1 0 1

, 0

, 0;

a y a a y a A a a

d y b d y b B d d

(1.5)

iv. Periodic boundary conditions,

, .y a y b y a y b (1.6)

The homogenous BVP associated with (1.1)-(1.2) takes the form

0 1 2 0, , ,p x y x p x y x p x y x x a b (1.7)

1 20, 0.l y l y (1.8)


4

Theorem 1.1 (Agarwal and O'Regan [5]) Let 1y x and 2y x be any two linearly

independent solutions of the DE (1.7). Then, the homogeneous boundary value problem

(1.7)-(1.8) has only the trivial solution if and only if

1 1 1 2

2 1 2 2

0.l y l y

l y l y

(1.9)

Theorem 1.2 (Agarwal and O'Regan [5]) The non-homogeneous boundary value

problem (1.1)-(1.2) has a unique solution if and only if the homogeneous boundary

value problem (1.7), (1.8) has only the trivial solution.

Consider the second-order nonlinear DE

, , , , ,y f x y y x a b (1.10)

subject to the boundary conditions

1 20 ,y a A y b A (1.11)

where constants 1A and 2A are real finite and arbitrary.

The following theorem given in (Isaacson and Keller [52]) guarantees the existence of

a unique solution of the BVP (1.10)-(1.11).

Theorem 1.3 (Isaacson and Keller [52]) Suppose the function f in BVP (1.10)-(1.11)

satisfies the following conditions

(i). f and its partial derivatives y

ff

y

and y

ff

y

are all continuous on

, , , , ;D x y y a x b y y

(ii). , , 0yf x y y on some D for some ;

(iii). constant K exists, with

, ,

for all , ,f x y y

K x y y Dy

.

Then solution of BVP (1.10)-(1.11) exists and is unique.

For existence and uniqueness of BVPs of 6-th order we refer to (Agarwal [4]); which

provides a detailed description of the existence and uniqueness theorems for the

solution of n-the order linear and nonlinear boundary value problems in ordinary

differential equations (ODEs).


5

1.3 Polynomial and non-polynomial spline functions

Ordinary and partial differential equations are useful in describing mathematical

models for various physical processes. While there are many theoretical results on

existence, uniqueness, and properties of solutions of such equations, usually only the

simplest specific problems can be solved explicitly, especially when the nonlinear

terms are involved and we usually construct approximate solutions. Since only limited

classes of the equations are solved by analytical means, numerical solution of these

differential equations is of practical importance. Polynomials have long been the

functions most widely used to approximate other functions mainly because of their

simple mathematical properties. However, it is well-known that polynomials of high

degree tend to oscillate strongly and in many cases they are liable to produce very poor

approximations. With a spline function we combine low degree and hence weakly

oscillating polynomials in such a way as to obtain a function which is as smooth as

possible in the sense that it has maximal continuity without being globally a

polynomial. Spline functions can be integrated and differentiated due to being

piecewise polynomials and can be easily stored and implemented on digital computers.

Thus, spline functions are adapted to numerical methods to get the solution of the

differential equations. Numerical methods with spline functions in getting the

approximate solution of the differential equations lead to band matrices which are

solvable easily with algorithms having low cost of computation.

Definition 1.1 Let be the partition 0 1 ... 1nx x x n m of the interval ,a b .

Then a spline s of degree m with knots at is a function possessing the following

two properties

(i). In each subinterval 1[ , )i ix x of ,a b , s is a polynomial of degree m or less.

(ii). s and its derivatives of order 1,2,..., 1m are continuous on ,a b , i.e.

1 ,ms C a b .

The space all such functions is denoted by mS .

Thus a spline function is a series of polynomial arcs of degree m or less, joined

together in such a way that the function and its derivatives of orders 1m or less are


6

continuous everywhere. The spline is, in general, a different polynomial in each of the

subintervals 1[ , )i ix x and the continuity constraint 1 ,ms C a b imposes maximal

continuity on this piecewise defined function.

Piecewise non-polynomial splines are a blend of trigonometric as well as

polynomial basis functions which form a complete extended Chebyshev space. This

approach ensures enhanced accuracy and general form to the existing spline based

algorithms. A parameter is introduced in the trigonometric part of the spline function.

The C -differentiability of the trigonometric part of non-polynomial splines

compensates for the loss of smoothness inherited by polynomial splines. It is well

known that the Bézier basis is a basis for the space of degree-n algebraic polynomials

21, , ,..., .nnP span x x x (1.12)

A new basis, called the C-Bézier basis, is constructed in (Chen and Wang [30]) for the

space

2 21, , ,..., , cos ,sin ,nn span x x x x x (1.13)

in which 1nx and nx in (1.12) are replaced by cos x and sin x .

A part of this thesis presents numerical methods for the numerical solution of linear

and nonlinear second and sixth-order ordinary differential equations as well as system

of BVPs based on the use of non-polynomial spline functions. Consider the numerical

solution of Nth-order linear and non-linear ordinary differential equations of the form

, ,... 0,N

N

d yF x y

dx

(1.14)

In order to develop the numerical method for approximating solution of boundary value

problems like the one given in Eq. (1.14), the interval [ , ]a b is partitioned into 1n

uniformly spaced points ix such that 0 1 1... and n n

b aa x x x x b h

n

. For

each i-th segment, the non-polynomial spline function xPi , equivalent to m-th degree

polynomial, has the form,

2 2

0 1 2 ( 2)

( 1) ( ) ,

0,1,..., 1,

...

sin cos

mi i i ii i i m i

i im i m i

i n

P x a a x x a x x a x x

b k x x c k x x

(1.15)

where the coefficients are arbitrary real and finite constants and k is free parameter.

iP x defined in Eq. (1.15) reduces to usual m-th degree spline in ,a b when 0k .


7

The continuity conditions for non-polynomial spline iP x and its derivatives upto

order 1m , are exploited at the point ,i ix y to get the main scheme for the numerical

solution of the BVPs. The main advantage of non-polynomial based schemes is to

obtain methods of different order with higher accuracy. Computational cost of non-

polynomial spline based methods is the same as that of polynomial splines. In order to

get a unique solution, we require more equations at each end of the range of integration.

These equations are developed by Taylor series and method of undetermined

coefficients. The schemes generally lead to systems of algebraic equations involving

banded matrices and can easily be solved. Newton method is used in the case of

nonlinear system.

1.4 B-Splines

The term B-spline was coined by Isaac Jacob Schoenberg and is short for basis spline.

A B-spline is a spline function that has minimal support with respect to a given degree,

smoothness, and domain partition. A detailed description of B-splines can be found in

(deBoor [35], Prenter [102]). We specifically consider the type of basis for which the

knots are equidistant. In such a case the B-spline is said to be uniform. A fundamental

theorem states that every spline function of a given degree, smoothness, and domain

partition can be represented as a linear combination of B-splines of that same degree

and smoothness, and over that same partition. This feature immediately implies the

band structure of matrices that appear in interpolation and collocation problems;

moreover some of the properties of B-splines imply total positivist. In consequence,

these matrices can be efficiently stored in a computer and inverted by Gaussian

elimination without pivoting. The problem of stable calculation of polynomial B-

splines and their derivatives has been completely solved since 1975. The derivation of

B-splines may be found, for example, in (Prenter [102]).

The B-spline functions are used as a basis functions in finite element method,

Galerkin method and collocation method to construct numerical methods for the

approximate solutions of BVPs occurring in various engineering applications.

Application of the Galerkin method accompanied by higher-degree polynomials results

in a higher-degree matrix system. That brings a burden for numerical analysis, and the

computational cost of the matrix system increases in the evaluation of both linear and


8

nonlinear systems. On the other hand, the collocation method together with B-spline

approximations represents an economical alternative since it only requires the

evaluation of the unknown parameters at the grid points. Papers on solving ODEs and

PDEs using B-splines are quite numerous of which we mention some with quartic and

quintic B-splines. In Refs. (Caglar et al. [26], Saka and Dag [117]) quartic B-splines

are used and solution of various PDEs is obtained by employing quintic B-spline

collocation method in (Ramadan and El-Danaf [103], Raslan [113], Saka [116], Saka

and Dag [118], Zaki [161]). In this thesis methods are developed for the numerical

solution of initial-BVPs in PDEs. The time derivative of given equation is discretized

by a first order accurate forward difference formula and the -weighted, (0 1)

scheme is applied to the space derivative at two adjacent time levels. The B-spline

collocation method over finite intervals is also applied to the space-split equations.

These methods provide satisfactory results in terms of accuracy and ease of

implementation.

1.5 Von Neumann stability analyses

The stability of numerical schemes is closely associated with numerical error. A

solution is said to be unstable if errors introduced at some stage in the calculations (for

example, from erroneous initial conditions or local truncation or round-off errors) are

propagated without bound throughout subsequent calculations. Thus a method is stable

if the difference between the theoretical and numerical solutions remains bounded at a

given time t , as time and space steps tend to zero or time step remains fixed at every

level and t . There are three common methods of investigating stability: the

energy method, the matrix method, and the Von Neumann method. A detailed

description of these methods may be found, for example, in (Mitchell and Griffiths

[89], Twizell [147]).

In chapters 4 and 5 the stability of numerical schemes is investigated by Von

Neumann method developed in the early 1940s. Here we mention this method briefly.

The method is linearized before applying Von Neumann method. The descritized

solution in the nonlinear term is replaced by a constant value. Following (Twizell

[147]) , suppose the errors are given by

,n n nm m mZ u U (1.16)

where nmu is the theoretical and n

mU is the approximate solution of the scheme so that


9

nmU contains approximate errors. The approach is to find the conditions under which

Eq. (1.16) remains bounded as n increases for fixed step size h and time step t . Von

Neumann method expresses the initial line of errors 0mZ in terms of finite Fourier series

and finds the criterion governing the growth of a function which reduces to this series

for 0t . The finite Fourier series for the errors at the 2N points is given by

10

0

exp( ),N

m r rr

Z A i mh

(1.17)

where 0,1,..., 1, 1m N i and rA denote the Fourier coefficients. Suppose the

scheme is linear (If it is nonlinear, the approximate solution in the nonlinear term is

replaced by a constant value), then by the superposition principal it is only necessary to

consider the error propagation due to a single term. Consider the term associated with

the typical frequency s and the coefficient sA being constant can be neglect. To

find the propagation of this single, typical error as t increases, it is necessary to find the

solution of the scheme which will reduce to exp i mh at 0t . Such a solution is

given by

exp exp exp exp ,t i x n t i mh (1.18)

where is a function of and is complex. It is clear that error term exp i mh will

not grow with time when

exp 1 .t (1.19)

Introducing exp t , we have

1. (1.20)

Condition (1.20) is called von Neumann criterion for stability. Here is called the

amplification factor.

1.6 Convergence of two-point BVPs

Consider the solution of nth-order ordinary differential equations of the form

2

2, , , ,... 0

n

n

dy d y d yF x y

dx dx dx

(1.21)

with the prescribed boundary conditions. Let the error between the approximate and

exact solutions be denoted by


10

, 0,1,..., .i i ie y y x i n (1.22)

A numerical procedure is said to be convergent if

0lim 0,ih

e

(1.23)

where h is the step size.

The following definitions and some basic results are needed to establish

convergence of the numerical methods.

Definition 1.2 A matrix norm . : m n is a nonnegative real valued function

which satisfies the following axioms:

0 when 0 and 0 iff 0;

for all ;

;

( ) .

i A A A A

ii A A

iii A B A B

iv AB A B

Definition 1.3 Suppose that is a linear space over the field of real numbers. The

nonnegative real-valued function . is called a norm on the space if it satisfies the

following axioms:

0 when 0 and 0 iff 0 in ;

for all and in ;

for all and in (Triangle inequality).

i v v v v

ii v v v

iii u v u v u v

A linear space , equipped with a norm, is called a normed linear space.

Definition 1.4 The linear space 2 ,L a b is the collection of all measurable functions

f x defined on ,a b for which

2

,b

af x dx

where integration is Lebesgue integration.

Definition 1.5 Let ,f x C a b (The linear space of all continuous functions on

,a b ) or 2 ,f x L a b . The real valued function 2

. defined by

122

2

b

af f x dx


11

is called 2L norm of f .

In this thesis we have used the following approximation for 2L error norm:

12 22

21

,Nb

i iai

L u U dx h u U

where u and U are exact and approximate solutions respectively.

Definition 1.6 A vector norm . : n is a nonnegative function satisfying the

following axioms:

0 when 0 and 0 iff 0 in ,

,

.

n

n

i

ii

iii

v v v v

v v

u v u v u,v

The following vector norms are in common use in numerical linear algebra: the 1-norm

1. , the 2-norm ( Euclidean norm)

2. , norm .

. If T

1 2, ,..., nnv v v v ,

these norms are defined as follows:

Definition 1.7 The 1-norm 1

. of the vector v is defined by

11

.n

ii

v

v

Definition 1.8 The 2-norm ( or Euclidean norm) 2

. of the vector v is defined by

12

12

2T

21

.n

ii

v

v v v

Definition 1.9 The norm .

of the vector v is defined by

1

max .n

ii

v

v

Definition 1.10 A matrix A is convergent to zero if the sequence of matrices

2 3, , ,...A A A converges to the null matrix 0.

Theorem 1.4 (Twizell [147]) For a given matrix A , the necessary and sufficient

condition for the sequence 2, , ,...I A A to converge to the zero matrix is that every eigen

value of A should be less than unity in modulus.


12

Theorem 1.5 A necessary and sufficient condition for the series 2 3 ...I A A A

converges to 11 A

is that lim 0r

rA

.

Proof The necessity of the hypothesis is obvious. To establish sufficiency, Theorem

1.4 is used; this shows that if lim 0r

rA

, then all eigen values of A are less than unity

in modulus. This shows that the determinant of the matrix I A is not zero and thus the

matrix 1I A

exists.

Consider next the formula

2 3 1... .r rI A I A A A A I A

Since 1I A

exists, it follows that

12 3 1

1 1 1

1

...

,

r r

r

I A A A A I A I A

I A I A A

I A

since 1 0rA as .r

The series 2 3 ...I A A A thus converges to 1I A

if 0rA as .r

Lemma 1.1 For any matrix norm with 1,I then the following inequality holds,

111 .I A A

Proof The proof follows from Theorem 1.5 by replacing A by .A

Lemma 1.2 A sufficient condition for convergence of the series 2 3 ...I A A A is

that 1A for any norm of a matrix A .

Lemma 1.3 Let A be some matrix; then, if 1A for some norm, then the following

inequality holds,

1 1... 1 .

rrI A I A A A A

Proof It is easy to see the from Theorem 1.5 that

1 1 2... ...r r rI A I A A A A


13

2

1 1 2

1

1 2

1

... ...

...

1 ...

if 1.

r

r r r

r

r

r

I A I A A A A

A A

A A A

A I A A

The previous results can be summarized in the following theorem.

Theorem 1.6 The following statements are equivalent:

(i) A is convergent matrix,

(ii) lim 0n

nA

for some norm . ,

(iii) 1A where A is spectral radius of A .

Definition 1.11 (Agarwal and O'Regan [5]) A problem consisting of a PDE in a domain

with a set of initial and / or boundary conditions is said to be well-posed if the

following three properties hold:

i. Existence: There exists at least one solution of the problem.

ii. Uniqueness: There is at most one solution of the problem.

iii. Stability: The unique solution depends continuously on data (initial and boundary

conditions): i.e., a slight change in the data leads to only a small change in the solution.

Definition 1.12 A class of nonlinear partial differential equations possesses a special

type of elementary solution. These solutions known as solitons, which have the form of

localized waves that conserve their properties even after interaction among them, and

then act somewhat like particles. These equations have a number of local conserved

quantities, like energy, mass and momentum. Solitons were first appeared in the

solution of the Korteweg-de Vriese equation. When a soliton encounters another

soliton of arbitrary size or velocity it can change beyond recognition for a short or long

period, but ultimately it will revert to its original shape.

1.7 Newton’s method

For solution of nonlinear systems, the well known Newton’s method is used to solve

the nonlinear system of equations. For the sake of completeness the following results


14

are quoted from (Suli and Mayers [137])

Definition 1.13 The recursion defined by

1

, 0,1, 2,...,k+1 k k kfJ k

x x x f x

(1.24)

where n0x , is called Newton’s method (or Newton’s iteration) for the system of

equations 0f x . It is implicitly assumed that the Jacobian matrix kfJ x exists

and is non-singular for each 0,1, 2,...k

Definition 1.14 Suppose that kx is a convergent sequence in n and lim k

kx .

We say that kx converges to with at least order 1q , if there exist a sequence

k of positive real numbers converging to 0, and 0 , such that

, 0,1, 2,...,kk k

x and 1lim .k

qkk

(1.25)

If (1.25) holds with , 0,1, 2,...,kk k k

x then the sequence kx is said

to converge to with order q. In particular, if 2q , then we say that the sequence

kx converges to quadratically.

The next theorem is concerned with the convergence of Newton’s method.

Theorem 1.7 (Suli and Mayers [137]) Suppose that 0f , that in some (open)

neighborhood N of , where f is defined and continuous, all the second-order

partial derivatives of f are defined and continuous, and that the Jacobian matrix

fJ of f at the point is non-singular. Then, the sequence kx defined by

Newton’s method (1.24) converges to the solution provided that 0x is sufficiently

close to ; the convergence of the sequence kx to is at least quadratic.

1.8 Summary

The research work presented in this thesis consists of six chapters. Following is the


15

chapter-wise brief summary:

In chapter 2 quartic non-polynomial spline functions are used to develop numerical

methods for computing approximations to the solution of general linear, nonlinear and

system of second-order BVPs associated with obstacle, unilateral, and contact

problems. Singularly perturbed BVPs are also considered. It is shown that the

suggested methods give approximations, which are better than those produced by other

collocation, finite-difference and spline methods. Convergence analysis of the method

for obstacle problem is discussed. Numerical examples from literature are given to

illustrate practical usefulness of new approach.

In chapter 3 Non-polynomial splines, which are equivalent to seventh-degree

polynomial splines, are used to develop a class of numerical methods for computing

approximations to the solution of linear and nonlinear sixth-order BVPs with two-point

boundary conditions. Second-, Fourth- and Sixth-order convergence is obtained by

using standard procedure. The first sections explain the development of numerical

methods of various orders while the rest of the sections discuss standard convergence

analysis and numerical validation. Numerical results are given to verify the order of

convergence as predicted by the analysis.

Chapter 4 covers construction of numerical scheme based on collocation method

using quartic B-spline functions for the numerical solution of the one-dimensional

modified equal width (MEW) wave equation. Using Von-Neumann approach the

scheme is shown to be unconditionally stable. Performance of the method is validated

through test problems including single wave, interaction of two waves and use of

Maxwellian initial condition. Using error norms L2 and L∞ and conservative properties

of mass, momentum and energy, accuracy and efficiency of the suggested method is

established through comparison with the existing numerical techniques.

In chapter 5 algorithms based on quartic and quintic B-spline collocation methods

are designed for the numerical solution of the modified regularized long wave (MRLW)

equation. Stability analysis is performed using Von-Neumann approach. Performance

of the method is validated through numerical examples. Using error norms L2 and L∞

and conservative properties, accuracy of the new method is established through

comparison with the existing techniques to prove its simple applicability and

superiority. Propagation of a solitary wave, interaction of multiple solitary waves, and

generation of train of solitary waves are also investigated.

In chapter 6 quartic and quintic B-spline functions have been used to develop


16

collocation methods for the numerical solution of a class of nonlinear partial

differential equations known as Kuramoto-Sivashinsky (KS) equation. Also, by the

virtue of splitting technique, the equation is reduced to a problem of second order in

space. Performance of the schemes is tested through application of the methods on

benchmark problems. Accuracy of the methods is established through error norms L2

and L∞.

***************************************************

Chapter 2

Non-Polynomial Splines Methods for

the Solution of Second-Order

Boundary-Value Problems

2.1 Introduction

In this chapter we investigate numerical solutions of the following three types (A, B, C)

of BVPs.

Type A:

Second-order linear and nonlinear differential equations of the type


Type B:

System of second-order BVPs of the type

with the boundary conditions (2.2) and the continuity conditions of yy and at c and

d. Here, f and g are continuous functions on ,a b and ,c d respectively. The

parameters 1 2, and r are real finite constants.

Type C:

The self-adjoint singularly perturbed two-point BVP of the form;

, , ,y g x y x f x y f x y a x b (2.1)

1 2,( ) and ( )y a y b (2.2)

,

f x a x c

y g x y x f x r c x d

f x d x b

(2.3)

Chapter 2 Second-order BVPs

18

where 0 1, are given constants, is a small positive parameter such that 0 1

and f and g are sufficiently smooth functions.

Type A belongs to general second second-order BVPs. Problems of this type occur

frequently in mechanical problems without dissipation (Henrici [47]). These special

BVPs also occur in other engineering context, for example Troesh’s problem relating to

the confinement of a plasma column by radiation pressure (Jones [67], Kubicek and

Hlavacek [79]). Methods of order two, four and six are described by various authors

(See (Al-Said [10], Chawala [28], Jain [63], Tirmizi [142], Tirmizi and Twizell [144],

Tirmizi and Twizell [145], Twizell and Tirmizi [150], Usmani [151], Usmani and

Warsi [154]) ).

Problems of type B arise in the study of obstacle problems. This model represents

an elastic obstacle lying over elastic string which is characterized by a sequence of

boundary value problems (2.3) with the values of y and/or y are unknown at points c

and d. The locations of free boundary become intrinsic part of the problem. The

analytical solution of the problems (2.1)-(2.3) cannot be obtained for arbitrary choices

of ,f x g x and ,f x y . In such cases, we usually resort to some numerical methods

to obtain an approximate solution. Numerical solution of problems of type B is

discussed in a series of papers (Al-Said [9], Al-Said [10], Aziz et al. [16], Islam et al.

[55], Islam et al. [59], Noor and Khalifa [95], Noor and Tirmizi [96]) using splines,

finite difference and collocation methods.

