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ISSN 1749-3889 (print), 1749-3897 (online)

International Journal of Nonlinear Science

Vol.10(2010) No.2,pp.222-230

Numerical Solutions of Fourth Order Boundary Value Problems by Galerkin

Method with Quintic B-splines

K.N.S. Kasi Viswanadham1 ∗, P. Murali Krishna1, Rao S. Koneru2

1 Department of Mathematics, National Institute of Technology Warangal -506 004 INDIA.2 Retired Professor of Mathematics, Indian Institute of Technology Bombay, Mumbai - 400 076 INDIA.

(Received 11 June 2010, accepted 5 July 2010)

Abstract: In this paper, Galerkin method with quintic B-splines as basis functions is presented to solve a

fourth order boundary value problem with two different cases of boundary conditions. In the method, the ba-

sis functions are redefined into a new set of basis functions which vanish at the boundary where the Dirichlet

type of boundary conditions are prescribed. The proposed method is tested on several numerical examples

of fourth order linear and nonlinear boundary value problems. Numerical results obtained by the proposed

method are in good agreement with the exact solutions available in the literature.

Keywords: Galerkin method; quintic B-splines; basis functions; fourth order boundary value problems; band

matrix; absolute error

1 Introduction

Generally, fourth order boundary value problems arises in the mathematical modeling of viscoelastic and inelastic flows,

deformation of beams and plates deflection theory, beam element theory and many more applications of engineering and

applied mathematics. Solving such type of boundary value problems analytically is possible only in very rare cases. Many

researchers worked for the numerical solutions of fourth order boundary value problems [1–5]. In this paper, we try to

present a simple finite element method which involves Galerkin approach with quintic B-splines as basis functions to solvethe fourth order boundary value problems with two different cases of boundary conditions. In this paper, we consider a

general fourth order linear boundary value problem given by

0()(4) + 1()′′′ + 2()′′ + 3()′ + 4() = (), < < (1)

subject to boundary conditions

() = 0, () = 0, ′() = 1, ′() = 1, (2a)

or

() = 0, () = 0, ′′() = 2, ′′() = 2, (2b)

where 0, 1, 2, 0, 1 and 2 are finite real constants and 0(), 1(), 2(), 3(), 4() and () are all

continuous functions defined on the interval [, ]. The boundary value problem (1) is solved with either cases of the

boundary conditions (2a) or (2b).In section 2 of this paper, the justification for using the Galerkin method has been mentioned. In section 3, the

definition of quintic B-splines has been described. In section 4, the description of the Galerkin method with quintic B-

splines as basis functions has been presented. In particular, the proposed method with the boundary conditions (2a) is

presented in section 4.1, where as the proposed method with the boundary conditions (2b) is presented in section 4.2. In

section 5, solution procedure to find the nodal parameters has been presented. In section 6, the proposed method is tested

on four linear and three nonlinear boundary value problems. The solution of non-linear problem has been obtained as

the limit of sequence of linear problems generated by the quasilinearization technique [6]. Finally in the last section the

conclusions of the paper are presented.

∗Corresponding author. E-mail address:kasi [email protected]

Copyright c⃝World Academic Press, World Academic Union

IJNS.2010.10.15/410



K.N.S. Kasi Viswanadham, P. Murali Krishna, Rao S. Koneru: Numerical Solutions of Fourth Order Boundary ⋅ ⋅ ⋅ 223

2 Justification for using Galerkin method

In finite element method(FEM) the approximate solution can be written as a linear combination of basis functions which

constitute a basis for the approximation space under consideration. FEM involves methods like Rayleigh Ritz, Galerkin,

Least Squares and Collocation etc.,.

In a Galerkin method, a weak form of approximate solution for a given differential equation is existing and is unique

under appropriate conditions [7, 8] irrespective of properties of a given differential operator and weak solution is also

a classical solution of given differential equation provided sufficient attention is given to boundary conditions[9]. That

means the basis functions should vanish on the boundary where the Dirchlet type of boundary conditions are prescribed.

It is given that when a differential equation is approximated by quintic B-splines, the method yields sixth order accurate

results [10]. Therefore, in this paper we intend to use the Galerkin method with quintic B-splines as basis functions to

approximate the solution of a given problem.