Type C belongs to singular perturbation problems (SPPs) which have applications

in various disciplines of science and engineering, for instance fluid mechanics, fluid

dynamics, elasticity, quantum mechanics, chemical reactor theory, convection-diffusion

processes, optimal control etc. SPPs are thoroughly discussed in the books (Doolan et

al. [36], Miller et al. [88], Morton [92]). The solution of SPPs exhibits a multi-scale

character; that is, there are thin layer(s) where the solution varies rapidly, while away

from the layer(s) the solution behaves regularly and varies slowly. So the numerical

treatment of SPPs give major computational difficulties and the standard methods do

not yield accurate results for all x when is very small. A variety of numerical

methods are available in the literature to solve SPPs for second-order ordinary

differential equations. For details, one may refer to survey articles by (Kadalbajoo and

0 1,

'' , 0, 0 1,

(0) and (1)

y g x y x f x g x x

y y

(2.4)


19

Patidar [70], Kumar and Singh [81]). These methods include finite difference method

(Natesan and Ramanujam [93], Niijima [94], Shishkin [125]), finite element method

(Boglaev [22], O'Riordan and Stynes [97], Vukoslavcevic and Surla [156]), boundary

value approach (BVM) (Kumar et al. [80]) and various forms of splines based methods

(Aziz and Khan [14], Aziz and Khan [15], Kadalbajoo and Bawa [69], Khan et al. [77],

Mohanty and Jha [90], Rashidinia et al. [108], Surla et al. [138], Surla and Stojanovic

[139], Surla and Vukoslavcevic [140]).

In the present chapter, we have developed a class of methods based on quartic non-

polynomial spline functions for the numerical solution of the aforementioned types of

problems. The spline function proposed here has the form:

24 1, , , cos , sin ,T Span x x kx kx

where k is the frequency of the trigonometric part of the splines function which can be

real or pure imaginary and which will be used to raise the accuracy of the method. Thus

in each subinterval 1i ix x x , we have:

21, , , cos , sin ,Span x x kx kx

21, , , cosh , sinh ,Span x x kx kx or

2 3 41, , , , , when 0 .Span x x x x k

The above correlation can be better explained by the following equation:

24

2

23 4

1, , , sin( ), cos( )

6 241, , , -sin ( ) , cos( ) 1 .

2

T Span x x kx kx

kxSpan x x kx kx kx

k k

(2.5)

Spline solution has its own advantages. For example, once the solution has been

computed, the information required for spline interpolation between mesh points is

available. This is particularly important when the solution of the BVP is required at

different locations in the interval [a, b]. This approach has the added advantage that it

not only provides continuous approximations to xy , but also to y and higher

derivatives at every point of the range of integration. Also, the C -differentiability of

the trigonometric part of non-polynomial splines compensates for the loss of

smoothness inherited by polynomial splines.

The organization of this chapter is as follows. In Section 2.2 the numerical methods

are described and methods of different orders are categorized. Convergence analysis of


20

problems of first two types is given in section 2.3 and numerical results are reported in

section 2.4. Some conclusions are drawn in section 2.5.

2.2 Numerical Methods

Let the partition of the space interval ,a b into uniformly spaced points mx be such

that 0 1 1... N Na x x x x b and b a

hN

. For simplicity we take

3

4

a bc

and 3

4

a bd

. For each i-th segment, the quartic non-polynomial spline function

xPi has the form,

2,

0,1,..., 1,

cos sini i i i i i i i i i

i n

P x a k x x b k x x c x x d x x e

(2.6)

where , , , ,i i i i ia b c d e are arbitrary real and finite constants and k is free parameter.

iP x defined in Eq. (2.6) reduces to usual quartic spline in ,a b when 0k as

mentioned in Eq. (2.5).

To derive expression for the coefficients of Eq. (2.6) in terms of

, , , ,i j i j i j i j i jy D M T S , the following relations are defined for the relevant values of j:

, , ,

, .

i i j i j i i j i j i i j i j

ivi i j i j i i j i j

P x y P x D P x M

P x T P x S

(2.7)

We obtain the following expressions from Eq. (2.6):

1 2 1/ 2 1/ 2

4 3 3 2

221/ 2

1/ 2 1/ 22 4 2 2

tan 2, , ,

cos 2 2 2

1, , .

8 8 2 2

i i i i ii i i

i i i ii i i i i

S T T M Sa b c

k k k k

T h M hT hDhd D e y S kh

k k k k

(2.8)

The continuity conditions for non-polynomial spline iP x and its first, second and

third derivatives are exploited at the point ,i ix y to get the following consistency

relations:

2

1 1/ 2 1/ 2 1/ 2 1/ 2 13 2

2 2

1 2 1/ 24 2 4 4 2 4

tan / 2( ) 3

2 8 2

cos1 1 3 1

cos 2 8 cos( / 2) 8,

i i i i i i i i

i i

h h hD D y y M M T T

k k

h hS S

k k k k k k

(2.9)


21

1 1/ 2 1 23 2

2sin / 2,i i i i

hD D hM S

k k

(2.10)

1 1/2 1/2 1/2 1/22 2 2 2

tan /2 1 1 cos( ) 1

cos( /2) cos( /2)i i i i i iT T M M S Sk k k k k

(2.11)

1 1/ 2

2sin / 2.i i iT T S

k

(2.12)

Eqs. (2.9)-(2.11) yield the following equation,

2

1/ 2 1/ 2 1/ 2 1/ 22

2

1/ 2 1/ 23 3 2

1

2 tan / 2 8

.2 sin / 2 2 tan / 2 8

i i i i i

i i

h hhD y y M M

k k

h h hS S

k k k

(2.13)

Likewise from Eqs. (2.11) and (2.12) it follows that,

1/ 2 1/ 2 1/ 2 1/ 2

1 1.

2 tan / 2 2 tan / 2 2 sin / 2i i i i i

kT M M S S

k k

(2.14)

Elimination of T’s from Eqs. (2.12) and (2.14), D’s from Eqs. (2.10) and (2.13) yield

1/ 2 1/ 2 3 / 2 3 / 2 1/ 2

1/ 2

1 12

2 tan / 2 2 tan / 2 2 sin / 2

2sin / 21 1

sin / 2 tan / 2,

i i i i i

i

kM M M S S

k k

Sk k k

(2.15)

2 2 2

1/ 2 1/ 2 1/ 2 3/ 2 1/ 22 2

2

1/ 2 1/ 22

1 cos / 2 12 2

2 8(1 cos / 2 2

1

8(1 cos / 2

i i i i i

i i

h h hS y y y M

k k

hM M

k

(2.16)

When 0k , formula derived in (Usmani [152]) becomes special case of Eq. (2.15);

that is,

4

21/ 2 1/ 2 3/ 2 1/ 2 1/ 2 3/ 22 6 .

8i i i i i i

hh M M M S S S

(2.17)

Eliminating 'iS s from Eqs. (2.16) and (2.17), we get


22

3/ 2 1/ 2 2

1/ 2 3/ 2 5/ 2 3/ 22 2

5/ 2

2 2 2 2

3/ 2 1/ 22 2

2 2 2 2

1/ 22 2

41

10 416sin / 4

4 3 2 4 cos / 2 4cos

4 sin / 4

8 19 24( 1 )cos / 2 16cos.

8 sin / 4

i i

i i i i

i

i i

i

y yh

y y M Mk

y

h k h kM M

k

h k h kM

k

(2.18)

Eq. (2.18) in simpler form renders the following equation,

5/ 2 3/ 2

25/ 2 3/ 2 3/ 2 1/ 2 1/ 2 3/ 2 1/ 2

1/ 2

4 10

i i

i i i i i i i

i

M M

y y y y y h M M

M

(2.19)

where

2 2

2 2

2 2

2 2

2 2

1 1,

16sin / 4

4 3 2 4 cos / 2 4cos,

4 sin / 4

8 19 24( 1 )cos / 2 16cos,

8 sin / 4

3, 4,..., 2.i n

(2.20)

According to Eq. (2.5), 2

0lim , , 1,76,230

48k

h

which is the formula based on

quartic polynomial spline functions derived in (Usmani [152]).

The local truncation errors , 2, 4,..., 2,it i n associated with scheme (2.19) are,

2 (2) 3 (3) 4 (4) 5 (5) 6 (6) 7 (7) 8 (8) 9

2 3 4 5 6 7 8 ,i i i i i i i it C h y C h y C h y C h y C h y C h y C h y O h (2.21)

2

3

4

5

6

7

8

8 2 2 ,

18 2 2 ,

21

64 3 34 10 ,241

48 98 26 ,48

12408 15 706 82 ,

57601

504 2882 242 ,3840

113232 7 1635 730 .

322560

C

C

C

C

C

C

C


23

Thus for different choices of , , in scheme (2.19), methods of different order are

obtained. The Eq. (2.19) gives 4n equations in n unknowns 1/ 2 , 0,1,..., 1.iy i n We

require four more equations, two at each end of the range of integration, for the direct

computation of 1/ 2iy . These equations are developed by Taylor series and method of

undetermined coefficients. The general form of initial boundary equations for the main

scheme is as follows:

250 0 0 1 2 3 41/2 3/2 5/2 1/2 3/2 5/2 7/2 9/2 010 14 3 ,y y y y h M M M M M Mh h h h h h (2.22)

250 0 0 1 2 3 41/2 3/2 5/2 7/2 1/2 3/2 5/2 7/2 9/2 0.2 3 10 4y y y y y h M M M M M Mk k k k k k (2.23)

The remaining two equations at the other end can be obtained from above equations by

writing them in reverse order. The constants ,i ia b are parameters to be determined.

2.2.1 Second-order convergence

For second-order convergence we put

2 3 271, ,

55 2 55 in the main Eq. (2.19), which gives

2 3 4

70,

330C C C .

The values of the coefficients of the first boundary equation are given by

0 1 2 3 4 5

5, 5, 1, 0.

4h h h h h h

(2.24)

The values of the coefficients of the second boundary equation are given by

0 1 2 3 4 5

11, 0, 1, 1, 0.

4k k k k k k

(2.25)

2.2.2 Fourth-order convergence

For fourth-order convergence we put

1 13, , 5

18 9r in the main Eq. (2.19), which gives

2 4 6

10,

180C C C .


0 1 2 3 4 5

2 154 75 1, , , , 0.

48 48 48 48h h h h h h

(2.26)


24


0 1 2 3 4 5

10 75 230 76 1, , , , , 0.

48 48 48 48 48k k k k k k

(2.27)

2.2.3 Sixth-order convergence

For sixth-order convergence we put

1 22 149

, ,20 15 30

r in the main Eq. (2.19), which gives

2 4 6 8

10,

15120C C C C .


0 1 2 3 4 5 1

1 1 1 1 1 1

608 40439 24059 2009 869 126, , , = , , 13440.,h h h h h h cd

cd cd cd cd cd cd

(2.28)


0 1 2 3 4 5 2

2 2 2 2 2 2

10784 157059 612759 170163 8649 406, , , = , , 120960.,k k k k k k cd

cd cd cd cd cd cd

(2.29)

2.3 Convergence analysis

Now we investigate the error analysis of the quartic non-polynomial spline method

described in section 2.2 for problems of first two types. To do so we let, ,i ic t C T

and 1/2ieE be n-dimensional column vectors. Here ~

1/ 2 1/ 2 1/ 2i i ie y y is the

discretization error for 0,1,..., 1i n . Thus, we can write our method in the matrix

form:

~

AY = C + T

A Y = C

AE = T,

(2.30)

where

20 0 , , , , ij ijh p m A A BD A PM P M (2.31)

and


25

3, 1, ,

2, 2,3,..., 1,

1, 1,

0, otherwise,

ij

i j n

i j np

i j

(2.32)

5, 1, ,

6, 2, 3,..., 1,

1, 1,

0, otherwise.

ij

i j n

i j nm

i j

(2.33)

The five-diagonal matrix 0A is given by

14 3 1

3 10 4 1

1 4 10 4 1

1 4 10 4 1

1 4 10 3

4 3 1

0A

(2.34)

and the matrix B is given by

40439 24059 2009 869 126

157059 612759 170163 8649 406

406 8649 170163 612759 157059

126 869 2009 24059 40439

B

(2.35)

and 1/2 1/2diag , 1,2,..., with 0i ig i n g D for /4 3 /4n i n . The vector C is defined by


26

21 1

21 2

2

2

2

2

110 , 1,13440

12 , 2,120960

3, 2 and 1 14 4

3, 1 and 2,4 4

3, and 1,4 4

, 1 an4

i

i

i

i

c

h F i

h F i

n nh F i i n

n nh F r i

n nh F r i

nh F r i

2

2

22 1

22

3d ,432 , 2 1 ,

4 432 2 , 3 2 ,

4 412 , 1,

120960110 , ,

13440

i

i

n

n

n

n nh F r i and

n nh F r i i

h F i n

h F i n

(2.36)

where

5 3 1 1 32 22 2 2

0 1/ 2 3/ 2 5/ 2 7/ 2 9/ 2

9/ 2 7/ 2 5/ 2

0 1/2 3/2 5/2 7/2 9/2

608 40439 24059 2009 869 126 , 1,

, 2,

, 3 2,

406 8649 170163 6

10784 157059 612759 170163 8649 406

i i ii i i

n n n

f f f f f f i

i

F f f f f f i n

f f f

f f f f f f

3/ 2 1/ 2

9/2 7/2 5/2 3/2 1/2

12759 157059 10784 1,

, .

,

126 869 2009 24059 40439 608n n n

nn n n n n

f f f i n

i nf f f f f f

(2.37)

The analysis of the convergence for our method depends on the properties of the

matrix 0A . It has been shown in (Al-Said [10]) and (Usmani [152]) that the matrix 0A

is nonsingular and that its inverse satisfies the inequality

2 21

2

3.

64

b a h

h

0A (2.38)

Thus from Eqs (2.30) and (2.31) , we have

.

2 2

2

h h

h

-1 -1-1 -1 -10 0 0

-1-1 -10 0

E = A T = A + BD T = I + A BD A T,

E I + A BD A T

Using the result 1I and 11 1

I + M M with M 1, we get the

following expression

2h

-1

0

-10

A TE .

1- A B D

(3.39)


27

We consider the following two cases for T :

(i) For 1 13

, , 518 9

, LTE is 66

1

180h MT

(ii) For 1 22 149

, ,20 15 30

, LTE is 88

1

15120h MT

where 6

6 max xxM y and

88 max xx

M y .

Using Eq. (2.38) and the fact that

B is some finite number, it follows from standard

procedure, that:

Case i 2h

E , hence the method (2.19) with second-order boundaries is second-

order convergent,

Case ii 4h

E , hence the method (2.19) with fourth-order boundaries is fourth-

order convergent,

Case ii 6h

E , hence the method (2.19) with sixth-order boundaries is sixth-

order convergent.

It is remarked that in the case of problems of type (2.3), established numerical

methods (Al-Said [9], Al-Said [10], Aziz et al. [16], Islam et al. [55], Islam and

Tirmizi [56], Noor and Khalifa [95], Noor and Tirmizi [96]) lag behind theoretical

convergence irrespective of the order of the method. The reason for this failure is that

in most of the cases, maximum error lies around the breakup points c and d of the

interval ,a b where solution and its first derivative satisfy extra continuity conditions.

Since there is no break up point in the domain set of problems of type (2.1), all the

numerical methods of various orders perform well and exhibit accuracy predicted

theoretically.

2.4 Numerical results and discussion

In order to test the efficiency of the new scheme developed in section 2.2, we apply it

on a number of problems from the literature. We use double precision arithmetic in

order to reduce the round-off errors to a minimum.


28

2.4.1 Linear problems

Problem 2.1 Consider the linear BVP

2 2 , 2 3, 2 0 and 3 0,x y y x x y y (2.40)

for which the analytic solution is

21 365 19 .

38y x x x

x

(2.41)

The above problem is the type where both f x and g x in Eq. (2.1) are variables.

Table 2.1 contains results for second-order method 2.2.1. The results using the fourth-

order method 2.2.2 are summarized in Table 2.2 and compared with quartic and quintic

spline methods (Al-Said [10], Usmani and Warsi [154]) of the same order. It is seen

that our method performs better than these methods. Table 2.3 contains results for

sixth-order method 2.2.3 which is compared with sixth-order method given in (Usmani

[151]) that uses a quindiagonal scheme. It is clear from the table that method 2.2.3 is far

superior to this method. Fig. 2.1 shows the exact and approximate solution for 1

64h .


with analytical solution

The above example is the type where only f x in Eq. (2.1) is variable. The results

using the fourth-order method 2.2.2 are summarized in Table 2.4 and compared with

quintic spline method (Usmani and Warsi [154]) of the same order. It is seen that

present method produces better results. Exact and approximate solution is shown in Fig.

2.2 for 1

128h .


with analytical solution

4 , 0 1, 0 1 0,xy y xe x y y (2.42)

1 .xy x x x e (2.43)

2 2, 0 1, 0 1 1,y y x x y y (2.44)

22sinh.

sinh1

xy x x (2.45)


29

Table 2.5 contains the results for fourth-order method. We have compared our results

with fourth-order method based on (1,3) padeapproximant (Tirmizi [142]) . It is seen

that our method performs better than pade approximant. Fig. 2.3 shows the exact and

approximate solution for 1

64h .

2.4.2 Nonlinear problem

Problem 2.4 Consider the following nonlinear problem:

311 , 0 1

2y x y x , 0 1 0y y

(2.46)

for which the exact solution is

21.

2y x x

x

(2.47)

For the sake of comparison with fourth-order method, we have applied the quartic

splines based scheme given in (Al-Said [10]) to the nonlinear problem and the results

are summarized in Table 2.6. It is seen from the table that method 2.2.2 produces better

results. We have also compared the sixth-order method with higher order finite-

difference method (Tirmizi and Twizell [145]) and the results are given in Table 2.7. It

is observed that four iterations were enough for present methods to converge. The plot

of exact and approximate solution is shown in Figs. 2.4-2.5 for 1 1

,32 64

h .

2.4.3 System of differential equations

Problem2.5 Consider the system of differential equations (2.3) when 0f and

1r .

30, for 0 and ,

4 43

1, for ,4 4

x xu

u x

(2.48)

with the boundary conditions 0 0,u u and the condition of continuity of

3 and at and .

4 4u u x

The analytical solution for this problem is given by


30

1

2

1

1 2

4, 0

4

4 31 cosh ,

2 4 4

4 3, ,

4

where = +4coth and = sinh +4cosh , 4 4 4

x x

x xu x

x x

(2.49)

We have compared our method with different methods like cubic spline (Al-Said [9]),

quartic spline (Al-Said [10]), quintic spline (Aziz et al. [16]), non-polynomial quadratic

spline (Islam et al. [55]), non-polynomial cubic spline (Islam and Tirmizi [56]),

collocation based on cubic spline (Noor and Khalifa [95]) and finite difference method

(Noor and Tirmizi [96]) . The results are displayed in Table 2.8. It is clear from the

table that method 2.2.3 gives better results than the existing spline and finite difference

methods except in the case of non-polynomial quadratic spline (Islam et al. [55]).

Graph of exact and approximate solution is provided in Fig. 2.6 for 80

h

.

2.4.4 Singular perturbation problems

Problem 2.6 Consider the following problem with constant coefficients:

2 2'' cos 2 cos 2 , 0 1, 0 1 0.y y x x x y y (2.50)

The theoretical solution for this problem is

2exp (1 ) / exp /

cos .1 exp 1/

x xy x x

(2.51)

The results of maximum absolute errors for this problem are tabulated in Table 2.9 for

different values of the parameters and n . We have compared our method with

various methods from literature including cubic, quintic, sextic splines, splines in

tension, family of exponential spline difference scheme (Aziz and Khan [14], Aziz and

Khan [15], Khan et al. [77], Rashidinia et al. [108], Surla et al. [138], Surla and

Stojanovic [139], Surla and Vukoslavcevic [140]) and the results are summarized in

Tables 2.10-2.16. The Tables confirm better performance against the existing methods.

Figs. 2.7-2.26 show the exact and approximate solutions for various values of and n .


31

Problem 2.7 Consider the following equation with variable coefficients:

2

2

'' 1 1 1 1 2 1 exp 1 /

2 1 exp / ,

y x x y x x x x x

x x x

(2.52)

subject to the boundary conditions:

0 1 0.y y

The exact solution is given by

1 1 exp / exp (1 ) / .y x x x x x (2.53)

The above problem contains variable coefficients. We have solved it by fourth- and

sixth-order methods. The results of maximum absolute errors are tabulated in Table

2.17 and 2.18 for different values of the parameters and n .