3 Definition of Quintic B-splines

The cubic B-splines are defined in [11, 12]. The quintic B-splines with evenly spaced knots are defined in [10]. The

existence of quintic spline interpolate () to a function in a closed interval [, ] for spaced knots (need not be evenly

spaced) = 0 < 1 < . . . < −1 < = is established by constructing it. The construction of () is done withthe help of the quintic B-splines. Introduce ten additional knots −5, −4, −3, −2, −1, +1, +2, +3, +4 and

+5 such that

−5 < −4 < −3 < −2 < −1 < 0

and

< +1 < +2 < +3 < +4 < +5.

Now the quintic B-splines () are defined by

() =

⎧⎨

⎩

+3∑=−3

( − )5+′()

, for ∈ [−3, +3]

0, otherwise

where

( − )5+ =

{( − )5, for ≥

0, for ≤

and

() =

+3∏=−3

(− )

Here the set {−2(), −1(), 0(), . . . , (), +1(), +2()} forms a basis for the space 5() of fifth degree

polynomial splines. The quintic B-splines are the unique nonzero splines of smallest compact support with the knots at

−5 < −4 < −3 < −2 < −1 < 0 < .. . < < +1 < +2 < +3 < +4 < +5.

4 Description of the method

To solve the boundary value problem (1) by the Galerkin method with quintic B-splines as basis functions, we approximate

() as

() =+2∑=−2

() (3)

where ’s are the nodal parameters to be determined.

In Galerkin method the basis functions should vanish on the boundary where the Dirichlet type of boundary conditions

are specified. In the set of quintic B-splines {−2(), −1(), 0(), 1(),. . ., −1(), (), +1(), +2()},

the basis functions −2(), −1(), 0(), 1(), 2(), −2(), −1(), (), +1() and +2() are not

vanishing at one of the boundary points. So, there is a necessity of redefining the basis functions into a new set of basis

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224 International Journal of Nonlinear Science, Vol.10(2010), No.2, pp. 222-230

functions which vanish on the boundary where the Dirichlet type of boundary conditions are specified. For this, we

proceed in the following manner.

Using the quintic B-splines described in section 3 and the Dirichlet type of boundary conditions prescribed in (2), we

get the approximate solution at the boundary points as

0 = (0) = −2−2(0) + −1−1(0) + 00(0) + 11(0) + 22(0) (4)

0 = () = −2−2() + −1−1() + () + +1+1() + +2+2() (5)

Eliminating −2 and +2 from the equations (3), (4) and (5), we get

() = () ++1∑=−1

() (6)

where

() =0

−2(0)−2() +

0

+2()+2() (7)

and

() =

⎧⎨⎩

()− (0)−2(0) −2(), = −1, 0, 1, 2

(), = 3, 4, . . . , − 3

()−()

+2()+2(), = − 2, − 1, , + 1

(8)

Applying Galerkin method for (1) with the basis functions (), = −1, 0, 1, . . . , , + 1, we get

0

0()(4) + 1()′′′ + 2()′′ + 3()′ + 4()

() =

0

()()

for = −1, 0, 1, . . . , , + 1. (9)

4.1 Method with boundary conditions (2a)Integrating by parts the terms upto second derivative on the left hand side of (9), we get each term after applying the

Neumann conditions prescribed in (2a) as

0

0()(4)() = −

0()()

′′0

+

2

2

0()()

1 −

2

2

0()()

0

1

−

0

3

3

0()()

′() (10)

0

1()′′′() =

1()()

0

1

−

1()()

1 +

0

2

2

1()()

′() (11)

0

2()′′() = −

0

2()()

′() (12)

Substituting (10), (11) and (12) in (9) and using the approximation for () given in (6), and after rearranging the

terms for the resulting equations we get a system of equations in the matrix form as

A = b (13)

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where

A = [ ];

=∫ 0

− 3

3

0()()

+ 2

2

1()()

−

2()()

+ 3()()

′

() + 4()()()}

+

0()()

0′′

(0)−

0()()

′′

() = −1, 0, 1, . . . , , + 1; = −1, 0, 1, . . . , , + 1

(14)

b = [];

=∫ 0

()() +

3

3

0()()

− 2

2

1()()

+

2()()

− 3()()

′() − 4()()()

}

+1

1()()

− 2

2

0()()

−1

1()()

− 2

2

0()()

0

+

0()()

′′()−

0()()

0′′(0)

= −1, 0, 1, . . . , , + 1;

(15)

and = [−1 0 1 . . . +1] .