32

Table 2.1

Observed maximum absolute errors for linear problem 2.1

h Second-order method

1/8 3.8203E-3

1/16 1.0646E-3

1/32 2.8473E-4

1/64 7.3859E-5

1/128 1.8824E-5

1/256 4.7523E-6

1/512 1.1940E-6

1/1024 2.9924E-7

Table 2.2


h Method 2.2.2 (Al-Said [10]) (Usmani and Warsi [154])

1/16 8.74E-10 9.56E-9 2.92E-9

1/32 9.45E-11 5.99E-10 2.16E-10

1/64 6.81E-12 3.75E-11 1.40E-11

1/128 4.40E-13 2.34E-12 8.91E-12

Table 2.3


h Method 2.2.3 (Usmani [151])

1/8 1.1779E-9 4.97E-8

1/16 7.6235E-12 1.03E-9

1/32 3.7773E-14 1.86 E-11

1/64 4.3021E-16 3.13E-13


33

Table 2.4

Observed maximum absolute errors for the linear problem 2.2

h Method 2.2.2 (Usmani and Warsi [154])

1/8 1.36E-6 1.10E-6

1/16 3.81E-8 1.09E-7

1/32 3.49E-9 7.51E-9

1/64 2.36E-10 4.81E-10

1/128 1.51E-11 3.03E-11

1/256 1.01E-12 1.85E-12

1/512 6.00E-13 6.48E-13

1/1024 2.72E-13 7.64E-13

Table 2.5

Observed maximum absolute errors for the linear problem 2.3

h Method 2.2.2 (1,3) pade approximant (Tirmizi [142])

1/8 2.71E-8 6.25E-8

1/16 6.98E-10 3.92E-9

1/32 6.31E-11 2.46E-10

1/64 4.29E-12 1.53E-11

1/128 2.69E-13 7.75E-12

Table 2.6

Observed maximum absolute error for the nonlinear problem 2.4

h Method 2.2.2 (Al-Said [10])

1/8 8.69E-6 1.43E-5

1/16 2.20E-7 9.2E-7

1/32 7.08E-9 5.77E-8

1/64 6.14E-10 3.62E-9

1/128 4.18E-11 2.26E-10


34

Table 2.7

Observed maximum absolute errors for the nonlinear problem 2.4

h Method 2.2.3 (Tirmizi and Twizell [145])

1/8 4.96E-7 6.32E-7

1/16 4.94E-9 6.33E-9

1/32 3.20E-11 1.57E-10

1/64 1.58E-13 2.72E-12

Table 2.8

Observed maximum absolute error for problem 2.5

Method / 20h / 40h / 80h

Method 2.2.3 3.35E-4 1.01E-4 2.14E-5

(Al-Said [9]) 1.26E-3 3.29E-4 8.43E-5

(Al-Said [10]) 8.74E-4 1.19E-4 2.98E-5

(Aziz et al. [16]) 2.57E-3 7.31E-4 1.94E-4

(Islam et al. [55]) 2.39E-4 6.23E-5 1.62E-5

(Islam and Tirmizi [56]) 6.43E-4 1.83E-4 4.87E-5

(Noor and Khalifa [95]) 1.42E-2 7.71E-3 4.04E-3

(Noor and Tirmizi [96]) 2.50E-2 1.29E-3 6.58E-3

Table 2.9

Maximum absolute errors for problem 2.6 method 2.2.3

n=16 n=32 n=64 n=128 n=256

1/16 1.19E-07 6.74E-10 4.26E-12 6.18E-14 2.09E-14

1/32 4.08E-08 2.19E-10 3.00E-12 4.61E-14 9.60E-15

1/64 5.84E-07 4.92E-09 2.81E-11 1.27E-13 1.62E-14

1/128 6.48E-06 6.4E-08 4.24E-10 2.1E-12 2.84E-14


35

Table 2.10

Maximum absolute errors for problem 2.6 reported in (Aziz and Khan [14])

n=16 n=32 n=64 n=128 n=256

1/16 1.57E-05 8.79E-07 5.32E-08 3.30E-09 2.05E-10

1/32 8.27E-06 4.41E-07 2.62E-08 1.62E-09 1.00E-10

1/64 1.84E-05 8.67E-07 6.65E-08 4.39E-09 2.78E-10

1/128 1.03E-04 2.61E-06 2.23E-07 1.54E-08 9.44E-10

Table 2.11

Maximum absolute errors for problem 2.6 reported in (Aziz and Khan [15])

n=16 n=32 n=64 n=128 n=256

1/16 4.074E-05 2.532E-06 1.581E-07 9.878E-09 6.172E-10

1/32 2.005E-05 1.242E-06 7.746E-08 4.839E-09 3.023E-10

1/64 5.456E-05 3.429E-06 2.146E-07 1.343E-09 8.397E-10

1/128 1.834E-04 1.226E-05 7.689E-07 4.814E-08 3.011E-09

Table 2.12

Maximum absolute errors for problem 2.6 reported in (Khan et al. [77])

n=16 n=32 n=64 n=128 n=256

1/16 1.224E-06 6.458E-09 3.407E-11 1.037E-12 8.881E-15

1/32 2.004E-06 1.683E-08 1.367E-10 1.093E-12 1.901E-14

1/64 8.898E-06 1.168E-07 1.203E-09 1.083E-11 9.076E-14

1/128 5.746E-05 9.984E-07 1.182E-08 1.141E-10 9.919E-13

Table 2.13

Maximum absolute errors for problem 2.6 reported in (Rashidinia et al. [108])

n=16 N=32 n=64 n=128 n=256

1/16 4.07E-05 2.53E-06 1.58E-07 9.87E-09 6.17E-10

1/32 2.00E-5 1.24E-06 7.74E-08 4.83E-09 3.02E-10

1/64 5.45E-05 3.42E-06 2.14E-07 1.34E-08 8.39E-10

1/128 1.83E-04 1.22E-05 7.68E-07 4.81E-08 3.01E-9


36

Table 2.14

Maximum absolute errors for problem 2.6 reported in (Surla et al. [138])

n=16 n=32 n=64 n=128 n=256

1/16 4.14E-03 1.02E-03 2.54E-04 6.35E-05 1.58E-05

1/32 3.68E-03 9.03E-04 5.61E-05 1.40E-05 3.50E-06

1/64 3.45E-03 8.40E-04 2.08E-04 5.20E-05 1.30E-05

1/128 3.43E-03 8.21E-04 2.03E-04 5.06E-05 1.26E-05

Table 2.15

Maximum absolute errors for problem 2.6 reported in (Surla and Stojanovic [139])

n=16 n=32 n=64 n=128 n=256

1/16 8.06E-03 2.02E-03 5.08E-04 1.27E-04 3.17E-05

1/32 7.11E-03 1.79E-03 4.48E-04 1.12E-04 2.80E-05

1/64 6.58E-03 1.66E-03 4.15E-04 1.04E-04 2.60E-05

1/128 6.36E-03 1.61E-03 4.03E-04 1.01E-04 2.52E-05

Table 2.16

Maximum absolute errors for problem 2.6 reported in (Surla and Vukoslavcevic [140])

n=16 n=32 n=64 n=128 n=256

1/16 1.20E-04 7.47E-06 4.67E-07 2.90E-08 4.39E-09

1/32 1.28E-04 8.00E-06 5.00E-07 3.14E-08 1.99E-09

1/64 1.60E-04 1.00E-05 6.26E-07 3.92E-08 2.31E-09

1/128 2.34E-04 1.47E-05 9.23E-07 5.77E-08 3.72E-09

Table 2.17

Maximum absolute errors for problem 2.7, 1 13

, , 518 9

n=16 n=32 n=64 n=128 n=256

1/16 1.64E-06 6.81E-08 4.84E-09 3.19E-10 2.02E-11

1/32 8.96E-06 1.79E-07 1.24E-08 8.73E-10 5.63E-11

1/64 4.41E-05 1.62E-06 3.46E-08 2.75E-09 1.84E-10

1/128 1.92E-04 6.78E-06 1.34E-07 9.05E-09 6.42E-10


37

Table 2.18

Maximum absolute errors for problem 2.7, 1 22 149

, ,20 15 30

n=16 n=32 n=64 n=128 n=256

1/16 1.63E-08 9.22E-10 4.14E-13 3.00E-15 1.08E-14

1/32 1.47E-07 1.00E-09 5.04E-12 2.38E-14 1.99E-14

1/64 1.25E-06 1.06E-08 6.07E-11 2.75E-13 1.22E-14

1/128 1.03E-05 1.05E-07 7.14E-10 3.58E-12 1.87E-14


38

0 10 20 30 40 50 60 700

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Grid Points

So

luti

on

Exact solutionCalculated values

Fig. 2.1: Plot of solution for problem 2.1 , 1

64h

0 20 40 60 80 100 120 1400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Grid Points

So

luti

on



128h


39

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Grid Points

So

luti

on



64h

0 5 10 15 20 25 30 35-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

Grid Points

So

luti

on



32h


40

0 10 20 30 40 50 60 70-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

Grid Points

So

luti

on



64h

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Grid Points

So

luti

on



h


41

0 2 4 6 8 10 12 14 16-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid Points

So

luti

on



, 1616

n

0 5 10 15 20 25 30 35-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid Points

So

luti

on


Fig. 2.8: Plot of solution for problem 2.6, 1

, 3216

n


42

0 10 20 30 40 50 60 70-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid Points

So

luti

on



, 6416

n

0 20 40 60 80 100 120 140-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid Points

So

luti

on



, 12816

n


43

0 50 100 150 200 250 300-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid Points

So

luti

on



, 25616

n

0 2 4 6 8 10 12 14 16-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid Points

So

luti

on



, 1632

n


44

0 5 10 15 20 25 30 35-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid Points

So

luti

on



, 3232

n

0 10 20 30 40 50 60 70-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid Points

So

luti

on



, 6432

n


45

0 20 40 60 80 100 120 140-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid Points

So

luti

on



, 12832

n

0 50 100 150 200 250 300-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Grid Points

So

luti

on



, 25632

n


46

0 2 4 6 8 10 12 14 16-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Grid Points

So

luti

on



, 1664

n

0 5 10 15 20 25 30 35-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Grid Points

So

luti

on



, 3264

n


47

0 10 20 30 40 50 60 70-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Grid Points

So

luti

on



, 6464

n

0 20 40 60 80 100 120 140-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Grid Points

So

luti

on



, 12864

n


48

0 50 100 150 200 250 300-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Grid Points

So

luti

on



, 25664

n

0 2 4 6 8 10 12 14 16-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Grid Points

So

luti

on



, 16128

n


49

0 5 10 15 20 25 30 35-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Grid Points

So

luti

on



, 32128

n

0 10 20 30 40 50 60 70-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Grid Points

So

luti

on



, 64128

n


50

0 20 40 60 80 100 120 140-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Grid Points

So

luti

on



, 128128

n

0 50 100 150 200 250 300-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Grid Points

So

luti

on



, 256128

n


51

2.5 Conclusion

Quartic non-polynomial spline functions are used to develop a class of numerical

methods for solving linear, nonlinear, system of two point BVPs and SPPs. Second-,

fourth- and sixth-order accurate methods have been obtained. Numerical results

confirm the orders of the methods except for the system of second-order BVPs due to

extra continuity conditions associated with the system. Numerical results show better

accuracy of the new methods over some existing methods of the same order reported in

the literature. The methods are computationally efficient and the algorithm can easily

be implemented on a computer.

***************************************************

Chapter 3

Non-Polynomial Splines Approach to the

Solution of Sixth-Order

Boundary-Value Problems

3.1 Introduction

In this chapter, non-polynomial spline functions are applied to develop numerical

methods for obtaining smooth approximations for the following BVP:

6 , , , / ,D y x f x y a x b D d dx (3.1)


2 40 2 4

2 40 2 4

, , ,

, , .

y a A D y a A D y a A

y b B D y b B D y b B

(3.2)

where y x and ( , )f x y are continuous functions defined in the interval ,x a b . It

is assumed that 6, [ , ]f x y C a b is real and that iA , , 0, 2, 4,iB i are finite real

numbers. The literature on the numerical solution of sixth-order BVPs is sparse. Such

problems are known to arise in astrophysics; the narrow convicting layers bounded by

stable layers, which are believed to surround A-type stars, may be modeled by sixth-

order BVPs (Toomre et al. [146]). Also in (Glatzmaier [44]) it is given that dynamo

action in some stars may be calculated by such equations. (Chandrasekhar [27])

determined that when an infinite horizontal layer of fluid is heated from below and is

under the action of rotation, instability sets in. When this instability is an ordinary

convection, the ordinary differential equation is sixth order.

Theorems, which list the conditions for the existence and uniqueness of solutions of

sixth-order BVPs, are thoroughly discussed in the book by Agarwal (Agarwal [4]).

Non-numerical techniques for solving such problems are contained in papers (Baldwin

[18], Baldwin [19]). Numerical methods of solution are contained implicitly in

Chapter 3 Sixth-order BVPs

53

(Chawala and Katti [29]), although those authors concentrated on numerical methods

for fourth-order problems. E. H. Twizell (Twizell [148]) developed a second-order

method for solving special and general sixth-order problems and in his later work

(Twizell and Boutayeb [149]) developed finite-difference methods of order two, four,

six and eight for solving such problems. The authors in (Siddiqi and Twizell [132])

used sixth-degree splines, where spline values at the mid knots of the interpolation

interval and the corresponding values of the even order derivatives are related through

consistency relations. Sinc-Galerkin method for the solutions of sixth order BVPs was

used in (Gamel et al. [41]) whereas decomposition and modified domain

decomposition methods were used in (Wazwaz [157]) to investigate solution of the

sixth-order BVPs.

The spline function proposed in this chapter has the form:

2 3 4 57 =Span 1, , , , , , cos , sinT x x x x x kx kx

where k is the frequency of the trigonometric part of the splines function. Thus in each

subinterval 1i ix x x , we have:

2 3 4 5

2 3 4 5

2 3 4 5 6 7

Span 1, , , , , , cos , sin ,

Span 1, , , , , , cosh , sinh or

Span 1, , , , , , , , when 0 .

x x x x x kx kx

x x x x x kx kx

x x x x x x x k

The above correlation can be better explained by the following equation:

2 3 4 5

7

3 5

2 3 4 5

7

2 4

6

1, , , , , , sin( ), cos( )

7!1, , , , , - sin ( )

6 120Span

6!, 1 cos( )

2 24

Span

,

T x x x x x kx kx

kx kxx x x x x kx kx

k

kx kxkx

k

(3.3)

In the next section the special sixth-order, BVP to be solved is transformed using

non-polynomial spline techniques into linear or non-linear algebraic system. In section

3.3 methods of different orders are categorized. In section 3.4, properties of the inverse

of coefficient matrix are discussed. In section 3.5, convergence of the method is

established and numerical results are reported in section 3.6. In the last section some

conclusions are drawn.


54

3.2 Numerical Methods

To develop spline approximation to the problem (3.1)-(3.2), the interval ,a b is divided

into n equal subintervals, using the grid points , 0,1,..., ,ix a ih i n where b a

hn

. For

each segment 1,i ix x , the polynomial 1/ 2 ( )iP x has the form:

51/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

4 3 21/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

( ) sin ( ) cos ( ) ( )

( ) ( ) ( ) ( )

i i i i i i i

i i i i i i i i i

P x a k x x b k x x c x x

d x x e x x f x x g x x r

(3.4)

where 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2, , , , , , andi i i i i i i ia b c d e f g r are constants and k is free

parameter. Let 1/ 2iy be an approximation to 1/ 2( )iy x , obtained by the segment

1/ 2 ( )iP x of the mixed splines function passing through the points

1/ 2 1/ 2 1/ 2 1/ 2( , ) and ( , ).i i i ix y x y To determine the coefficients in (3.4) at the common

nodes 1/ 2 1/ 2,i ix y , we first define:

( )1/ 2 1/ 2 1/2 1/ 2( ) , ( ) , ( ) , ( ) , 1,2,..., , 1/ 2.vi

i i j i j i i j i j i i j i j i i j i jP x y P x Z P x M P x S i n j

From algebraic manipulation we get the following expressions for coefficients:

5 6

1/ 2 1/ 2 1/ 26

1/ 21/ 2 6

2 2 2 2 2 6

1/ 2 1/ 2 1/ 2

2 6

1/ 2 1/ 2 1/ 2 1/ 2

1/ 2

1/ 2 1

csc( )

2

1( cos( ) )csc( ),

,

sin( ) sin( ) sin( )

sin( ) 6 6 cos( ) 6 cos( )

6 6

i i i

ii

i i i

i i i i

i

i ih k

a S Sk

Sb

k

h k S h S k h M k

h k M hS k hk S hkSc

hkS hZ

4 6

6 6

/ 2 1/ 2 1/ 2

6 6

1/ 2 1/ 2 1/ 2

2 2 2 2 2 6

1/ 2 1/ 2 1/ 2

2 6

1/ 2

1/ 2

csc( )

2

,sin( ) 6 sin( ) 12 sin( )

12 sin( ) 12 sin( ) 12 sin( )

2 sin( ) 3 sin( ) 3 sin( )

2 sin( ) 14

i i

i i i

i i i

i

ih k

k hk Z S

S y k y k

h k S h S k h M k

h k M hSd

1/ 2 1/ 2 1/ 2

6 6

1/ 2 1/ 2 1/ 2 1/ 2

6 6

1/ 2 1/ 2 1/ 2

6

1/ 2

1/ 2 3 6

14 cos( ) 16 cos( ),

16 16 sin( ) 14 sin( ) 30 sin( )

30 sin( ) 30 sin( ) 30 sin( )

20 sin(

csc( )

2

i i i

i i i i

i i i

i

i

k hk S hkS

hkS hZ k hk Z S

S y k y k

y k

eh k

6 2 6

1/ 2 1/ 2

2 2 6

1/ 2 1/ 2 1/ 2 1/ 2

2 6 2 2

1/ 2 1/ 2 1/ 2 1/ 2

6

1/ 2 1/ 2 1/

) 20 sin( ) sin( )

sin( ) 12 8 cos( ) 8 sin( )

8 3 sin( ) 3 sin( ) 12 cos( )

12 sin( ) 20 sin( ) 20

i i

i i i i

i i i i

i i i

y k h k M

h k S hkS hk S hk Z

hS k h M k h S k hkS

hZ k S S

2

,

sin( )

(3.5)


55

4

5

41/ 2 -1/ 2 1/ 2

51/ 2 1/ 2 1/ 2 1/ 2

61/ 2 1/ 2

1/ 2 6

1

2csc( )

,

cos( ) sin( ) ,

whereby .,

i i i

i i i i

i ii

k

k

f S M k

g S S Z k

S y kr kh

k

Using the continuity condition of the third, fourth and fifth derivatives at 1/2 1/2( , )i ix y , i.e.,

( ) ( )1 1 2 1 2( ) ( )n n

i i i iP x P x where 3,4,5n , we obtain,

1/2 1/2 1/23 2 5 2 5

3/2 1/2 1/22 5 3 3

1/2 1/2 1/2 1/2 1/24 4 2 5

1/2 12 5 2

1 72 24sin( ) cot( ) cot( )

24 1 2cot( ) cos( )cot( ) cot( )

18 3 3 18 36csc( )

48 24csc( )

i i i

i i i

i i i i i

i i

S S Sk h k h k

S S Sh k k k

S S M M Shk hk h h h k

S Zh k h

/2 1/2 1/2 1/2 1/23 6 3 6 3 3

3/2 3/2 3/2 3/2 3/2 3/22 5 3 4 2 3 6

3/23

120 60 60 120

36 60 3 3 24 60csc( )

1csc( )

0,

i i i i

i i i i i

i

S S y yh k h k h h

S y M S Z Sh k h h hk h h k

Sk

(3.6)

3/ 2 1/ 2 3/ 2 3/ 23 5 3 5 2 4 2

3/ 2 3/ 2 3/ 2 3/ 2 1/ 2 1/ 23 5 3 4 6 4 2 4 2

1/ 2 1/ 2 1/ 23 5 3 3 4

168 168 24 24cot( ) 168cot( )

192 168 360 360 24 24csc( )

192 168 384 360csc( ))

i i i i

i i i i i i

i i i

S S S Mh k h k h k h

S Z S y S Mh k h h k h h k h

S Z Zh k h h h

1/ 2 1/ 26 4

360

0,

i iS yk h

(3.7)

1/2 1/2 1/2 1/23 4

1/2 -1/2 1/2 1/2 1/23 4 3 3 4 5 4 5

5 61/2 1/2 1/2 1/24 5 6 5

1 1 2 120csc( ) cos( )cot( ) cot( )

60 60 120 360 720csc( ) csc( )

360 720 720 1441440/( )

i i i i

i i i i i

i i i i

S S S Sk k k h k

S M M S Sh k h h h k h k

Z h k S S yh h k k

1/25

3/2 3/2 3/2 3/2 3/23 4 3 4 5 4 5 6

3/2 1/2 1/2 3/25 4 5 4 5 4 5

3/2

0

0.60 60 360 360 720

csc( )

720 720 360 360cot( ) cot( ) cot( )

1csc( )

i

i i i i i

i i i i

i

yh

S M S Z Sh k h h k h h k

y S S Sh h k h k h k

Sk

(3.8)

Replacing i by 1, 1, 2, 2i i i i , in Eqs. (3.6)-(3.8) we get the following equations:


56

1 / 2 3 / 2 1 / 23 2 5 2 5

5 / 2 1 / 2 3 / 22 5 3 3

3 / 2 1 / 2 1 / 2 3 / 2 1 / 24 4 2 5

3 / 2 12 5 2

1 72 24sin( ) cot( ) cot( )

24 1 2cot( ) cos( ) cot( ) cot( )

18 3 3 18 36csc( )

48 24csc( )

i i i

i i i

i i i i i

i i

S S Sk h k h k

S S Sh k k k

S S M M Shk hk h h h k

S Zh k h

/ 2 3 / 2 1 / 2 1 / 2 3 / 23 6 3 6 3 3

5 / 2 5 / 2 5 / 2 5 / 2 5 / 2 5 / 22 5 3 4 2 3 6

5 / 23

0,120 60 60 120

36 60 3 3 24 60csc( )

1csc( )

i i i i

i i i i i i

i

S S y yh k h k h h

S y M S Z Sh k h h hk h h k

Sk

(3.9)

3 / 2 1 / 2 3 / 23 2 5 2 5

1 / 2 3 / 2 1 / 22 5 3 3

1 / 2 3 / 2 3 / 2 1 / 2 3 / 2 1 / 24 4 2 5 2 5

32

1 72 24sin( ) cot( ) cot( )

24 1 2cot( ) cos( ) cot( ) cot( )

18 3 3 18 36 48csc( ) csc( )

24

i i i

i i i

i i i i i i

i

S S Sk h k h k

S S Sh k k k

S S M M S Shk hk h h h k h k

Zh

/ 2 1 / 2 3 / 2 3 / 2 1 / 2 1 / 23 6 3 6 3 3 2 5

1 / 2 1 / 2 1 / 2 1 / 2 1 / 2 1 / 23 4 2 3 6 3

0,

120 60 60 120 36csc( )

60 3 3 24 60 1csc( )

i i i i i

i i i i i i

S S y y Sh k h k h h h k

y M S Z S Sh h hk h h k k

(3.10)

3 / 2 5 / 2 3 / 23 2 5 2 5

7 / 2 3 / 2 5 / 22 5 3 3

5 / 2 3 / 2 3 / 2 5 / 2 3 / 2 5 / 24 4 2 5 2 5

32

1 72 24sin( ) cot( ) cot( )

24 1 2cot( ) cos( ) cot( ) cot( )

18 3 3 18 36 48csc( ) csc( )

24

i i i

i i i

i i i i i i

i

S S Sk h k h k

S S Sh k k k


Zh

/ 2 5 / 2 3 / 2 3 / 2 5 / 2 7 / 23 6 3 6 3 3 2 5

7 / 2 7 / 2 7 / 2 7 / 2 7 / 2 7 / 23 4 2 3 6 3

0,

120 60 60 120 36csc( )

60 3 3 24 60 1csc( )

i i i i i

i i i i i i



(3.11)

5 / 2 3 / 2 5 / 23 2 5 2 5

1/ 2 5 / 2 3 / 22 5 3 3

3 / 2 -5 / 2 5 / 2 3 / 2 5 / 2 3 / 24 4 2 5 2 5

52

1 72 24sin( ) cot( ) cot( )

24 1 2cot( ) cos( ) cot( ) cot( )

18 3 3 18 36 48csc( ) csc( )

24

i i i

i i i

i i i i i i

i

S S Sk h k h k

S S Sh k k k


Zh

/ 2 3 / 2 5 / 2 5 / 2 3 / 2 1/ 23 6 3 6 3 3 2 5

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 23 4 2 3 6 3

0,

120 60 60 120 36csc( )

60 3 3 24 60 1( )csc

i i i i i

i i i i i i



(3.12)


57

5/ 2 1/ 2 5/ 2 5/ 23 5 3 5 2 4 2

5/ 2 5/ 2 5/ 2 5/ 2 1/ 2 1/ 23 5 3 4 6 4 2 4 2

1/ 2 1/ 2 3/ 23 5 3 3 4 6

168 168 24 24cot( ) cot( )

192 168 360 360 24 24csc( )

192 168 384 360csc( )

i i i i

i i i i i i

i i i i


S Z S y S Mh k h h k h h k h

S Z Z Sh k h h h k

1/ 2 1/ 24

0,

360iy

h

(3.13)

1/ 2 3/ 2 1/ 2 1/ 23 5 3 5 2 4 2

1/ 2 1/ 2 1/ 2 1/ 2 3/ 23 5 3 4 6 4 2 4

3/ 2 3/ 2 3/ 2 1/ 22 3 5 3 3 4 6

168 168 24 24cot( ) cot( )

192 168 360 360 24csc( )

24 192 168 384 360sin( )

i i i i

i i i i i

i i i i i


S Z S y Sh k h h k h h k

M S Z Z Sh h k h h h k

-3/ 2 3/ 24

0,

360iy

h

(3.14)

7 / 2 3/ 2 7 / 2 7 / 23 5 3 5 2 4 2

7 / 2 7 / 2 7 / 2 7 / 2 3/ 23 5 3 4 6 4 2 4

3/ 2 3/ 2 3/ 2 5/ 22 3 5 3 3 4 6

168 168 24 24cot( ) cot( )

192 168 360 360 24csc( )

24 192 168 384 360csc( )

i i i i

i i i i i

i i i i i




3/ 2 3/ 24

0,

360iy

h

(3.15)

-1/ 2 5/ 2 1/ 2 1/ 23 5 3 5 2 4 2

1/ 2 1/ 2 1/ 2 1/ 2 5/ 23 5 3 4 6 4 2 4

5/ 2 5/ 2 5/ 2 3/ 22 3 5 3 3 4 6

168 168 24 24cot( ) cot( )

192 168 360 360 24csc( )

24 192 168 384 360csc( )

i i i i

i i i i i

i i i i i




-5 / 2 5/ 24

0,

360iy

h

(3.16)

1/ 2 1/ 2 3/ 2 3/ 23 4

1/ 2 1/ 2 3/ 2 1/ 2 3/ 23 4 3 3 4 5 4 5

1/ 2 3/ 2 1/ 2 1/ 24 5 6 5 6 5 5

1 1 2 120sin( ) cos( )cot( ) cot( )

60 60 120 360 720sin( ) csc( )

360 1440 720 720 1440

i i i i

i i i i i

i i i i

S S S Sk k k h k


Z S S yh h k h k h h

3/ 2 5/ 23 4

5/ 2 5/ 2 5/ 2 5/ 2 5/ 23 4 5 4 5 6 5

3/ 2 1/ 2 5/ 2 5/ 24 5 4 5 4 5

600,

60 360 360 720 720sin( ))

720 360 360 1cot( ) cot( ) cot( ) csc( )

i i

i i i i i

i i i i

y Sh k

M S Z S yh h k h h k h

S S S Sh k h k h k k

(3.17)


58

3/ 2 3 / 2 1/ 2 1/ 23 4

3/ 2 3 / 2 1/ 2 3/ 2 1/ 23 4 3 3 4 5 4 5

3/ 2 1/ 2 3 / 2 3/ 24 5 6 5 6 5 5

1 1 2 120sin( ) cos( ) cot( ) cot( )

60 60 120 360 720csc( ) csc( )

360 1440 720 720 1440

i i i i

i i i i i

i i i i

S S S Sk k k h k


Z S S yh h k h k h h

1/ 2 1/ 23 4

1/ 2 1/ 2 1/ 2 1/ 2 1/ 23 4 5 4 5 6 5

1/ 2 3/ 2 1/ 2 1/ 24 5 4 5 4 5

600,

60 360 360 720 720csc( )

720 360 360 1cot( ) cot( ) cot( ) csc( )

i i

i i i i i

i i i i

y Sh k



(3.18)

3/ 2 3/ 2 5 / 2 5 / 2 3/ 23 4 3 4

3/ 2 5 / 2 3/ 2 5 / 2 3 / 23 3 4 5 4 5 4

5 6

5 / 2 3/ 25 6 5 6 5

1 1 2 120 60sin( ) cos( ) cot( ) cot( )

60 120 360 720 360csc( ) csc( )

1440 720 7201440 /( )

i i i i i

i i i i i

i i i

S S S S Sk k k h k h k

M M S S Zh h h k h k h

h k S S yh k h k h

3/ 2 5 / 2 7 / 25 3 4

7 / 2 7 / 2 7 / 2 7 / 2 7 / 23 4 5 4 5 6 5

5 / 2 3 / 2 7 / 2 7 / 24 5 4 5 4 5

1440 600,

60 360 360 720 720csc( )

720 360 360 1cot( ) cot( ) cot( ) csc( )

i i

i i i i i

i i i i

y Sh h k



(3.19)

5 / 2 5 / 2 3 / 2 5 / 23 4 3 4

5 / 2 3 / 2 5 / 2 3 / 2 5 / 23 3 4 5 4 5 4

3/ 2 5 / 2 5 / 2 3 / 25 6 5 6 5 5

1 1 2 120 60sin( ) cos( ) cot( ) cot( )

60 120 360 720 360sin( ) csc( )

1440 720 720 1440

i i i i

i i i i i

i i i i

S S S Sk k k h k h k

M M S S Zh h h k h k h

S S y yh k h k h h

1/ 2 1/ 23 4 3

1/ 2 1/ 2 1/ 2 1/ 2 3/ 24 5 4 5 6 5 5 6

5 / 2 1/ 2 1/ 24 5 4 5

60 600.

360 360 720 720 720csc( ) cot( )

360 360 1cot( ) cot( ) csc( )

i i

i i i i i

i i i

S Mh k h


S S Sh k h k k

(3.20)

We eliminate Z’s and M’s by solving Eqs. (3.6)-(3.19) simultaneously. The values of

' and 'Z s M s obtained from these equations are used in Eq. (3.20), which results, after

lengthy calculations the following recurrence relation,

7 / 2 5/ 2 3/ 2 1/ 2 1/ 2 3/ 2 5/ 2

67 / 2 5/ 2 5/ 2 3/ 2 3/ 2 1/ 2 1/ 2

6 15 20 15 6

,

i i i i i i i

i i i i i i i

y y y y y y y

h S S S S S S S

(3.21)

where

6 3 5

1 120 20 1 120,

120 sin sin sin

(3.22)


59

3 6 5 5 3

3 5 5 6 3

6 5 3

1 40 2cos 720 480 240 cos 26 40 cos,

120 sin sin sin sin sin sin

1 100 67 960cos 840 1800 52cos 80cos,

120 sin sin sin sin sin sin

1 2400 960 80

120 sin sin

3 5

132cos 240 cos 52 1440 cos,

sin sin sin sin

for 4,5, 6,..., 3.i n

Here 0

1lim , , , 1, 120, 1191, 2416

5040

, which is the polynomial case as mentioned

in Eq. (3.3).

The local truncation error ,it associated with the scheme developed in (3.21) is,

6 (6) 7 (7) 8 (8) 9 (9) 10 (10) 11 (11)

6 7 8 9 10 11

12 (12) 13 (13) 14 (14) 1512 13 14 ,

i i i i i i i

i i i

t C h y C h y C h y C h y C h y C h y

C h y C h y C h y O h

(3.23)

where

6

7

8

9

10

11

12

13

1 2 2 2 ,

1 2 2 2,

23 74 34 10

,8

7 218 98 26,

48121 5(3026 706 82

,1920

77 13682 2882 242,

38406227 21(133274 16354 730

,967680

3353 3(7454

C

C

C

C

C

C

C

C

14

18 75938 2186,

19353604681 6155426 397186 6562

.10321920

C

(3.24)

Thus for different choices of , , , in scheme (3.21), methods of different order

are obtained.

The relation (3.21) gives 6n algebraic equations in n unknowns 1/ 2my ,

0,1,..., 1.m n We require six more equations, three at each end of the range of

integration, for the direct computation of 1/ 2my . These equations are developed by

Taylor series and method of undetermined coefficients. General form of boundary

equations for the main scheme is as follows.


60

0 1/ 2 3/ 2 5 / 2 7 / 2 9 / 2

(2) (4) 6

1 0 1 0 1 1 1 1 1 1

2 4 (6) (6) (6) (6) (6) (6)1/ 2 3/ 2 5/ 2 7 / 2 9 / 2 11/ 2

20 35 21 7

,

y y y y y y

y y hh h a y b y c y d y e y f y

(3.25)

(2) (4) 6

2 0 2 0 2 2 2 2 2 2

0 1/ 2 3/ 2 5/ 2 7 / 2 9/ 2

2 4 (6) (6) (6) (6) (6) (6)1/ 2 3/ 2 5/ 2 7 / 2 9 / 2 11/ 2

10 21 21 15 6

,y y h

y y y y y y

h h a y b y c y d y e y f y

(3.26)

(2) (4) 6

3 0 3 0 3 3 3 3 3 3

0 1/ 2 3/ 2 5/ 2 7 / 2 9/ 2 11/ 2

2 4 (6) (6) (6) (6) (6) (6)1/ 2 3/ 2 5/ 2 7 / 2 9 / 2 11/ 2 .

2 7 15 20 15 6

y y h

y y y y y y y

h h a y b y c y d y e y f y

(3.27)

The remaining three equations at the other end can be obtained from (3.25)-(3.27) by

writing them in reverse order. The constants , , , , , , ,i i i i i i i ia b c d e f and are

parameters which must be chosen so that the local truncation errors of (3.25)-(3.27) are

identical with (3.21).

3.3 Numerical methods of different orders

3.3.1 Second-order convergence

1 722 10543 23548, , ,

46080 46080 46080 46080 in the main Eq. (3.21) gives

6 7 8

10, .

24 C C C

The values of the coefficients for these methods are:

1 1 1 1 1 1 1 1

2 2 2 2 2 2

2 2 3 3 3 3 3

3 3

7 77 13005 9821 721 1, , , , , , 0, 0,

2 963 25 9821 23547 10543 722

, , , , , ,4 64

1 1 1 721 10543 23548, 0, , , , , ,

4 19210543 722

, ,

a b c d e fcd cd cd cd

a b c dcd cd cd cd

e f a b ccd cd cd cd

d e fcd cd 3

1, 0, 46080. cd

cd

(3.28)

3.3.2 Fourth-order convergence

7 3 2 17 8 9,

20 5 5 20 5 10

in the main Eq. (3.21) gives

6 7 8 9 10

151 30, 2

120 2C C C C C

for arbitrary constants , .


61

The values of the coefficients of boundary equations are given by

1 1 1 1 1 1

1 1 2 2 2 2

2 2 2 2 3 3

3 3

7 77 2685830 2704842 166502 50582, , , , ,

2 96

3 25 12161085 353329650, 0, , , , ,

4 64

64014771 41166795 10321920 1 1, , , 0, , ,

4 192

51608615,

,

a b c dcd cd cd cd

e f a bcd cd

c d e fcd cd cd

a bcd

3 3

3 3

193535007 273530281 141926257, , ,

10321920 10321920, 10321920.0,,

c dcd cd cd

e f cdcd cd

(3.29)

3.3.3 Sixth-order convergence

11 47 3 151 3 , and

400 20 300 10 240 4

in the main Eq. (3.21) gives

6 7 8 9 10 11 12

41530,

24 151200

C C C C C C C for arbitrary constant .

The values of the coefficients for these methods are:

1 1 1 1 1

1 1 1 2 2

2 2 2 2

2 2

7 77 119271901 128562861 18967984, , , , ,

2 96

11537036 3523149 483175 3 25, , , , ,

4 64

80655078 277312338 77419113 19393863, , , ,

5790447 920775, ,

a b ccd cd cd

d e fcd cd cd

a b c dcd cd cd cd

e fcd cd

3 3 3

3 3 3 3 3

1 1 3815581, , ,

4 192

100731801 255355121 100719251 3839346 5140, , , ,

464486400.

,

1,

acd

b c d e fcd cd cd cd cd

cd

(3.30)

3.3.4 Eight-order convergence

1 41 2189 4153, , ,

30240 5040 10080 7560 in the main Eq. (3.21) gives

6 7 8 9 10 11 12 13 140, 0, 0, 0. C C C C C C C C C

Boundary equations for eight order convergence are not constructed; however these

equations can be obtained following the procedure adapted in the previous cases.


62

3.4 Properties of the Coefficient matrix A0

Consider the following seven-band matrix A0 of order n :

0

35 21 0 0

21 21 0 0

,

0 0 21 21

0 0 21 35

a

b

A a b M b a

b

a

T T TR TR

R

R

(3.31)

Where [ 7,1,0,...,0], [15, 6,1,0,...,0].a b Here T denotes the operation of

transposition and for a row vector 1 2 1 1, ,..., , , ,..., . v vRn n nv v v v v v Further, a, b

are 4n dimensional row vectors. The matrix ,M i jm is a seven-band matrix of

order 4n given by,

,

20, 1,2,..., 4,

15, 1,

6, 2,

1, 3,

0, 3.

i j

i j n

i j

m i j

i j

i j

(3.32)

In this section we find matrix 1A 0 and 1A

0 . The matrix 1A 0 is likewise of the

form,

10 ,

c

d

A c d N d c

d

c

T T TR TR

R

R

(3.33)

where ,i i= c dc d are 4n dimensional row vectors. T and R have the same

meaning as defined above and , , , , , , , are scalars. The matrix

,N i jn is of order 4n . Using 0 0A A I-1n , we get the following equations:

35 21 1

35 21 0

21 21 0

21 21 0

T

TR

T T TR T TR

T

TR

i

ii

iii

iv

v

0

ac

ac

M c a a b b

bc

bc

(3.34)


63

4

21 21 1

21 21 0

35 21 0

35 21 0

.

T

TR

T

TR

T T TR T TR

T TR R T TR Rn

vi

vii

viii

ix

x

xi

0

bd

bd

ad

ad

Md a a b b

MN a c a c b d b d I

Here 0 denotes a 4n dimensional null column vector.

In order to find the unknowns in above equations, we need to use 1, .M i jm

From (Hoskins and Ponzo [49]) it is given by,

,

3 2 1 1 2

240 1 1 2 3

2 2 1 3 3 1,

1 2 2 1 1 1 3 4 3 2

3 2 1 1 2

240 1 1 2 3

2 2 1 3 3 1

1 2 2 1 1 1 3 4 3 2

i j

n i n i n i j j j

n n n n n

i i j j n ni j

i i j j n n i i j j n nm

n j n j n j i i i

n n n n n

j j i i n n

j j i i n n j j i i n n

, .i j

(3.35)

Algebraic manipulation of Eqs. (3.34(i-v)) yields the following expressions. 4 2

4 2

4 2

4 2

16 40 11,

2880

16 40 9,

960

14 20 11,

2880

14 20 9,

960

n n

n

n n

n

n n

n

n n

n

(3.36)

and

3 2 2

3 2 4 3 2

16 24 16 48 362 2 3,

2880 24 108 112 6 6 36 56 6 9

1, 2, ..., 4 .

i

i n i i nn i

n i i i n i i i ic

i n

(3.37)

Eqs. (3.34(vi-x)) yield the following results:

4 2

4 2

16 40 9,

960

16 120 160 51,

320

n n

n

n n n

n

(3.38)


64

4 2

4 2

14 20 9,

960

14 60 51,

320

n n

n

n n

n

and

3 2 2

3 2 4 3 2

16 24 16 48 362 2 3,

960 24 108 192 126 6 36 96 126 51

1,2,..., 4.

i

i n i i nn id

n i i i n i i i i

i n

(3.39)

From Eq. (3.34(xi)), we have

= + ,-1N M U (3.40)

where

1 .U M a c + b d + b d a c T T TR R TR R (3.41)

Hence ,U i ju is given by:

2

5 41 2

4 3 23

2 34 5 6

,1

2880 3 2 1

12 30 3

3 5 5 5 6 ,

60 3 20 4

i jn n n n

i z i n z

n n n n z

i n z i z iz

u

(3.42)

where

5 4 3 2 2 21

4 3 2 4 3 2

5 4 3 22

2 2 4 3 2

4 3 2

5 4 23

12 30 3 20 8 12 30 2 9 9

2 5 15 75 60 3 5 5 5 6 ,

12 88 32 12 2 14 19

8 9 35 31 2 2 7 20 79 54

3 2 7 4 7 6 ,

153 12 90 60 480 360 48

z j j n j n n j n n

j n n n n n n n n

z j j n j n n

j n n j n n n n

n n n n

z j j n n n

4

2 2 3 2

4 2

5 4 24

2 3 2 3 3 2

2 4 3 2

5 4 3 2

180 2 5 3 20 4 18 15

8 4 75 135 60 ,

6 2 3 4 9 35 31

9 1 2 5 6 6 5 38 81 51

6 69 420 679 338

12 64 41 238 413 186 ,

n

j n n j n n

j n n n

z j n j n n

n n n n j n n n

j n n n n

j n n n n n


65

5 2 4 3 25

2 4 3 2

3 4 3 2

6 5 4 3 2

6 5 4 3 2

5 4 3 26

4 5

12 8 12 18 2 20 61 57

18 5 53 195 294 153

4 8 103 490 959 645

3 4 38 125 145 39 183 90

4 2 19 49 67 489 711 315 ,

6 5 15 75 60

15 2 13

z j n n j n n n

j n n n n

j n n n n

n n n n n n

j n n n n n n

z j n n n n

j n n

3 6 5 4 3 2

2 6 5 4 3 2

8 7 6 5 4 3

2

8 7 6 5 4 3

4 3 2

20 2 19 49 67 489 711 315

15 12 100 233 115 1127 1425 558

4 20 55 530 1134 6006

825 1110 360

8 40 200 1825 4230 16502

64

139 291 162

j n n n n n n

j n n n n n n

n n n n n n

n n

n n n n n nj

n n n

2.

80 8595 3150n n

We summarize the results of this section in the following lemma.

Lemma 3.1 The matrix A0 is nonsingular and if 1,A 0 i ja , then , 0i ja and ,i ja

are given by ((3.36)-(3.40)).

In order to find 1A

0 , we first find the sum of elements in the i-th row of 1A 0 .

The sum of elements in the i-th row of 1M is given by,

11 2 1 2 3 ,

720 i i i n i n i n i

(3.43)

and the sum of the elements of i-th row of U is:

4 3 3 2 2

4 3 2

4 3 2

4 6 16 24 8 34 52 291

4 4 12 104 192 116 .960

18 108 258 288 120

i n i n i i i n

n i i i i n

i i i i

(3.44)

Let iS designate the sum of the elements in the i-th row of 1A 0 in modulus, then

, ,1 1

n n

i i j i jj j

S a a

is given by

5 3 2 3

5 4 3 2

6 5 4 3 2

12 6 20 30 20 151

12 30 40 90 16 242880

4 12 20 60 16 48 .

.

i

i n i i i n

i i i i i n

i i i i i i

S

(3.45)


66

Consider iS as a function of the real variable i. It turns out that iS is maximum for

11

2i n . The infinity norm of 1A

0 must be restricted to an integral value of i and

therefore

1( 1) / 2maxA

0 i i nS S .

Here equality will hold only if n is odd. We substitute 11

2i n into (3.45) to get

( 1) / 2nS . Hence,

6 4 21

( 1) / 2

6 4 22 4 66

6

61 175 259 225

46080

61 175 259 225.

46080

A

0 n

n n nS

b a h b a h b a hh

h

(3.46)

3.5 Convergence

Clearly, the family of numerical methods is described by the Eqs. (3.21), boundary

equations and the solution vector 1/ 2 3/ 2 1/ 2, ,...,Y T

ny y y . T denoting transpose, is

obtained by solving a nonlinear algebraic system of order n which has the form,

6 , .h f x 00A Y B Y C = (3.47)

Now we investigate the error analysis of the seven-degree non-polynomial spline

method described in section 3.2. To do so we let,

1/ 2 1/ 2( ) , , ,y Y C T i i i iy x y c t and 1/ 2E ie ,be n-dimensional column

vectors. Here 1/ 21/ 2 1/ 2ii ie y x y is the discretization error for 4,..., 4i n . Thus,

we can write our method in the matrix form:

The matrix B is given by


67

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

3 3 3 3 3 3

2 2 2 2 2 2

1 1 1 1 1 1

.B

a b c d e f

a b c d e f

a b c d e f

f e d c b a

f e d c b a

f e d c b a

(3.48)

The column vector C is given by

0 1 1

0 2 2

0 3 3

0 3 3

0 2 2

0 1 1

2 42 4

2 42 4

2 42 4

2 42 4

2 42 4

2 42 4

20

10

2

0

.

0

2

10

20

C

=

A A A

A A A

A A A

B B B

B B B

B B B

h h

h hh h

h h

h hh h

(3.49)

The vector 1/ 2 3/ 2 1/ 2, ,...,y T

ny x y x y x satisfies

6 , ,h f x 0 - 0A y B y C T =

where 1 2, ,...,T T

nt t t is the vector of local truncation errors, and a conventional

convergence analysis shows that the norm of the vector

E y Y

satisfies:

6

6 4 22 4 6

6 7 8 9 10 11 12

6 6 7 7 8 8 9 9 10 10 11 11 12 12

46080 61 175 259 225

... ,

Eb a

b a h b a h b a h B F

C h V C h V C h V C h V C h V C h V C h V

(3.50)

where max for 1, 2,..., , and max .i

i ia x b a x b

d y x fV i B F

dx y x

B


68

The order of convergence of the numerical method is, thus p, if 6pC is the first non-

vanishing constant on the right hand side of (3.23) provided

*

6 4 22 4 6

46080

61 175 259 225F

b a h b a h b a h

(3.51)

3.6 Numerical Results and Discussion

The numerical methods outlined in the previous sections were tested on the following

nonlinear and linear problems.

3.6.1 Non-linear problems Problem 3.1 Consider the nonlinear BVP

6 2 , 0 1 xD y x e y x x (3.52)


2 4 2 40 0 0 1, 1 1 1 .y D y D y y D y D y e (3.53)

The theoretical solution for this problem is

.xy x e (3.54)

The results of maximum absolute errors for this problem are listed in Table 3.1.

Problem 3.2 We consider the nonlinear BVP

66 20exp 36 40 1 , 0 1 D y x y x x x (3.55)

with boundary conditions:

2 4 2 41 1 1 10 0, 0 , 0 1, 1 ln 2, 1 , 1 .

6 6 24 16 y D y D y y D y D y

(3.56)

for which the theoretical solution is

1ln 1 .

6 y x x

(3.57)

The results of maximum absolute errors for this problem are listed in Tables 3.2-3.3.

3.6.2 Linear problems



69

6 324 11 exp , 0 1 D y x xy x x x x (3.58)

with boundary conditions:

2 2 4 40 0 1 , 0 0, 1 4 , 0 8, 1 16 . y y D y D y e D y D y e (3.59)

The analytical solution of the above differential equation is

1 exp . y x x x x (3.60)

The maximum errors (in absolute value) for different orders are shown in Table 3.4.

Problem 3.4 Finally we consider the linear BVP

6 6 2 cos 5sin , 1 1 D y x y x x x x x (3.61)

with boundary conditions,

2 2

4 4

1 0 1 , 1 4cos 1 2sin 1 , 1 4cos 1 2sin 1 ,

1 8cos 1 12sin 1 , 1 8cos 1 12sin 1 .

y y D y D y

D y D y

(3.62)

The analytical solution of the above differential equation is

2 1 sin . y x x x (3.63)

The maximum errors (in absolute value) are listed in Table 3.5.

The interval 0 1x was divided into n equal subintervals each of width 2 mh

with 3,4,5m so that 7,15,31n respectively. The value of y Y , where Y is

numerical solution, was computed for each value of n . The results for all second-,

fourth-, and sixth-order methods are given in Tables 3.1-3.5. In Table 3.1 the new

method (3.21) is applied to problem 3.1 and the results are compared with (Wazwaz

[157]), where as in Table 3.2 the new method is applied to problem 3.2 and the results

are compared with (Boutayeb and Twizell [24]). The new methods of different orders

are applied to problems 3.2-3.4 and the results are shown in Tables 3.3-3.5 respectively.

Performance of the new method is better than (Wazwaz [157]) and it is comparable in

case of (Boutayeb and Twizell [24]). In (Akram and Siddiqi [8]) problem 3.4 is solved

with the same solution but different boundary conditions. These results are reported in

Table 3.6. This table shows improved performance of our method. Figs. 3.1-3.4 show

graphs of exact and approximate solutions for various values of n. Error graphs are

shown in Figs. 3.5-3.8.


70

Table 3.1

Maximum absolute errors corresponding to problem 3.1

n Sixth-order method (3.3.3) (Wazwaz [157])

10 3.577×10-13 1.299×10-4

Table 3.2


n Sixth-order method (3.3.3) (Boutayeb and Twizell [24])

7 4.55×10-7 2.41×10-7

15 5.96×10-9 7.56×10-10

31 2.68×10-11 2.25×10-11

Table 3.3


n Second-order method (3.3.1) Fourth-order method (3.3.2) Sixth-order method (3.3.3)

7 2.3×10-3 3.0×10-3 4.55×10-7

15 3.4×10-4 1.38×10-4 5.96×10-9

31 1.54×10-4 4.30×10-6 2.68×10-11

Table 3.4



7 2.99×10-2 2.39×10-4 1.15×10-9

15 7.00×10-3 3.43×10-6 3.95×10-12

31 1.80×10-3 7.34×10-8 4.41×10-11


71

Table 3.5



7 1.23×10-2 6.97×10-4 1.78×10-8

15 2.80×10-3 3.60×10-5 1.37×10-10

31 1.6×10-3 7.44×10-7 9.45×10-11

Table 3.6

Maximum absolute errors corresponding to problem 3.4 and results of (Akram and Siddiqi [8])

n (Akram and Siddiqi [8]) Sixth-order method (3.3.3)

8 1.04×10-4 9.2×10-9

16 6.58×10-6 8.45×10-11

32 6.86×10-7 2.75×10-11

64 1.51×10-7 1.97×10-9


72

1 2 3 4 5 6 7 8 9 101

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Grid Points

So

luti

on


Fig. 3.1: Plot of solution for problem 3.1 , 10n

0 5 10 15 20 25 30 350

0.02

0.04

0.06

0.08

0.1

0.12

Grid Points

So

luti

on




73

0 5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Grid Points

So

luti

on



0 2 4 6 8 10 12 14 16-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Grid Points

So

luti

on




74

1 2 3 4 5 6 7 8 9 100.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

-13

Grid points

Err

or

Fig. 3.5: Error graph for problem 3.1 , 10n

0 5 10 15 20 25 30 35-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-11

Grid points

Err

or



75

0 5 10 15 20 25 30 35-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-11

Grid points

Err

or


0 5 10 15 20 25 30 35-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-11

Grid points

Err

or



76

3.7 Conclusion

Non-polynomial spline functions are used to develop a class of numerical methods for

approximate solution of sixth-order linear and non-linear BVPs, with two point

boundary conditions. Second-, Fourth- and Sixth- order convergence is obtained. It has

been shown that the relative errors in absolute value confirm the theoretical

convergence.

***************************************************

Chapter 4

Collocation Method using

Quartic B-Spline for Numerical Solution

of the Modified Equal Width Equation

4.1. Introduction

We consider a model for nonlinear waves, the modified equal width (MEW) equation

2 0 t x xxtu u u u (4.1)

subject to the following physical boundary conditions

1 2, , , , u a t u b t (4.2a)

along with collocation boundary conditions necessary for unique quartic B-spline

solution

, , 0, , , 0, x x xx xxu a t u b t u a t u b t (4.2b)

and initial condition

,0 , .u x f x a x b (4.3)

The parameter is a positive constant and is an arbitrary constant, ( )f x is a

localized disturbance inside the interval [ , ]a b and 0 asU x . Here the

subscripts t and x denote differentiation with respect to t and x respectively. The

MEW equation was introduced by (Morrison et al. [91]) as a model for nonlinear

dispersive waves and is related to the modified regularized long wave (MRLW)

equation (Abdulloev et al. [3]) and the modified Korteweg-de Vries (MKdV) equation

(Gardner et al. [42]). Many authors have investigated numerical solution of the problem

(4.1)-(4.3). These include finite difference method given by (Esen and Kutluay [39]),

He's variational iteration method (Junfeng [68]), tanh and sine-cosine methods

(Wazwaz [159]), mesh free methods (Haq et al. [46]) and various forms of finite

element methods including collocation and Galerkin methods, (see (Esen [38], Evans

Chapter 4 MEW equation

78

and Raslan [40], Raslan [114], Saka [116], Zaki [162]) and the references therein).

In Refs. (Evans and Raslan [40], Raslan [114], Saka [116], Zaki [162]) the MEW

equation is solved numerically by collocation methods based on quadratic, cubic and

quintic B-splines. The present method solves Eqs. (4.1)-(4.3) by using quartic B-spline

collocation method. In Section 4.2, a new numerical method is developed. The stability

analysis of the method is established in section 4.3 and test problems are reported in

section 4.4 to validate performance of the method. Some conclusions are drawn in

section 4.5.

4.2 Quartic B-spline solution

In order to develop the numerical method for approximating solution of boundary value

problems like the one given in Eqs. (4.1)-(4.3), the interval [ , ]a b is partitioned into

1N uniformly spaced points mx such that 0 1 1... N Na x x x x b and

b a

hN

. The quartic B-splines , 2, 3,... 1mB x m N at the knots mx are defined

as (See (Prenter [102])):

4

1 2 2 1

4

2 1 1 1

4

3 2 14 4 4

3 2 1 2

4

3 2 3

, , ,

5 , , ,

10 , , ,1

5 , , ,

, , ,

0 otherwise,

m m m

m m m

m m mm

m m m m

m m m

d x x x x

d d x x x x

d d x x x xB x

h x x x x x x

x x x x

(4.4)

and the set 2 1 1, ,..., NB B B of quartic B-splines forms a basis over the interval ,a b .

The numerical solution ,U x t to ,u x t is given as:

1

2

, ,

N

m mm

U x t t B x (4.5)

where m t are time dependent parameters to be determined at each time level. The

nodal values , and m m mU U U at the knots mx are derived from Eqs. (4.4)-(4.5) in the

following form


79

2 1 1

2 1 1

2 1 12

11 11 ,

43 3 ,

12.

m m m m m

m m m m m

m m m m m

U

Uh

Uh

(4.6)

where dashes represent differentiation with respect to space variable. Eq. (4.1) can be

rewritten as

2 0. xx xtu u u u (4.7)

We discretize the time derivative of Eq. (4.7) by a first order accurate forward

difference formula and apply the -weighted, (0 1) scheme to the space

derivative at two adjacent time levels to obtain the equation

1 1

12 21 0,

n n n nn nxx xx

x x

U U U UU U U U

t

(4.8)

where t is time step and the superscripts n and n+1 are successive time levels. In this

work we take 1/ 2 . Hence Eq. (4.8) takes the form

11 1 2 2

0,2

n nn n n nxx xx x xU U U U U U U U

t

(4.9)

The nonlinear term in Eq. (4.9) is approximated using the Taylor series:

2 1 1 2 1 1 2( ) ( ) 2 2( ) . n n n n n n n n nx x x xU U U U U U U U U (4.10)

At the nth time step, we denote , and m m mU U U at the knots mx by the following

expressions:

1 2 1 1

2 2 1 1

3 2 1 12

11 11 ,

43 3 ,

12.

n n n nm m m m m

n n n nm m m m m

n n n nm m m m m

L

Lh

Lh

(4.11)

Using the knots , 0,1,.., ,mx m N as collocation points, the following recurrence

relation at point mx is obtained using Eq. (4.9)-(4.11):

21 1 1 1 21 2 2 1 3 4 1 1 1 2 32 2n n n n

m m m m m m m m m m m mha a a a L tL L L (4.12)

where


80

2 21 0 1 2 0 1

2 23 0 1 4 0 1

20 1 2

4 24 , 11 12 24 ,

11 12 24 , 4 24 ,

L 2 1 and 0,1,..., .

m m m m m m

m m m m m m

m m m

a L thL a L thL

a L thL a L thL

h L L m N

The Eq. (4.12) relates parameters at adjacent time levels and gives 1N equations in

4N unknowns , 2, 1,..., 1.i i N In order to get a unique solution; we eliminate

the parameters 2 1 1, , N . Using Eq. (4.6) and the boundary conditions (4.2), the

values of the parameters take the form

12 0 1

11 0 1

1 2 1 2

333 7,

4 4 87 1

,4 4 8

11 11 .N N N N

(4.13)

Elimination of the above parameters from Eq. (4.12) yields a 4-banded linear system of

(N+1) equations in (N+1) unknown parameters , 0,1,...,i i N . The linear system can

be solved by a four-diagonal solver successively for , 1, 2,..., ;ni n once we calculate

the initial parameters 0m . Finally the approximate solution ,U x t will be obtained

from Eq. (4.6).

Using the initial and boundary conditions, the values of the initial parameters 0m at

the initial time are determined with the help of the following expressions:

0 0 0 0

2 1 1

0 0 0 00 2 1 1 2

0 0 0 00 2 1 0 12

0 0 0 02 1 1 2

0 0 0 02 1 12

,0 ,0 11 11 , 0,1,..., ,

4,0 3 3 0,

12,0 0,

4,0 3 3 0,

12,0 0.

m m m m m m m

N N N N N

N N N N N

U x u x f x m N

U xh

U xh

U xh

U xh

(4.14)

Eq. (4.14) consists of a linear 4 4N N system which can also be solved by a

four-diagonal solver.


81

4.3 Stability Analysis

In this section we apply the Von-Neumann stability method (Mitchell and Griffiths

[89]) for the stability of scheme developed in the previous section. Since this method is

applicable to linear schemes, the nonlinear term 2xU U is linearized by taking U as a

constant value k . The linearized form of proposed scheme takes the form

1 1 1 11 2 2 1 3 4 1 4 2 3 1 2 1 1,

n n n n n n n nm m m m m m m mpp p p p p p p (4.15)

where

2 21 2

2 22 24 4 , 22 24 12 ,p h h t k p h h t k

2 23 4

2 222 24 12 , 2 24 4 , 0,1,..., .p h h t k p h h t k m N

Substitution of exp , 1,n nm i mh i into Eq. (4.15) leads to

1 2 3 4

4 3 2 1

exp 2 exp exp

exp 2 exp exp .

p i h p i h p p i h

p i h p i h p p i h

(4.16)

Simplifying Eq. (4.16), we get

2 2 2

2 2

2 2

2 2

2 2 2

2 2

2 2

2

2

22

22

where

A= 12 24 24 8 cos

2 4 24 cos 2 ,

20 16 48 sin

2 4 24 sin 2 ,

= 12 24 24 8 cos

2 4 24 cos 2 ,

20 16

h

C h

A iB

C iD

dth k h dth k h

h dth k h

B h dth k h

h dth k h

dth k h dth k h

h dth k h

D h dth k

2 2

48 sin

2 4 24 sin 2 .

h

h dth k h

(4.17)

After simplification, we obtain same expressions for 2 2A B and 2 2C D in the

following form:


82

2 2 2 2

4 2 2 2 4 2 2

4 2 2 2 4 2

4 2 2 2 2 4

4 2 2 2 4 2 2

122 40 240 2888 cos

143 12 2 144

8 22 288 24 10 cos 2

8 4 24 144 cos 3 ,

A B C D

h dt h k hh

h h dt k

h h dt k h

h dt h k h h

(4.18)

so that 2

1 and the linearized numerical scheme for the MEW equation is

unconditionally stable.

4.4 Test problems and discussion

In this section the numerical method outlined in the previous section is tested for a

single solitary wave and interactions of two solitary waves. Moreover, the Maxwellian

initial condition is also considered. The accuracy of the scheme is measured in terms of

the following discrete forms of 2L and L error norms:

2

21

, ,N

i i i ii

i

L Max u U L h u U

(4.19)

where u and U are exact and approximate solutions respectively. The exact solitary

wave solution of MEW equation is given in (Esen and Kutluay [39]):

0

2

where

( , ) sec ,

1 , ,

6

u x t A h k x x ct

Ac k

(4.20)

and A, c represent the amplitude and velocity of a single solitary wave initially centered

at 0x . The initial condition for the above problem is given by

0( ,0) sec ,u x A h k x x (4.21)

and the boundary conditions are taken from Eq. (4.2a) with 1 2 0 . We examine

the conservation properties of the MEW equation related to mass, momentum and

energy by calculating the following three invariants (See (Zaki [162])):

2 2 41 2 3, , .

b b b

x

a a a

C udx C u u dx C u dx (4.22)


83

Integrals in Eq. (4.22) can be approximated by the trapezoidal rule. 4.4.1 A single solitary wave Problem 4.1 To compare our results with (Esen [38], Esen and Kutluay [39], Evans

and Raslan [40], Raslan [114]), we choose the following parameters:

00, 80, 3, 0.25, 0.1, 0.2,0.05, 1, 30.a b A h t x

In order to find error norms and the invariants 1 2 3, ,C C C at different times, the

computations are carried out for times upto 20.t We have compared present method

with earlier published papers (Esen [38], Esen and Kutluay [39], Evans and Raslan

[40], Raslan [114]) at t=20 and the results are reported in Table 4.1. At t=20 the error

norms of the present method are -4 -40.010451 10 , 0.009269 10 ,L -4

2 0.015789 10 ,L

-40.007867 10 for the time steps 0.2 and 0.05 respectively. It is clear from Table 4.1

that the errors in 1 2 3, ,C C C approach zero during the simulation, showing excellent

conservation properties of the new method. Hence the performance of the new method

is better than the above mentioned methods. Fig. 4.1 shows the graphs of the single

solitary wave solutions at 0 and 10t . Initially the centre of solitary wave of

amplitude 0.25 is located at 30x . At time 20t its magnitude is 0.249922 centered

at 30.6x . The absolute difference in amplitudes over the time interval 0, 20 is

observed to be 57.8 10 while it is 52 10 in the case of velocities. It can be

concluded that the solitary wave moves to the right with almost constant magnitude and

velocity. The error graph at time 20t is reported in Fig. 4.2. It can be observed from

the graph that the maximum errors occur around the central position of the solitary

wave.

The same problem is also considered for different values of the amplitude at time

step of 0.01. In Table 4.2 the error norms and invariants are summarized for

0.25,0.5,0.75,1.0A . It is observed that the errors are smaller and the invariants

remain constant during the simulation. The new method is compared with (Esen and

Kutluay [39]) and the comparison of error norms declares superiority of present

scheme. Fig. 3 shows the graphs of the solutions for 0.25,0.5,0.75,1.0A at 20t .


84

Problem 4.2 In order to compare our method with earlier work (Esen [38], Saka [116],

Zaki [162]) we choose the parameters 0, 70, 3, 0.25,0.5,1.0, 0.1, 0.05,a b A h t

01, 30x . The simulation is performed upto time t=20. Error norms and the

invariants 1 2 3, ,C C C are recorded for different values of t and tabulated in Table 4.3. It

is observed that the accuracy of the scheme in terms of error norms increases for

decreasing values of A. For 1,0.25A and t=20 the error norms of the present method

are found as -3 -71.095929 10 ,9.27 10L and -3 -72 1.747622 10 ,7.878 10L . The

invariant quantities 1 2 3, ,C C C are almost constant during the simulation. In Table 4.3

we have also compared our results with lumped Galerkin method using quadratic B-

spline functions (Esen [38]), collocation and petro-Galerkin methods using quintic B-

splines (Saka [116], Zaki [162]) at t=20. In this problem it is observed that the accuracy

of different schemes also depends on amplitude A. Performance of the present method

is better than the methods given in (Esen [38], Zaki [162]) when A=0.25 and

comparable with (Zaki [162]) for 0.5A . It is observed that error norms are less in the

methods given in (Saka [116], Zaki [162]) for 1.A

4.4.2 Interaction of two solitary waves

Problem 4.3 To study the interaction of two solitary waves we use the following initial

condition:

2

1

( ,0) sec ,i iu x A h k x x (4.23)

where

6 1, .i

i

cA k

Parameters 1 2 1 23, 1, 0.5, 15, 30, 1, 0.1, 0.2, 0 80A A x x h t x are chosen

for the sake of comparison with the results of (Esen and Kutluay [39], Evans and

Raslan [40], Saka [116]). These parameters give two solitary waves having amplitudes

of ratio 2:1 and their peak positions are located at 15 and 30.x The analytical values

of the invariants 1 2 3, ,C C C for the above parameters are given in (Evans and Raslan

[40], Saka [116]) as:


85

2 21 1 2 2 1 2

4 43 1 2

84.712389, 3.333333,

34

1.416667.3

C A A C A A

C A A

(4.24)

The calculations are performed from 0 to 80.t t Values of the invariant quantities

1 2 3, ,C C C are tabulated in Table 4.4 for the present method and are compared with

Refs. (Esen and Kutluay [39], Evans and Raslan [40]). It can be seen from the table that

the invariants remain satisfactorily constant throughout the simulation. The upper

bounds for absolute error in the invariants 1 2 3, ,C C C from t=0 to t=55 are less than

7 3 31.0 10 ,2.6 10 and 2.6 10 respectively. In Refs. (Esen and Kutluay [39], Evans

and Raslan [40]), the same errors are less than 6 3 32.0 10 ,3.1 10 ,2.6 10 and

3 3 21.8 10 ,8.7 10 ,1.8 10 . Fig. 4.4 shows the state of interaction and then separation

of solitary waves at times 30,35,40t and 55,80t in sequel. Initially the larger

wave of amplitude 1 is centered at 15x and the smaller one of amplitude 0.5 at

30x . Since the velocity of larger wave is 0.5 and that of the smaller 0.125, the larger

wave moves faster than the smaller and hence collides with later. At 80t the

amplitude of the larger wave is 0.9993 centered at 56.8x and that of smaller 0.4988

with peak position located at 37.7.x Hence during this interaction the amplitude is

almost unchanged. The absolute difference in amplitude for larger wave is 47.0 10

and that of smaller 31.2 10 , consequently the velocities of the waves are almost

maintained after the interaction. Thus the waves interact and then emerge from the

collision by preserving their shapes and velocities. We have solved the same problem

with 0.025t and the invariants are reported in Table 4.5 along with the invariants

reported in Ref. (Saka [116]). It is evident from the comparison of Tables 4.4-4.5 that

the conservation properties of the present method are excellent when time step is

reduced.

4.4.3 The Maxwellian initial condition

Problem 4.4 The birth of solitary waves is considered using the Maxwellian initial

condition

2,0 exp .u x x (4.25)


86

The parameters 3, 1,0.5,0.1,0.05,0.02,0.005, 0.1, 0.01, 20 20h t x used in

Refs. (Saka [116], Zaki [162]) are chosen. In the case of Maxwellian condition the

behavior of the solution depends on the values of . The Maxwellian does not break up

into solutions for c where c is some critical number, and exhibits rapidly

oscillating wave packets. When ,c a mixed type of solution is obtained consisting

of a leading soliton with an oscillating tail (See (Zaki [162])). The Maxwellian breaks

up into a number of solitons according to the value of when .c Simulations

are performed upto time 12.t For 1,0.5 the Maxwellian shows an oscillatory

behavior and no clean waves are obtained as shown in Fig. 4.5.

For 0.1,0.05,0.02,0.005 the number of observed solitary waves is 1, 2, 3 and 7

respectively as shown in Fig. 4.5. The graphs are in good agreement with earlier work

(Saka [116], Zaki [162]). It is also clear from Fig. 4.5 that the peaks of solitary waves

lie on the straight line. For various values of the conservative quantities are tabulated

in Table 4.6 which remain almost constant during the simulation.


87

Table 4.1

Invariants and error norms for single solitary wave, problem 4.1

t Time L 410 2L 410 1C 2C 3C

0.2 0 0.0 0.0 0.785398 0.166667 0.005208

5 0.002706 0.004075 0.785398 0.166667 0.005208

10 0.005377 0.008094 0.785398 0.166667 0.005208

15 0.007944 0.012009 0.785398 0.166667 0.005208

20 0.010451 0.015789 0.785398 0.166667 0.005208

20 (Esen and Kutluay [39]) 2.576377 2.701647 0.785398 0.166474 0.005208

20 (Evans and Raslan [40]) 1.569539 2.021476 0.785286 0.166582 0.005206

20 (Raslan [114]) 1.744330 1.958879 0.784668 0.166434 0.005194

0.05 0 0.0 0.0 0.785398 0.166667 0.005208

5 0.002357 0.002128 0.785398 0.166667 0.005208

10 0.004788 0.004186 0.785398 0.166667 0.005208

15 0.007141 0.006114 0.785398 0.166667 0.005208

20 0.009269 0.007867 0.785398 0.166667 0.005208

20 (Esen [38]) 0.465523 0.796940 0.785390 0.166761 0.005208

20 (Esen and Kutluay [39]) 2.569972 2.692812 0.785398 0.166474 0.005208

20 (Evans and Raslan [40]) 2.498925 2.905166 0.784955 0.166477 0.005200

3, 0.25, 0.1, 1, 30,0 80oA h x x


88

Table 4.2

Invariants and error norms for single solitary wave for various values of A, problem 4.1

A Time L 410 2L 410 1C 2C 3C

0.25 0 0.0 0.0 0.785398 0.166667 0.005208

5 0.002431 0.002174 0.785398 0.166667 0.005208

10 0.004952 0.004282 0.785398 0.166667 0.005208

15 0.007409 0.006267 0.785398 0.166667 0.005208

20 0.009651 0.008087 0.785398 0.166667 0.005208

20 (Esen and Kutluay [39]) 2.569562 2.692249 -- -- --

0.50 0 0.0 0.0 1.570796 0.666667 0.083333

5 0.018825 0.015873 1.570796 0.666667 0.083333

10 0.028470 0.026252 1.570796 0.666667 0.083333

15 0.028486 0.032155 1.570796 0.666667 0.083333

20 0.028502 0.036027 1.570796 0.666667 0.083333

20 (Esen and Kutluay [39]) 14.57568 18.26059 -- -- --

0.75 0 0.0 0.0 2.356194 1.500000 0.421875

5 0.035184 0.039624 2.356194 1.500000 0.421875

10 0.033608 0.047029 2.356194 1.500000 0.421875

15 0.035750 0.049007 2.356194 1.500000 0.421875

20 0.036156 0.051698 2.356194 1.500000 0.421875

20 (Esen and Kutluay [39]) 30.91793 43.95711 -- -- --

1.0 0 0.0 0.0 3.141593 2.666667 1.333333

5 0.095600 0.133043 3.141593 2.666667 1.333333

10 0.185455 0.261374 3.141593 2.666667 1.333333

15 0.276137 0.403847 3.141593 2.666667 1.333333

20 0.366993 0.549764 3.141593 2.666667 1.333333

20 (Esen and Kutluay [39]) 56.82131 82.85314 -- -- --

3, 0.1, 0.01, 1, 30,0 80oh t x x


89

Table 4.3

Invariants and error norms for single solitary wave, problem 4.2

A Time L 310 2L 310 1C 2C 3C

0.25 0 0.0 0.0 0.785398 0.166667 0.005208

5 0.000236 0.000213 0.785398 0.166667 0.005208

10 0.000479 0.000419 0.785398 0.166667 0.005208

15 0.000714 0.000611 0.785398 0.166667 0.005208

20 0.000927 0.000787 0.785398 0.166667 0.005208

20 (Esen [38]) 0.046009 0.080145 0.785397 0.166764 0.005208

20 (Saka [116]) 0.00032 0.00027 0.785398 0.166667 0.005208

20 (Zaki [162]) 0.00203 0.00345 0.78539 0.16667 0.00521

0.5 0 0.0 0.0 1.570796 0.666667 0.083333

5 0.002090 0.003158 1.570796 0.666667 0.083333

10 0.004018 0.005902 1.570796 0.666667 0.083333

15 0.005900 0.008422 1.570796 0.666667 0.083333

20 0.007787 0.010999 1.570796 0.666667 0.083333

20 (Saka [116]) 0.00640 0.00920 1.570796 0.666667 0.083333

20 (Zaki [162]) 0.00852 0.01172 1.57078 0.66666 0.08333

1.0 0 0.0 0.0 3.141593 2.666668 1.333333

5 0.266834 0.410369 3.141593 2.666663 1.333329

10 0.541117 0.845886 3.141593 2.666659 1.333324

15 0.817482 1.294345 3.141593 2.666654 1.333320

20 1.095929 1.747622 3.141593 2.666650 1.333316

20 (Saka [116]) 0.65318 1.04778 3.141593 2.666667 1.333334

20 (Zaki [162]) 0.08360 0.14465 3.14165 2.66676 1.33343

3, 0.1, 0.05, 1, 30,0 70oh t x x


90

Table 4.4

Invariant quantities for interaction of two waves, problem 4.3

Present method (Esen and Kutluay [39]) (Evans and Raslan [40])

Time 1C 2C 3C 1C 2C 3C 1C 2C 3C

0 4.712389 3.333337 1.416669 4.712388 3.329462 1.416669 4.712388 3.332357 1.416670

10 4.712389 3.332777 1.416108 4.712389 3.328927 1.416103 4.712022 3.324678 1.400768

20 4.712389 3.332191 1.415520 4.712387 3.328361 1.415523 4.711697 3.324210 1.401182

30 4.712389 3.330775 1.413861 4.712388 3.327818 1.413882 4.711242 3.346583 1.424847

40 4.712389 3.330942 1.414043 4.712385 3.327112 1.414050 4.711017 3.321250 1.398239

50 4.712389 3.330976 1.414314 4.712388 3.326632 1.414330 4.710804 3.320956 1.398729

55 4.712389 3.330701 1.414043 4.712386 3.326393 1.414062 4.710630 3.323628 1.399068

60 4.712389 3.330417 1.413763 4.712388 3.326228 1.413785 -- -- --

70 4.712389 3.329849 1.413199 4.712388 3.325891 1.413228 -- -- --

80 4.712389 3.329283 1.412635 4.712389 3.325434 1.412671

1 2 1 23, 1, 0.5, 15, 30, 0.1, 0.2, 0 80A A x x h t x

Table 4.5

Invariants for interaction of two solitary waves, problem 4.3

Present method (Saka [116])

Time 1C 2C 3C 1C 2C 3C

0 4.712389 3.333336 1.416669 4.7123884 3.3333358 1.4166697

5 4.712389 3.333336 1.416669 4.7123895 3.3333358 1.4166697

10 4.712389 3.333336 1.416669 4.7123896 3.3333358 1.4166697

15 4.712389 3.333335 1.416668 4.7123896 3.3333358 1.4166697

20 4.712389 3.333334 1.416667 4.7123896 3.3333358 1.4166696

25 4.712389 3.333332 1.416664 4.7123896 3.3333358 1.4166690

30 4.712389 3.333318 1.416647 4.7123896 3.3333359 1.4166648

35 4.712389 3.333295 1.416615 4.7123897 3.3333359 1.4166568

40 4.712389 3.333325 1.416655 4.7123896 3.3333358 1.4166669

45 4.712389 3.333332 1.416664 4.7123896 3.3333357 1.4166695

50 4.712389 3.333332 1.416665 4.7123896 3.3333357 1.4166698

55 4.712389 3.333332 1.416665 4.7123896 3.3333357 1.4166698

1 2 1 23, 1, 0.5, 15, 30, 0.1, 0.025, 0 80A A x x h t x


91

Table 4.6

Invariants for Maxwellian initial condition for various values of , problem 4.4

Time 1C 2C 3C

1 0 1.772454 2.506634 0.886227

3 1.772453 2.506637 0.886223

6 1.772456 2.506629 0.886231

9 1.772456 2.506627 0.886232

12 1.772456 2.506627 0.886232

0.5 0 1.772454 1.879974 0.886227

3 1.772452 1.879979 0.886219

6 1.772454 1.879972 0.886227

9 1.772454 1.879972 0.886227

12 1.772454 1.879972 0.886226

0.1 0 1.772454 1.378646 0.886227

3 1.772437 1.378721 0.886144

6 1.772435 1.378718 0.886138

9 1.772430 1.378715 0.886130

12 1.772427 1.378712 0.886123

0.05 0 1.772454 1.315980 0.886227

3 1.772355 1.316331 0.885265

6 1.772273 1.315714 0.884471

9 1.772224 1.315072 0.883746

12 1.772167 1.314636 0.883001

3, 0.1, 0.01, 20 20h t x


92

0 10 20 30 40 50 60 70 800

0.05

0.1

0.15

0.2

0.25

x

U

t=0

0 10 20 30 40 50 60 70 800

0.05

0.1

0.15

0.2

0.25

x

U

t=20

Fig. 4.1: Single solitary wave solution at t = 0, 20.


93

0 10 20 30 40 50 60 70 80-8

-6

-4

-2

0

2

4

6

8

10x 10

-7

x

Err

or

Fig. 4.2: Error graph at 20.t

0 20 40 60 800

0.05

0.1

0.15

0.2

0.25

x

U

A=0.25

0 20 40 60 80-0.2

0

0.2

0.4

0.6

x

U

A=0.50

0 20 40 60 800

0.2

0.4

0.6

0.8

x

U

A=0.75

0 20 40 60 800

0.5

1

1.5

x

U

A=1.0

Fig. 4.3: Solitary wave solution for various values of A at 20t .


94

0 20 40 60 800

0.2

0.4

0.6

0.8

1

t=0

0 20 40 60 800

0.2

0.4

0.6

0.8

1

t=30

0 20 40 60 800

0.2

0.4

0.6

0.8

1

t=35

0 20 40 60 800

0.2

0.4

0.6

0.8

1

t=40

0 20 40 60 800

0.2

0.4

0.6

0.8

1

t=55

0 20 40 60 800

0.2

0.4

0.6

0.8

1

t=80

Fig. 4.4: Interaction of solitary waves at selected times.

-20 -10 0 10 20-0.5

0

0.5

1=1

-20 -10 0 10 20-0.5

0

0.5

1=0.5

-20 -10 0 10 20

0

0.5

1

1.5=0.1

-20 -10 0 10 20

0

0.5

1

1.5=0.05

-20 -10 0 10 20

0

0.5

1

1.5=0.02

-20 -10 0 10 20

0

0.5

1

1.5=0.005

Fig. 4.5: Maxwellian initial condition for different values of at 12t .


95

4.5 Conclusion

Quartic B-spline collocation method is employed to simulate the motion and interaction

of solitary waves of MEW equation. Four test problems are chosen from literature to

validate performance of the proposed method. The Maxwellian initial condition is also

studied. The accuracy of the method is checked through 2 ,L L error norms and the

invariants 1 2 3, ,C C C . It has been observed that the errors are sufficiently small and the

invariants are almost kept constant during simulation. The results obtained from

numerical experiments are in agreement with some earlier results available in the

literature. Linear stability analysis proved that the new method is unconditionally stable

theoretically and this has been supported by the test problems as well.

***************************************************

Chapter 5

Solitary Wave Solutions of the Modified

Regularized Long Wave Equation

5.1 Introduction

The generalized regularized long wave (GRLW) equation has the form

0,pt x x xxtU U U U U (5.1)

where parameters ,p and are positive constants. This equation has a major role in

the propagation of nonlinear dispersive waves and many authors have investigated its

numerical solution. These include finite difference method (Zhang [163]), Adomian

decomposition method (ADM) (Kaya [71], Kaya and El-Sayed [72]), mesh free

methods (Ali [12]) and Quasi linearization method (Ramos [107]). Exact solution was

obtained by using He’s variational iteration method (Soliman [135]) and Cosine-

function algorithm (Ali et al. [13]). A special case of Eq. (5.1) for 1p is the

regularized long wave (RLW) equation which is used to model a large number of

problems in various areas of science. The equation was originally introduced to explain

behavior of the undular bore development (Pergrine [101]). The analytical solution of

the RLW equation for some initial and boundary conditions are given in (Benjamin et

al. [20], Bona and Bryant [23]). It has been solved numerically by finite difference

methods (Eilbeck and McGuire [37], Pergrine [101]), Meshfree methods (Islam et al.

[53]), Fourier pseudospectral methods (Gou and Cao [45]) and various forms of finite

element methods including collocation, Galerkin and least square methods, (see (Dag

et al. [33], Dag et al. [34], Saka et al. [119]) for the details). We consider another

special case of Eq. (5.1) for 2p , the modified regularized long wave (MRLW)

equation given by

2 0t x x xxtu u u u u (5.2)


Chapter 5 MRLW equation

97

0 as .u x

We will consider the following periodic boundary conditions

, , 0u a t u b t (5.3)

and the collocation boundary conditions in order to get a unique B-spline solution,

given by

, , 0, , , 0.x x xx xxu a t u b t u a t u b t (5.4)

The initial condition is taken as

,0 , ,u x f x a x b (5.5)

where ( )f x is a localized disturbance inside the interval [ , ]a b . In Eq. (5.2) the

subscripts t and x denote differentiation with respect to t and x respectively. Various

methods have been used for the numerical solution of MRLW equation; for instance

cubic B-spline finite element method due to (Gardner et al. [43]), finite difference

method by (Khalifa et al. [74]), Adomian decomposition method in (Khalifa et al. [76])

and collocation method by (Khalifa et al. [75]).

We solve Eqs. (5.2)-(5.5) by quartic and quintic B-Spline collocation methods. The

outline of this chapter is as follows. In Section 5.2, the numerical methods are

presented. In section 5.3, stability is established and numerical results are reported in

section 5.4 to validate performance of the new methods. Conclusion is given in section

5.5.

5.2 The B-spline collocation methods

Let the partition of the space interval ,a b into uniformly spaced points mx be such

that 0 1 1... and N N

b aa x x x x b h

N

.

5.2.1 Quartic B-spline collocation method

We use the quartic B-splines given in Eq. (4.4). The approximate solution ,U x t of

MRLW equation to the exact solution ,u x t is given as:

1

2

, ,N

m mm

U x t t B x

(5.6)


98

where m t are time dependent parameters which will be determined for each time

level. Eq. (5.2) can be rewritten as

2 0.xx x xtu u u u u (5.7)

Discretizing the time derivative of Eq. (5.7) by a forward difference formula and

applying the -weighted scheme to the space derivative at two adjacent time levels, we

obtain the equation

1 111 2

21 0.

n n n nnnxx xx

x x

nn

x x

U U U UU U U

t

U U U

(5.8)

Substituting 1/ 2 in Eq. (5.8), we get

11 1 2 21

0.2 2

n nn n n n n n

xx xx x xx xU U U U U U U UU U

t

(5.9)

The nonlinear term in Eq. (5.9) is approximated by the Eq. (4.10). Using the knots

, 0,1,.., ,mx m N as collocation points, the following recurrence relation at point mx is

obtained using above equation and Eqs. (4.10)-(4.11):

2 2

1 2 1 3

1 1 1 1

1 2 2 1 3 4 1 2 1 2 ,m m m m

n n n n

m m m m m m m m h L tL L La a a a (5.10)

where

2 21 0 1 2 0 1

2 23 0 1 4 0 1

20 1 2

4 1 24 , 11 12 1 24 ,

11 12 1 24 , 4 1 24 ,

L 2 1 and 0,1,..., .

m m m m m m

m m m m m m

m m m

a L h t L a L h t L

a L h t L a L h t L

h L L m N


4N unknowns , 2, 1,..., 1.i i N To assure a unique solution; we eliminate the

parameters 2 1 1, , N from Eq. (5.10) by making use of Eq. (4.13) with 1 2 0 .

After the elimination, Eq. (5.10) will give a 4-banded linear system of (N+1) equations

in (N+1) unknown parameters , 0,1,...,i i N . The linear system can be solved by a

four-diagonal solver successively for , 1, 2,...,ni n . Finally the approximate solution

,U x t will be obtained from Eq. (4.6). Before the commencement of the solution

process, we can find initial parameters 0m with the help of Eq. (4.14).


99

5.2.2 Quintic B-spline collocation method

Quintic B-spline functions , 2, 3,... 2mB x m N are defined at the knots mx by

(See (Prenter [102])):

5

1 3 3 2

5

2 1 2 2 1

5

3 2 1 1

5

4 3 15

5

5 4 1 1 2

5

6 5 2 2 3

, ,

6 , ,

15 , ,1

20 , ,

15 , ,

6 , ,

0 otherwise.

m m m

m m m

m m m

m m m m

m m m

m m m

e x x x x

e e x x x x

e e x x x x

B x e e x x x xh

e e x x x x

e e x x x x

(5.11)

The set of quintic B-splines 2 1 1 2, ,..., ,N NB B B B forms a basis over the interval

,a b . We take the approximate solution as

2

2

, ,N

m mm

U x t t B x

(5.12)

where m t are time dependent unknown real coefficients and mB x are five-degree

B-spline functions . Using Eqs. (5.11)-(5.12), the nodal values , and m m mU U U at the

knots mx can be written as:

2 1 1 2

2 1 1 2

2 1 1 22

26 66 26 ,

510 10 ,

202 6 2 .

m m m m m m

m m m m m

m m m m m m

U

Uh

Uh

(5.13)

At the nth time step, mU and its derivatives at the knots mx are denoted by the

following expressions

1 2 1 1 2

2 2 1 1 2

3 2 1 1 22

26 66 26 ,

510 10 ,

202 6 2 .

n n n n nm m m m m m

n n n nm m m m m


H

Hh

Hh

(5.14)

Using the knots , 0,1,.., ,mx m N as collocation points, we obtain the following


100

recurrence relation at point mx form Eq. (5.9) by making use of Eqs. (4.10) and (5.14):

2

1 2 3

1 1 1 1 1 2

1 2 2 1 3 4 1 5 2 12 1 2 ,m m m

n n n n n

m m m m m m m m m m mh H tH H Ha a a a a (5.15)

where

1 1 2 2 1 2

3 1 4 1 2

25 1 2 1 1 2

22 1

5 40 , 26 50 80 ,

66 240 , 26 50 80 ,

5 40 , 2 2 ,

1 and 0,1,..., .

m m m m m m

m m m m m

m m m m m m

m m

a K K a K K

a K a K K

a K K K h H H

K h t H m N


5N unknowns , 2, 1,..., 2.i i N In order to have a closed form system; we

need to eliminate the parameters 2 1 1 2, , ,N N from the Eq. (5.15). Using Eq.

(5.13) and the boundary conditions, we arrive at the following values of the parameters:

2 0 1 2 1 0 1 2

1 2 1 2 2 1

165 65 9 33 9 1, ,

4 2 4 8 4 81 9 33 9 65 165

, .8 4 8 4 2 4N N N N N N N N

(5.16)

Elimination of the above parameters from Eq. (5.15) yields five-diagonal linear system

of (N+1) equations in (N+1) unknown parameters , 0,1,..., .i i N The linear system

can be solved by the penta-diagonal solver successively for , 1, 2,..., ;ni n once we

calculate the initial parameters 0m .Consequently the approximate solution ,U x t will

be obtained from Eq. (5.13).

The values of the initial parameters 0m at the initial time are determined with the

help of initial conditions and derivatives at boundaries in the following manner:

0 0 0 0 0

2 1 1 2

0 0 0 00 2 1 1 2

0 0 0 0 00 2 1 0 1 22

0 0 0 02 1 1 2

0 02 12

, 0 , 0 26 66 26 , 0,1,..., ,

5,0 10 10 0,

20,0 2 6 2 0,

5,0 10 10 0,

20,0 2

m m m m m m m m

N N N N N

N N N

U x u x f x m N

U xh

U xh

U xh

U xh

0 0 01 26 2 0.N N N

(5.17)

Eq. (5.17) consists of a five-diagonal linear system of 5N equations in

5N unknowns 0 , 2, 1,..., 2,m m N which can also be solved by penta-diagonal


101

algorithm.

5.3 Stability of the proposed scheme

Von-Neumann stability method (Mitchell and Griffiths [89]) is used for the stability of

scheme developed in the previous section. Being applicable to only linear schemes, the

nonlinear term 2xU U is linearized by taking U as a locally constant value k .

5.3.1 Stability of scheme based on quartic B-spline collocation method

The linearized form of proposed scheme using quartic B-spline collocation method

is given as

1 1 1 11 2 2 1 3 4 1 4 2 3 1 2 1 1,

n n n n n n n nm m m m m m m mpp p p p p p p (5.18)

where

2 2

2 21 2

2 23 4

2 2

1 1

1 12 24 4 , 22 24 12 ,

22 24 12 , 2 24 4 ,

0,1,..., .

k k

k kp h h t p h h t

p h h t p h h t

m N


1 2 3 4

4 3 2 1

exp 2 exp exp

exp 2 exp exp .

p i h p i h p p i h

p i h p i h p p i h

(5.19)

Simplifying Eq. (5.19), we get

,A iB

C iD

(5.20)

where

2 2 2

2 2

2 2

2 2

2 2 2 2

2 2

2

22 2A = 1 2 1 2 4 2 4 8 1 c o s

2 4 1 2 4 c o s 2 ,

2 0 1 6 1 4 8 s i n

2 4 1 2 4 s i n 2 ,

= 2 2 1 2 1 2 4 2 4 8 1 c o s

2 4 1 2 4 c o s 2 ,

2 0 1 6 1

h d t h k h d t h k h

h d t h k h

B h d t h k h

h d t h k h

C h d t h k h d t h k h

h d t h k h

D h d t h

2

2 2

4 8 s i n

2 4 1 2 4 s i n 2 .

k h

h d t h k h

After algebraic manipulation, we obtain similar expressions for 2 2A B and 2 2C D in


102

the following form:

2 2 2 2

4 2 2 2 2 2 2 2 4 2 2 2

4 2 2 2 2 2 2 2 4 2 2 2

4 2 2 2 2 2 2 2 4 2 2 2

4 2 2 2 2 2 2

16 61 20 40 20 120 144

8 143 12 24 12 24 144 cos

16 11 12 24 12 120 144 cos 2

8 4 8 4

A B C D

h dt h dt h k dt h k h

h dt h dt h k dt h k h h

h dt h dt h k dt h k h h

h dt h dt h k dt

2 4 2 2 224 144 cos 3 ,h k h h

so that 2

1, which proves unconditional stability of the linearized numerical

scheme for the MRLW equation.

5.3.2 Stability of scheme based on quintic B-spline collocation method

The linearized form of proposed scheme using quintic B-spline collocation method

is given by

1 1 1 1 11 2 2 1 3 4 1 5 2

5 2 4 1 3 2 1 1 2 ,

n n n n nm m m m m

n n n n nm m m m mp

p p p p p

p p p p

(5.21)

where

2 2 21 2 3

2 24 5

2 2

2 2

2 40 5 (1 ), 52 80 50 (1 ), 132 240 ,

52 80 50 (1 ), 2 40 5 (1 ), 0,1,..., .

p h h t k p h h t k p h

p h h t k p h h t k m N


1 2 3 4 5

5 4 3 2 1

exp 2 exp exp exp 2

exp 2 exp exp exp 2

m m m m m

m m m m m

p i h p i h p p i h p i h

p i h p i h p p i h p i h

(5.22)

Simplification of Eq. (5.22) yields

A iB

A iB

(5.23)

where

2 2 2

2 2

A= 4 80 cos 2 104 160 cos 132 240 ,

10 (1 )sin 2 100 (1 )sin ,

h h h h h

B h t k h h t k h

so that 1 and the linearized numerical scheme for the MRLW equation is

unconditionally stable.


103

5.4 Numerical Tests and Results

The numerical methods outlined in the previous section are tested for single solitary

wave and interactions of multiple solitary waves. The Maxwellian initial condition

which generates a train of solitary waves is also considered. The accuracy is checked

using the error norms L and 2L . The analytical solution of MRLW equation is given as

(See (Khalifa et al. [74])):

0

6( , ) sec 1 ,

1

c cu x t h x c t x

c

(5.24)

where 0x is an arbitrary constant. The initial condition is given by

0

6( ,0) sec ,

1

c cu x h x x

c

(5.25)

and the boundary conditions are extracted from the exact equation. The conservation

properties of the MRLW equation related to mass, momentum and energy are

determined by finding the following three invariants (Gardner et al. [43], Olver [98]):

2 2 4 21 2 3

6, , ,

b b b

x x

a a a

C udx C u u dx C u u dx

(5.26)

5.4.1 Single solitary wave

Problem 5.1 We take 01, 0.03, 0.2, 0.025, 1, 0, 40 60c h t x x to

compare our results with (Khalifa et al. [74]). In order to find error norms and the

invariants 1 2 3, ,C C C at different times, the computations are carried out for times upto

10t . The results are given in Table 5.1. At 10t the error norms are

-4 0.033611 10L and -42 0.048674 10L . The error in invariant 3C approaches

zero during the simulation and maximum absolute errors in 1 2,C C remain less than

3 63.55 10 and 2.0 10 throughout the simulation. In Table 5.1 the performance of

the new method is compared with finite difference method due to (Khalifa et al. [74]) at

10t . It is observed that errors of the method in (Khalifa et al. [74]) are considerably

larger than those obtained with method 5.2.1. Fig. 5.1(a) shows the solutions at

0,2,4,6,8,10t and illustrates the motion of solitary wave to the right along the


104

interval 40 60x when 0 10t . Initially the centre of the solitary wave of

amplitude 0.424264 is located at 0x . At time 10t its amplitude is 0.424202

centered at 10.2x . The absolute difference in amplitudes over the time interval

0,10 is found to be 56.2 10 . It can be inferred that the solitary wave moves to the

right with constant amplitude and velocity. Error graph is shown in Fig. 5.1(b).

Problem 5.2 For the sake of comparison with earlier work of authors (Gardner et al.

[43], Khalifa et al. [75]), we choose the following parameters for the solution of

MRLW equation using method 5.2.2:

00, 100, 6, 1, 0.2, 0.025, 1, 40.a b c h t x In order to find error norms

and the invariants 1 2 3, ,C C C at different times, the computations are carried out for

times upto 10.t The amplitude of the solitary wave is 0.9999644 located at 60.x

The absolute difference between numerical and exact peak values is 53.56 10 . The

results are summarized in Table 5.2. At 10t the error norms are -32.0248 10L

and -32 3.9314 10L . It is clear from Table 5.2 that 1C is constant and 2 3,C C remain

almost constant with the maximum error of 55.0 10 . In Table 5.2 the new method is

compared with (Gardner et al. [43], Khalifa et al. [75]) which shows superiority of our

scheme. Fig. 5.2(a) shows the graphs of the solutions at 0, 2,5,10t and depicts the

motion of solitary wave from left to the right along the interval 0 100.x The error

graph is reported in Fig. 5.2(b).

Problem 5.3 The parameters 06, 0.3, 0.1, 0.01, 1, 40,0 100c h t x x are

chosen to enable comparison with collocation method based on cubic splines (Khalifa

et al. [75]). The results are given in Table 5.3. At 20t the error norms are

-5 2.222848 10L and -52 5.089274 10L . The absolute error in invariants

1 2 3, ,C C C approach zero during the simulation. In Table 5.3 the new method is

compared with collocation method based on cubic splines given in (Khalifa et al. [75]).

Again errors are less in the present method, however both methods 5.2.1 and (Khalifa

et al. [75]) produce excellent conservation properties. Fig. 5.3(a) shows the solutions at

0,5,10,15,20t and illustrates the motion of solitary wave to the right along the

interval 0 100x when 0 20t . Initially the amplitude of solitary wave is

0.547723 and its peak position is located at 40x . At 20t its amplitude is recorded


105

as 0.547721 with center 66x . Hence the absolute difference in amplitudes over the

time interval 0, 20 is observed as 62.0 10 Error graph is displayed in Fig. 5.3(b).

The same problem is also solved by method 5.2.2 and the results are reported in

Table 5.4. Fig. 5.4(a) shows the solution at 0,6,12, 20t and the error graph is shown

in Fig. 5.4(b).

5.4.2 Two solitary waves

Problem 5.4 We consider interaction of two solitary waves using the following initial

condition:

2

1

6( ,0) sec .

1i i

ii i

c cu x h x x

c

(5.27)

In our numerical experiment, we choose 1 2 1 26, 4, 1, 25, 55, 1, 0.2,c c x x h

0.025,t 0 250.x The parameters give solitary waves of different amplitudes of

ratio 2:1 having centers at 25 and 55x x to make the interaction possible. For the

case of interaction of two solitary waves the analytical values of invariants are given in

(Khalifa et al. [75]) as 1 2 311.467698, 14.629243, 22.880466C C C . Computations

for both methods 5.2.1 and 5.2.2 are done up to time 20t and same values for

invariant quantities are obtained. The values of invariants 1 2 3, ,C C C are tabulated in

Table 5.5 and compared with (Khalifa et al. [75]) at 20t . The invariants are almost

constant , however it is observed that collocation method using cubic splines by

(Khalifa et al. [75]) has better conservative properties in this case. In Fig. 5.5 the

interaction of solitary waves is shown at different times 0,4,8,10,14,20t

respectively. It is clear from the figure that the two solitary waves interact at times

4,8,10t and then separate at 14,20t preserving their original shapes.

We have also taken 1 2 1 26, 0.03, 0.01, 18, 58, 40 180c c x x x to

compare our results with ADM method in (Khalifa et al. [76]). The results are tabulated

in Table 5.6. It is clear from the table that the conservation properties of present method

are better than ADM when smaller amplitudes and times are considered. Moreover, no

interaction of solitary waves is observed over the time interval 0, 2 .


106

5.4.3 Three solitary waves

Problem 5.5 Interaction of three solitary waves is considered using the initial

condition

3

1

6( ,0) sec ,

1i i

ii i

c cu x h x x

c

(5.28)

and the following parameters for the purpose of comparison.

1 2 3 1 2 36, 4, 1, 0.25, 15, 45, 60, 1, 0.2, 0.025, 1,0 250.c c c x x x h t x These parameters give solitary waves of different amplitudes 4, 2 and 1 having their

peaks located at 15,45 and 60x moving along the same direction. The analytical

values of invariants for the interaction of three waves are given in (Khalifa et al. [75])

as 1 2 314.9801, 15.8218, 22.9923C C C . Computations for both methods 5.2.1 and

5.2.2 are done from 0 to 45t t and same values for invariant quantities are

obtained. The values of the invariant quantities during the simulation are given in Table

5.7. It is evident from this table that the values produced by our method for 1 2,C C are

better when compared with the exact values whereas the performance of the method in

(Khalifa et al. [75]) is better in case of 3C . Fig. 5.6 shows the interaction of solitary

waves at times 0,5,8,15,20,40t respectively. The three solitary waves interact at

times 5,8,15t and then separate at 20,40t emerging unchanged.

The parameters 1 2 2 1 2 36, 0.03, 0.02, 0.01, 18, 48, 88, 40 180c c c x x x x are

chosen to compare our results with ADM of (Khalifa et al. [76]). The results are

tabulated in Table 5.8 along with those given in (Khalifa et al. [76]). It is again

observed that the conservation properties of present method are very good as compared

to ADM when smaller amplitudes and times are considered.

5.4.4 The Maxwellian initial condition

Problem 5.6 As the last problem, the Maxwellian initial condition

2,0 exp 40u x x , (5.29)

is used with boundary conditions given by Eq. (5.3) when 0a and 100b . The

numerical solutions of the MRLW equation are carried out for generation of a train of

solitary waves for various values of . We consider for the set of values


107

0.1,0.04,0.015,0.01, which is used in the earlier work(See (Gardner et al. [43], Khalifa

et al. [74])) and the simulations are performed up to time 15t . For 0.1 we

observe generation of a single solitary wave along with an oscillating tail. However

when is reduced, Maxwellian initial condition breaks up into a number of solitary

waves. For 0.04,0.015,0.01 the number of observed solitons is 2,3,4 respectively.

An oscillating wave behind the train of solitary waves is observed in each case. The

various cases for are shown in Fig. 5.7 and the graphs are in agreement with earlier

work (Gardner et al. [43], Khalifa et al. [74]). It is also clear from the Fig. 5.7 that the

peaks of solitary waves lie on the straight line. The conservative quantities are given in

Table 5.9 which remain almost constant during the simulation


108

Table 5.1 Accuracy test for problem 5.1 method 5.2.1 Time L

410 2L 410 1C 2C 3C

0 0.0 0.0 7.804400 2.129885 0.130251

1 0.001589 0.004743 7.805221 2.129885 0.130251

2 0.003456 0.009481 7.805897 2.129886 0.130251

3 0.005569 0.014214 7.806451 2.129886 0.130251

4 0.007960 0.018951 7.806898 2.129887 0.130251

5 0.010731 0.023719 7.807254 2.129887 0.130251

6 0.013944 0.028526 7.807528 2.129887 0.130251

7 0.017671 0.033387 7.807729 2.129887 0.130251

8 0.022125 0.038349 7.807864 2.129887 0.130251

9 0.027374 0.043428 7.807936 2.129887 0.130251

10 0.033611 0.048674 7.807948 2.129887 0.130251

10 (Khalifa et al. [74]) 1.99524 6.98280 7.80932 2.12988 0.130315

Invariants and error norms for single solitary wave, 1, 0.03, 0.2, 0.025,c h t 40 60x


109


310 2L 310 1C 2C 3C

0 0.0 0.0 4.44288 3.29983 1.41421

1 0.2209 0.4036 4.44288 3.29983 1.41421

2 0.4283 0.8041 4.44288 3.29982 1.41420

3 0.6289 1.2018 4.44288 3.29982 1.41420

4 0.8277 1.5953 4.44288 3.29981 1.41419

5 1.0261 1.9859 4.44288 3.29980 1.41419

6 1.2248 2.3749 4.44288 3.29980 1.41418

7 1.4239 2.7633 4.44288 3.29980 1.41418

8 1.6236 3.1520 4.44288 3.29979 1.41417

9 1.8239 3.5413 4.44288 3.29979 1.41417

10 2.0248 3.9314 4.44288 3.29978 1.41416

10 (Khalifa et al. [75]) 5.43718 9.30196 4.44288 3.29983 1.41420 10 (Gardner et al. [43]) 9.24 16.39 4.442 3.299 1.413

Invariants and error norms for single solitary wave, 6, 1, 0.2, 0.025, 1, 40,0 100oc h t xx


110


510 2L 510 1C 2C 3C

0 0.0 0.0 3.581967 1.345076 0.153723

2 0.267928 0.556873 3.581967 1.345076 0.153723

4 0.520815 1.103371 3.581967 1.345076 0.153723

6 0.768640 1.637097 3.581967 1.345076 0.153723

8 0.994829 2.156048 3.581967 1.345076 0.153723

10 1.209189 2.662302 3.581967 1.345076 0.153723

12 1.417087 3.158828 3.581967 1.345076 0.153723

14 1.621213 3.648135 3.581967 1.345076 0.153723

16 1.823033 4.132113 3.581967 1.345076 0.153723

18 2.023404 4.612153 3.581967 1.345076 0.153723

20 2.222848 5.089274 3.581967 1.345076 0.153723

20 (Khalifa et al.

[75])

29.6650 60.6885 3.58197 1.34508 0.153723

Invariants and error norms for single solitary wave, 6, 0.3, 0.1, 0.01, 40oc h t x


111

Table 5.4

Accuracy test for problem 5.2 method 5.2.2

Time L 510 2L 510 1C 2C 3C

0 0.0 0.0 3.58197 1.34508 0.15372

2 0.2683 0.5669 3.58197 1.34508 0.15372

4 0.5403 1.1236 3.58197 1.34508 0.15372

6 0.7902 1.6654 3.58197 1.34508 0.15372

8 1.0187 2.1912 3.58197 1.34508 0.15372

10 1.2354 2.7038 3.58197 1.34508 0.15372

12 1.4457 3.2064 3.58197 1.34508 0.15372

14 1.6523 3.7016 3.58197 1.34508 0.15372

16 1.8565 4.1914 3.58197 1.34508 0.15372

18 2.0593 4.6773 3.58197 1.34508 0.15372

20 2.2612 5.1602 3.58197 1.34508 0.15372

Invariants and error norms for single solitary wave, 6, 0.3, 0.1, 0.01, 1,c h t

40,0 100o xx

Table 5.5 Accuracy test for problem 5.4 Time

1C 2C 3C

0 (Analytical) 11.467698 14.629277 22.880432

2 11.467698 14.624259 22.860365

4 11.467698 14.619226 22.840279

6 11.467699 14.614169 22.820069

8 11.467700 14.606821 22.787857

10 11.467700 14.603687 22.771773

12 11.467699 14.603056 22.775766

14 11.467699 14.598059 22.756029

16 11.467700 14.593048 22.736127

18 11.467700 14.588061 22.716289

20 11.467701 14.583089 22.696510

20 (Khalifa et al. [75]) 11.4677 14.6292 22.8809

Invariants for two solitary waves, 1 2 1 2 0 2506, 4, 1, 25, 55, 0.2, 0.025, xc c x x h t


112

Table 5.6 Accuracy test for problem 5.4 Present method ADM (Khalifa et al. [76])


0 6.345403 0.592825 0.00548537 6.34540 0.592825 0.00548538

0.2 6.345374 0.592825 0.00548536 6.34532 0.592789 0.00548414

0.4 6.345377 0.592825 0.00548536 6.34508 0.592683 0.00548068

0.6 6.345380 0.592825 0.00548537 6.34466 0.592511 0.00547517

0.8 6.345383 0.592825 0.00548537 6.34408 0.592280 0.00546787

1.0 6.345385 0.592825 0.00548537 6.34333 0.591998 0.00545912

1.2 6.345388 0.592825 0.00548537 6.34241 0.591678 0.00544929

1.4 6.345390 0.592825 0.00548537 6.34132 0.591334 0.00543879

1.6 6.345392 0.592825 0.00548537 6.34007 0.590985 0.00542807

1.8 6.345394 0.592825 0.00548537 6.33864 0.590648 0.00541753

2 6.345395 0.592825 0.00548537 6.33705 0.590348 0.00540758

Invariants for interaction of two solitary waves, 1 2 1 26, 0.03, 0.01, 18, 58,c c x x 40 180x


113

Table 5.7 Invariant quantities for problem 5.5 Time

1C 2C 3C

0 14.980099 15.837528 23.008136

5 14.980105 15.824928 22.957891

10 14.980109 15.807025 22.877972

15 14.980106 15.807032 22.885947

20 14.980106 15.795022 22.837454

25 14.980107 15.782840 22.788852

30 14.980107 15.770634 22.740419

35 14.980108 15.758480 22.692279

40 14.980108 15.746389 22.644448

45 14.968030 15.734374 22.596591

45 (Khalifa et al.

[75])

13.7043 15.6563 22.9303

Invariants for three solitary waves, 1 2 3 1 2 3 , 0.2, 0.025,4, 1, 0.25, 15, 45, 60 h tc c c x x x

0 250x


114

Table 5.8 Invariant quantities for problem 5.5 Present method ADM (Khalifa et al. [76])


0 9.517705 0.904129 0.00786304 9.51769 0.904129 0.00786310

0.1 9.517580 0.904130 0.00786272 9.51770 0.904193 0.00786515

0.2 9.517585 0.904130 0.00786286 9.51771 0.904383 0.00787064

0.3 9.517590 0.904130 0.00786293 9.51772 0.904700 0.00787956

0.4 9.517594 0.904130 0.00786294 9.51772 0.905144 0.00789191

0.5 9.517598 0.904130 0.00786296 9.51773 0.905715 0.00790768

0.6 9.517602 0.904130 0.00786297 9.51774 0.906413 0.00792688

0.7 9.517606 0.904130 0.00786298 9.51775 0.907239 0.00794952

0.8 9.517610 0.904130 0.00786299 9.51776 0.908192 0.00797559

0.9 9.517613 0.904130 0.00786300 9.51776 0.909270 0.00800510

1.0 9.517616 0.904130 0.00786300 9.51777 0.910476 0.00803807

Invariants for interaction of three solitary waves, 1 2 3 16, 0.03, 0.02, 0.01, 18,c c c x 2 348, 88, 40 180x x x


115

Table 5.9 Invariant quantities for problem 5.6 Time

1C 2C 3C

0.1 0 1.77245 1.37865 0.76089 3 1.77246 1.37877 0.76158 6 1.77246 1.37867 0.76139 9 1.77246 1.37857 0.76125 12 1.77246 1.37846 0.76111 15 1.77246 1.37836 0.76097 0.04 0 1.77245 1.30345 0.83609 3 1.77264 1.30454 0.84638 6 1.77240 1.30191 0.83980 9 1.77236 1.30110 0.83961 12 1.77218 1.29905 0.83504 15 1.77216 1.29843 0.83549 0.015 0 1.77245 1.27211 0.86743 3 1.77244 1.27112 0.87222 6 1.77236 1.27003 0.87212 9 1.77224 1.26859 0.87036 12 1.77189 1.26478 0.85872 15 1.77222 1.26423 0.85410 0.01 0 1.77245 1.26585 0.87369 3 1.77238 1.26583 0.89749 6 1.77167 1.26155 0.89028 9 1.77027 1.25148 0.85771 12 1.76957 1.24724 0.85094 15 1.76864 1.24121 0.83689 Conservative quantities for MRLW equation using Mawellian initial condition


116

-40 -30 -20 -10 0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x

U

t=6

t=4

t=2

t=0

t=10

t=8

Fig. 5.1(a): Single solitary wave solution at 0, 2, 4, 6,8,10t

0 10 20 30 40 50 60 70 80 90 100-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t=0 t=2 t=5 t=10

Fig. 5.2(a): Single solitary wave solution


117

-40 -30 -20 -10 0 10 20 30 40 50 60-2

-1

0

1

2

3

4x 10

-6

x

Err

or

Fig. 5.1(b): Error graph at 10t

0 10 20 30 40 50 60 70 80 90 100-0.0025

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

X

Err

or



118

0 10 20 30 40 50 60 70 80 90 100-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x

U

t=0 t=5 t=10 t=15 t=20

Fig. 5.3(a): Single solitary wave solution method 5.2.1

0 10 20 30 40 50 60 70 80 90 100-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-5



119

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

t=0 t=6 t=12 t=20

Fig. 5.4(a): Single solitary wave solution method 5.2.2

0 10 20 30 40 50 60 70 80 90 100-0.000025

-0.00002

-0.000015

-0.00001

-0.000005

0

0.000005

0.00001

0.000015

0.00002

0.000025

x

Err

or



120

0 50 100 150 200 250-1

0

1

2

0 50 100 150 200 250-1

0

1

2

0 50 100 150 200 250-1

0

1

2

0 50 100 150 200 250-1

0

1

2

0 50 100 150 200 250-1

0

1

2

0 50 100 150 200 250-1

0

1

2

t=0 t=4

t=8 t=10

t=14 t=20

Fig 5.5: Interaction of two solitary waves at selected times

0 50 100 150 200 250-1

0

1

2

0 50 100 150 200 250-1

0

1

2

0 50 100 150 200 250-1

0

1

2

0 50 100 150 200 250-1

0

1

2

0 50 100 150 200 250-1

0

1

2

0 50 100 150 200 250-1

0

1

2

t=0

t=8

t=20

t=5

t=40

t=15

Fig. 5.6: Interaction of three solitary waves at selected times


121

0.00 20.00 40.00 60.00 80.00 100.00-0.50

0.00

0.50

1.00

1.50

0 .00 20.00 40.00 60.00 80.00 100.00-0.50

0.00

0.50

1.00

1.50

0.00 20.00 40.00 60.00 80.00 100.00-0.05

0.00

0.50

1.00

1.50

0 .00 20.00 40.00 60.00 80.00 100.00-0.50

0.00

0.50

1.00

1.50

= 0.1 =0.04

=0.015 =0.01

Fig. 5.7: The Maxwellian initial condition at 14.5t


122

5.5 Conclusion

Collocation method based on quartic and quintic B-splines is applied to investigate

propagation of nonlinear dispersive solitary waves of the MRLW equation. The method

is validated by choosing six test problems from literature. The accuracy of the method

is examined through 2 ,L L error norms and the invariant quantities 1 2 3, ,C C C . It has

been observed that the error norms are sufficiently small and the invariants are almost

constant during the simulation. The obtained results are in agreement with some earlier

results from literature. Linear stability analysis and numerical tests proved that the new

schemes are unconditionally stable.

***************************************************

Chapter 6

Collocation Method using B-Splines for

Numerical Solution of

Kuramoto-Sivashinsky Equation

6.1 Introduction

The Kuramoto-Sivashinsky (KS) equation is given by

0,t x xx xxxxu uu u u (6.1)

subject to the following boundary conditions:

1 2, , , , 0 ,u a t g t u b t g t t (6.2)

and initial condition

,0 , .u x f x a x b (6.3)

The parameters and are some arbitrary constants. KS equation is a canonical

nonlinear evolution equation having various applications, e.g., long waves on thin

films, long waves on the interface between two viscous fluids (Hopper and Grimshaw

[48]) , unstable drift waves in plasmas, reaction diffusion systems (Kuramoto and

Tsuzuki [83]) and flame front instability (Sivashinsky [133]). It also describes the

fluctuations of the position of a flame front, the motion of a fluid going down a vertical

wall, or a spatially uniform oscillating chemical reaction in a homogeneous medium

(Conte [31]). Its well- posedness was established in (Tadmor [141]).

A variety of numerical and approximate methods are available in the numerical

literature to solve KS equation. Some of those are given in the references (Abdel-

Gawad and Abdusalam [1], Hyman and Nicolaenko [50], Hyman et al. [51], Jolly et al.

[66], Kevrekidis et al. [73], Khater and Temsah [78], Lopez-Marcos [84], Manickam et

al. [87], Papageorgiou and Smyrlis [99], Smyrlis and Papageorgiou [134], Xu and Shu

[160]). Exact solution through the balance of the dispersive and dissipative effects has

been investigated within (Abdel-Hamid [2]). For more details about the exact methods,

Chapter 6 Kuramoto-Sivashinsky equation

124

we refer to (Baldwin et al. [17], Parkes and Duffy [100], Wazwaz [158]).

We solve Eqs. (6.1)-(6.3) by using quartic and quintic B-Spline collocation methods

and splitting techniques. The organization of this chapter is as follows. In Section 6.2,

the numerical methods are presented. The numerical results are given in section 6.3 to

examine performance of the new methods and section 6.4 includes conclusion.

6.2 The B-Spline Collocation Methods

In this section we construct the numerical methods for approximating solution of

boundary value problem (6.1)-(6.3). The spatial domain [ , ]a b is divided into N

subintervals of equal length using the knots mx a mh , 0,1,...,m N , 0 , Nx a x b

and .b a

hN

6.2.1 Quintic B-Spline Collocation Method 1

We use the collocation method with quintic B-splines given by Eq. (5.11) as basis and

derive some consistency relations, which lead us to find numerical solution of the

problem under consideration. To get approximate solution we proceed as follows. Let

2

2

, ,N

m mm

U x t t B x

(6.4)

be the approximate solution, where m t are time dependent unknown real

coefficients and mB x are fifth-degree B-spline functions . From Eqs. (5.11) and (6.4),

the values of U and its space derivatives at the knots mx can be obtained as

2 1 1 2

2 1 1 2

2 1 1 22

2 1 1 24

26 66 26 ,

510 10 ,

202 6 2 ,

1204 6 4 .

m m m m m m

m m m m m

m m m m m m

m m m m m m

U

Uh

Uh

Uh

(6.5)

Application of weighted scheme with 12 to Eq. (6.1) yields


125

1 1 11

0,2 2 2

n n n n n nn n

x x xx xx xxxx xxxxUU UU U U U UU U

t

(6.6)

where t is time step and the superscripts n and n+1 denote the time step labels. The

nonlinear term in Eq. (6.6) is approximated by Taylor series

1 1 1 .n n n n n n n

x x x xUU U U U U U U (6.7)

At the nth time step, we denote mU and its space derivatives at the knots mx by the

following expressions:

1 2 1 1 2

2 2 1 1 2

3 2 1 1 22

4 2 1 1 24

26 66 26 ,

510 10 ,

202 6 2 ,

1204 6 4 .


n n n nm m m m m



H

Hh

Hh

Hh

(6.8)

Using the knots , 0,1,.., ,mx m N as collocation points, we obtain the following

recurrence relation at point mx form Eqs. (6.5)-(6.8):

4

1 3 4

1 1 1 1 11 2 2 1 3 4 1 5 2 2 m m m

n n n n nm m m m m m m m m m h H t H Ha a a a a

(6.9)

where

3 21 0 1

3 22 0 1

23 0

3 24 0 1

3 25 0 1

5 20 120 ,

26 50 40 480 ,

66 120 720 ,

26 50 40 480 ,

5 20 120 ,

m m m

m m m

m m

m m m

m m m

a H h H t h t t

a H h H t h t t

a H h t t

a H h H t h t t

a H h H t h t t

40 2 2 and 0,1,..., .m mH h tH m N


5N unknowns , 2, 1,..., 2i i N . In order to get a unique solution the parameters

2 1 1 2, , ,N N are eliminated from Eq. (6.9). Using Eqs. (6.2) and (6.5), the

following values of the parameters are obtained:


126

12 0 1 2

11 0 1 2

21 2 1

22 2 1

5165 65 9,

4 2 4 833 9 1

,8 4 8 161 9 33

,8 4 8 16

59 65 165.

4 2 4 8

N N N N

N N N N

g

g

g

g

(6.10)

Elimination of the above parameters from Eq. (6.9) yields a five-diagonal linear system

of (N+1) equations in (N+1) unknown parameters , 0,1,..., .m m N After calculating

the initial parameters 0m , this linear system can be solved by the penta-diagonal solver

successively for , 1, 2,..., .nm n The approximate solution ,U x t will then be

obtained from Eq. (6.5).

The initial parameters 0m are determined with the help of derivatives at boundaries

and initial conditions:

0 0 0 0 0

2 1 1 2

0 0 0 00 2 1 1 2 0

0 0 0 0 00 2 1 0 1 2 02

0 0 0 02 1 1 2

0 0 02 12

,0 ,0 26 66 26

5,0 10 10 ,

20,0 2 6 2 ,

,

5,0 10 10 ,

20,0 2 6

m m m m m m m m

N N N N N N

N N N N

U x u x f x

U x uh

U x uh

U x uh

U xh

0 01 22 ,

0,1,..., .

N N Nu

m N

(6.11)

Eq. (6.11) consists of a five-diagonal linear system of 5N equations in

5N unknowns 0 , 2, 1,..., 2,m m N which can also be solved by penta-diagonal

solver.

6.2.2 Quintic B-Spline Collocation Method 1I

We use splitting technique in space for the numerical solution of the KS equation.

Splitting methods in time and in space are used to get economy and simplicity in the

technique. For that purpose we introduce a new variable defined by xxu v and split


127

Eq. (6.1) as follows:

0,t x xxu uu v v (6.12)

0,xxu v (6.13)

subject to the following boundary conditions:

1 2, , , , , 0, , 0, 0 ,u a t g t u b t g t v a t v b t t (6.14)

and initial conditions

,0 , ,0 , .u x f x v x f x a x b (6.15)

The approximations ,U V to the exact solutions u and v are given by

2 2

2 2

, , , ,N N

m m m mm m

U x t t B x V x t t B x

(6.16)

where ,m mt t are time dependent parameters to be determined and mB x are

fifth-degree B-spline functions.

The values of U, V and their space derivatives at the knots mx are obtained from

Eqs. (5.11) and (6.16), in the following manner

2 1 1 2

2 1 1 2

2 1 1 2

2 1 1 2

2 1 1 22

26 66 26 ,

510 10 ,

26 66 26 ,

510 10 ,

202 6 2 .

m m m m m m

m m m m m

m m m m m m

m m m m m

m m m m m m

U

Uh

V

Vh

Vh

(6.17)

We apply the implicit midpoint approximation rule for the unknown parameters in Eqs.

(6.12)-(6.13) and discretize the time derivative in Eq. (6.12) by finite-difference forward

formula at two adjacent time levels and 1n n to get:

1 11 1

1 1

0,2 2 2

0.2 2

n n n nn n n nx x xx xx

n n n nxx xx

UU UU V VU U V V

t

U U V V

(6.18)

At the nth time step, the approximate solutions ,m mU V and their space derivatives at the

knots mx are denoted by the following expressions:


128

1 2 1 1 2

2 2 1 1 2

3 2 1 1 22

1 2 1 1 2

2 2 1 1 2

3 22

26 66 26 ,

510 10 ,

202 6 2 ,

26 66 26 ,

510 10 ,

202


n n n nm m m m m



n n n nm m m m m

nm m

H

Hh

Hh

K

Kh

Kh

1 1 26 2 .n n n nm m m m

(6.19)

Let , 0,1,.., ,mx m N be the collocation points in the interval , .a b Using Eqs. (6.7),

(6.17) and (6.19) in Eq. (6.18), the following system of coupled equations is obtained:

1 1 1 1

1 2 2 1 3 4 1

1 1 1 1 2

5 2 1 2 2 1 3 1 1 3

1 1

2 1 1 2

1 1 1 1 1

2 1 1 2

2 1 1

2 1

2

20 40 120 40 20

26 66

n n n n

m m m m m m m m

n n n n

m m m m m m m m m m m

n n

m m m m

n n n n n

m m m m m

n n

m m

b b b b

b c c c h H t K K

c c

h

2

1 31 1 1

1 2

1 0 1 2 0 1 3 0

4 0 1 5 0 1

2 2 2

1 2 3

20

26

where

5 , 26 50 , 66 ,

26 50 , 5 ,

20 , 26 40 , 66 120 ,

with 2

m mn n n

m m m

m m m m m m m m

m m m m m m

m m m

m m

h K H

b H hH t b H hH t b H

b H hH t b H hH t

c th t c th t c th t

H h tH

2 and 0,1,..., .m N

(6.20)

Eq. (6.20) contains 2 2N equations in 2 10N unknowns. To obtain a unique solution

to this system, we eliminate the parameters 2 1 1 2 2 1 1 2, , , , , , ,N N N N from

it. The values of 2 1 1 2, , ,N N are already given in Eq. (6.10) whereas the

remaining ones are calculated and listed as follows:

2 0 1 2

1 0 1 2

15 35 ,2 23 3 1 ,4 2 4

(6.21)

1 2 1

2 2 1

1 3 3 ,4 2 4

3 1 55 .2 2

NN N N

NN N N

Elimination of above parameters from Eq. (6.20) gives a linear system of (2N+2)


129

equations in (2N+2) unknown parameters , , 0,1,..., ,i i i N which can be solved

successively to find , , 1, 2,...,n ni i n provided we find the initial parameters 0 0,m m

first. Consequently the approximate solution ,U x t will be obtained from Eq. (6.17).

We find initial parameters under the following constraints:

0 0 0 0

0 0 0 0

,0 ,0 , ,0 ,0 ,

,0 , ,0 ,

,0 , ,0

,0 , ,0 ,

,0 , ,0 ,

0,1,..., .

m m m m m m

N N N N

N N N N

U x u x f x V x u x f x

U x f x V x f x

U x f x V x f x

U x f x V x f x

U x f x V x f x

m N

(6.22)

6.2.3 Quartic B-Spline Collocation Method

Using Eq. (4.4) the approximate solutions are given by

1 1

2 2

, , , ,N N

m m m mm m

U x t t B x V x t t B x

(6.23)

where ,m mt t are time dependent parameters to be determined and mB x are

fourth-degree B-spline functions. Once again splitting method is used to obtain the

results given in the Eqs. (6.12)-(6.15) and (6.18). Eqs. (4.4) and (6.23) provide the

following values of ,U V and their space derivatives at the collocation points:

2 1 1

2 1 1

2 1 12

2 1 1

2 1 1

2 1 12

11 11 ,

43 3 ,

12,

11 11 ,

43 3 ,

12.

m m m m m

m m m m m

m m m m m

m m m m m

m m m m m

m m m m m

U

Uh

Uh

V

Vh

Vh

(6.24)

For the n-th time level the variables ,m mU V and their space derivatives at the knots mx

are denoted by:


130

1 2 1 1 2 2 1 1

3 2 1 12

1 2 1 1

2 2 1 1

3 2 1 12

411 11 , 3 3 ,

12,

11 11 ,

43 3 ,

12.

n n n n n n n nm m m m m m m m m m

n n n nm m m m m

n n n nm m m m m

n n n nm m m m m

n n n nm m m m m

L Lh

Lh

M

Mh

Mh

(6.25)

Substitution of Eqs. (6.24)-(6.25) into Eq. (6.18) gives rise to the following system of

coupled equations:

1 1 1 11 2 2 1 3 4 1 2

1 1 31 1 1 11 2 2 1 2 1 1

1 1 1 12 1 1 2

1 32 1 1 1 12 1 1

2

12 12 12 12

11 11

n n n nm m m m m m m m

m m mn n n nm m m m m m m m

n n n nm m m m

m mn n n nm m m m

d d d dh L t M M

e e e e

h M Lh

(6.26)

where

1 0 1 2 0 1 3 0 1

2 24 0 1 1 2

20 2

4 , 11 12 , 11 12 ,

4 , 12 , 11 12 ,

with 2 and 0,1,..., .

m m m m m m m m m

m m m m m

m m

d L hL t d L hL t d L hL t

d L hL t e th t e th t

L h tL m N

Eq. (6.26) consists of 2 2N equations in 2 8N unknowns. Imposition of boundary

conditions (6.14) gives the following relations:

12 0 1

11 0 1

1 2 1 2

2 0 1

1 0 1

1 2 1

333 7,

4 4 87 1

,4 4 8

11 11 ,

3 1,

2 21 1

,2 2

11 11 .

N N N N

N N N N

g

g

g

(6.27)

Elimination of above parameters from Eq. (6.26) with the help of Eq. (6.27) yields

2 2 2 2N N linear system of unknown parameters , , 0,1,..., ,i i i N which

can be solved successively to find , , 1, 2,...,.n ni i n In order to initiate the solution, we


131

first find the initial parameters 0 0,m m which are obtained with the help of Eq. (6.22).

Consequently the approximate solution ,U x t can be obtained from Eq. (6.24).

Following the procedure adopted in the previous chapters the stability of the

methods can be established.

6.3 Numerical validation

To demonstrate the effectiveness of the numerical methods we consider some test

problems. The accuracy of the schemes is measured in terms of the following discrete

error norms L and 2L .

Problem 6.1 For the sake of comparison with an earlier work (Xu and Shu [160]), we

Consider the following KS equation

0t x xx xxxxu uu u u (6.28)


3

0 0

3

0 0

30,

30,

15 119 tanh 30 11tanh 30 ,

19 19

15 119 tanh 30 11tanh 30 ,

19 19

u t

u t

c k ct x k ct x

c k ct x k ct x

(6.29)

and the initial condition

30 0

15 11,0 9 tanh 11tanh .

19 19u x c k x x k x x

(6.30)

The theoretical solution for this problem is given by

30 0

15 11, 9 tanh 11tanh .

19 19u x t c k x ct x k x ct x

(6.31)

We take 00.1, 1/ 2 11/19, 10.c k x Table 6.1 contains the 2L and L error

norms at time 1.t In Table 6.2 we re-produce the results reported in local

discontinuous Galerkin (LDG) method (Xu and Shu [160]) for the sake of comparison

with pk elements, (k=0,1,2). It is clear from the comparison of the Tables 6.1 and 6.2

that our methods 6.2.2 and 6.2.3 give less maximum absolute errors than (Xu and Shu

[160]) , however the L2 errors are comparable. Method 6.2.1 can be compared for

0,1k . Fig. 6.1 shows the regular shock profile for the above problem. From the

figure and computed results, it is observed that the moving shock is resolved very well.


132

Problem 6.2 Consider the following KS equation;


with the exact solution

30 0

15, 3 tanh tanh

19 19u x t c k x ct x k x ct x (6.33)

and the boundary conditions;

1 250, , 50, , 0,u t f t u t f t t (6.34)

where if t are obtained from Eq. (6.35). The initial condition is given by

30 0

15 11,0 9 tanh 11tanh .

19 19u x c k x x k x x

(6.35)

We take 00.2, 1/ 2 19, 10c k x and compare our results with (Xu and Shu [160]).

Table 6.3 contains the 2L and L error norms at time 1.t Table 6.4 contains the results

for LDG method (Xu and Shu [160]) with pk elements, (k=0,1,2) . Comparison of

tables 6.3 and 6.4 provides similar conclusion as discussed in problem 6.1. Fig. 6.2

shows the regular shock profile for the above problem. From the figure and numerical

computation it is again observed that the moving shock is resolved very well.

Problem 6.3 We finally consider the KS equation;


with the Gaussian initial condition

2,0 exp ,u x x (6.37)

and the boundary conditions;

30, 30, 0.u t u t (6.38)

Here the simulations are done upto 30.t Fig. 6.3 shows the solutions at 0,5,20,30t

and illustrates the chaotic solution for the above problem. From the figure it is observed

that the numerical results are convergent for the chaotic nature.


133

Table 6.1 Accuracy test for problem 6.1 Methods N L∞ error L2 error

Method 6.2.1 40 1.56×10-1 3.58×10-1

80 7.03×10-2 1.30×10-1

160 1.86×10-2 3.37×10-2

320 4.69×10-3 8.48×10-3

Method 6.2.2 40 1.44×10-1 3.01×10-1

80 3.12×10-3 5.59×10-3

160 1.44×10-4 2.69×10-4

320 1.95×10-5 3.32×10-5

Method 6.2.3 80 7.94×10-3 1.38×10-2

160 4.37×10-4 8.20×10-4

320 4.29×10-5 6.81×10-5

640 2.44×10-5 3.72×10-5

00.1, 11/19, 10 at 11/ 2c k x t

Table 6.2 Accuracy test for problem 6.1 (Xu and Shu [160]) Methods N L∞ error L2 error

P0 40 1.37 2.38×10-1

80 8.81×10-1 1.58×10-1

160 5.21×10-1 9.41×10-2

320 2.91×10-1 5.24×10-2

P1 40 6.64×10-1 6.08×10-2

80 1.82×10-1 1.49×10-2

160 4.64×10-2 3.78×10-3

320 1.19×10-2 9.57×10-4

P2 40 1.49×10-1 9.15×10-3

80 1.73×10-2 1.06×10-3

160 2.43×10-3 1.32×10-4

320 3.05×10-4 1.65×10-5

00.1, 11/19, 10 at 11/ 2c k x t


134

Table 6.3 Accuracy test for problem 6.2

Methods N L∞ error L2 error

Method 6.2.1 40 1.42×10-4 3.44×10-4

80 3.06×10-5 7.41×10-5

160 7.36×10-6 1.78×10-5

320 1.81×10-6 4.39×10-5

Method 6.2.2 40 3.22×10-5 7.46×10-5

80 1.45×10-6 3.41×10-6

160 6.91×10-8 1.76×10-7

320 7.00×10-8 1.87×10-7

Method 6.2.3 40 6.72×10-5 1.60×10-4

80 4.06×10-6 9.77×10-6

160 3.18×10-7 7.19×10-7

320 8.95×10-8 2.16×10-7

00.2, , 10 at 11/ 2 19c k x t

Table 6.4 Accuracy test for problem 6.2 (Xu and Shu [160]) Methods N L∞ error L2 error

P0 40 7.56×10-2 1.33×10-2

80 3.87×10-2 6.69×10-3

160 1.95×10-2 3.35×10-3

320 9.80×10-3 1.68×10-3

P1 40 1.04×10-2 1.29×10-3

80 2.59×10-3 3.26×10-4

160 6.54×10-4 8.17×10-5

320 1.64×10-4 2.05×10-5

P2 40 7.93×10-4 6.48×10-5

80 1.06×10-4 8.04×10-6

160 1.34×10-5 1.00×10-6

320 1.67×10-6 1.25×10-7

01/ 2 190.2, , 10 at 1c k x t


135

-30 -20 -10 0 10 20 303

4

5

6

7

x

u

t=0

-30 -20 -10 0 10 20 303

4

5

6

7

x

u

t=2

-30 -20 -10 0 10 20 303

4

5

6

7

x

u

t=5

-30 -20 -10 0 10 20 303

4

5

6

7

x

u

t=6

Fig 6.1: The shock profile wave propagation of the KS Eq.(6.28) with initial condition

(6.30), 05, 1/ 2 11/19 12, 160 in [ 30,30],c k x N

-50 -40 -30 -20 -10 0 10 20 30 40 504.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

x

U

t=0

-50 -40 -30 -20 -10 0 10 20 30 40 504.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

x

U

t=12

-50 -40 -30 -20 -10 0 10 20 30 40 504.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

x

U

t=5

-50 -40 -30 -20 -10 0 10 20 30 40 504.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

x

U

t=10

Fig 6.2: The shock profile wave propagation of the KS Eq. (6.32) with initial condition

(6.35), 05, , 25, 200 in 1/ 2 19 [ 50,50]c k x N


136

-30 -20 -10 0 10 20 30-0.5

0

0.5

1

1.5

-30 -20 -10 0 10 20 30-0.5

0

0.5

1

1.5

-30 -20 -10 0 10 20 30-4

-2

0

2

4

-30 -20 -10 0 10 20 30-4

-2

0

2

4

t=0 t=5

t=20 t=30

-30 -20 -10 0 10 20 30

0

5

10

15

20

25

30

-4

-2

0

2

4

t

x

U

Fig 6.3: The chaotic solution of the KS Eq. (6.36) with Gaussian initial condition (6.37) and boundary conditions

(6.38), 320N in [ 30, 30]


137

6.4 Conclusion

Quartic and quintic B-spline collocation methods are used to develop a class of

numerical methods for solving KS equation. Using splitting technique, the equation is

reduced to a problem of second order in space. These methods are computationally

efficient and the algorithm can easily be implemented on a computer. Theses methods

are compared with local discontinuous Galarkin method given in (Xu and Shu [160])

and seem to be effective and accurate.

***************************************************

Future Work

138

Future work

The research presented in the thesis is focused on the use of non-polynomial splins and

B-spline functions to obtain numerical solution of BVPs in ODEs and collocation

method for initial-BVPs in PDEs. This investigation has spawned a number of open

research problems. We will figure out some of them, and further investigation in

specified directions will certainly lead to the improvement and generalization of the

exiting algorithms designed for the numerical solution of initial and BVPs in ordinary

as well as partial differential equations.

(i). Non-polynomial spline functions based algorithms are relatively new methods

and are providing alternative ways of tackling the unilateral problems coupled

with the theory of variational inequalities. This approach can be extended to

nonlinear obstacle problem. Detail analysis of the methods both analytically and

numerically, will constitute an immediate and interesting subject of future

study.

(ii). The use of non-polynomial spline functions for the solution of BVPs can be

extended to eight and higher-order general as well as special linear and

nonlinear BVPs. Such types of problems have variety of applications in science

and engineering.

(iii). Non-polynomial spline functions can be used for the solution of singular BVPs

of third and fourth-orders.

(iv). The work done so far on the application of polynomial spline functions in

developing algorithms for the initial-value problems can be replaced by their

counter parts non-polynomial spline functions to improve the accuracy and

generalize the algorithms.

(v). The Literature on the use of non-polynomial splines for the numerical solution

PDEs is very limited. Hence this is another area open for future investigations.

(vi). High degree B-spline collocation methods can be used for the numerical

solution of evolution problems.

(vii). Numerical methods for the solution of generalized regularized long wave and

equal width wave equations can be investigated using quartic and quintic B-

splines.

Future Work

139

(viii). Numerical solution of the modified equations studied in this thesis can be

extended through the use of higher degree B-splines.

(ix). Numerical algorithms for solution of two dimensional PDEs can be developed

using collocation method based on B-splines.

(x). B-splines of various degrees can be used to find numerical solution of

Sivashinsky and Kawahara equation.

***************************************************

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140

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