4.2 Method with boundary conditions (2b)

Integrating by parts the terms upto second derivative on the left hand side of (9), we get each term after applying the

boundary conditions for second derivative prescribed in (2b) as

0

0()(4)() = −

0()()

2 +

0()()

0

2

+

2

2

0()()

′()

0

−

0

3

3

0()()

′() (16)

0

1()′′′() = −

1()()

′()

0

+

0

2

2

1()()

′() (17)

0

2()′′() = −

0

2()()

′() (18)

Substituting (16), (17) and (18) in (9) and using the approximation for () given in (6), and after rearranging the

terms for the resulting equations we get a system of equations in the matrix form as

A = b (19)

where

A = [ ];

=∫ 0

− 3

3

0()()

+ 2

2

1()()

−

2()()

+ 3()()

′

() + 4()()()}

+′

()

2

2

0()()

−

1()()

−′

(0)

2

2

0()()

0−

1()()

0

= −1, 0, 1, . . . , , + 1; = −1, 0, 1, . . . , , + 1.

(20)





b = [];

=∫ 0

()() +

3

3

0()()

− 2

2

1()()

+

2()()

− 3()()

′() − 4()()()

}

−′() 2

2 0()() − 1()()

+′(0)

2

2

0()()

0−

1()()

0

+2

0()()

−2

0()()

0

= −1, 0, 1, . . . , , + 1;

(21)

and = [−1 0 1 . . . +1] .

5 Solution procedure to find the nodal parameters

A typical integral element in the matrix A is−1

∑=0

where =∫ +1

()() () and (), () are the basis functions or their derivatives. It may be noted that

= 0 if (−3, +3) ∩ (−3, +3) ∩ (, +1) = . To evaluate each , we employed 6-point Gauss-Legendre

quadrature formula. Thus the stiff matrix A is an eleven diagonal band matrix. The nodal parameter vector has been

obtained from the system A = b using a band matrix solution package.

6 Numerical Results

To test the applicability of the proposed method, we considered three linear boundary value problems and three nonlinear

problems with boundary conditions of the type (2a) and one linear boundary value problem with boundary conditions

of the type (2b), because the exact solutions for these problems are available in the literature. For all the examples, the

solutions obtained by the proposed method are compared with the exact solutions.

Example 1 Consider the linear boundary value problem

(4) + 4 = 1, −1 < < 1 (22)

subject to (−1) = (1) = 0 , ′(−1) = −′(1) = −sinh 2− sin2

4(cosh 2 + cos 2).

The exact solution is

= 0.25

1− 2

sinh1sin1sinh sin + cosh1 cos 1 cosh cos

(cosh 2 + cos 2)

.

The proposed method mentioned in section 4.1 is tested on this problem where the domain [−1, 1] is divided into 10equal subintervals. Numerical results for this problem are given in Table 1. The maximum absolute error obtained by the

proposed method is 1.303× 10−6.


(4) + = −(8 + 7 + 3), 0 < < 1 (23)

subject to (0) = (1) = 1 , ′(0) = 1 , ′(1) = −.

The exact solution is = (1 − ). The proposed method mentioned in section 4.1 is tested on this problem where

the domain [0, 1] is divided into 10 equal subintervals. Numerical results for this problem are given in Table 2. The

maximum absolute error obtained by the proposed method is 5.990× 10−6.





Table 1: Numerical results for Example 1

Exact Solution Absolute error by proposed method

-0.8 3.976926E-02 1.788139E-07

-0.6 7.498498E-02 5.736947E-07

-0.4 1.023106E-01 1.288950E-06

-0.2 1.195382E-01 9.983778E-070.0 1.254157E-01 8.493662E-07

0.2 1.195382E-01 1.303852E-06

0.4 1.023106E-01 1.139939E-06

0.6 7.498498E-02 9.387732E-07

0.8 3.976926E-02 6.556511E-07



0.1 9.946539E-02 6.631017E-07

0.2 1.954244E-01 1.519918E-06

0.3 2.834704E-01 2.443790E-06

0.4 3.580379E-01 4.380941E-06

0.5 4.121803E-01 5.662441E-06

0.6 4.373085E-01 5.841255E-06

0.7 4.228881E-01 5.990267E-06

0.8 3.560865E-01 5.871058E-06

0.9 2.213642E-01 4.246831E-06


(4) − = −4(2 cos + 3 sin ), −1 < < 1 (24)

subject to (−1) = (1) = 0 , ′(−1) = ′(1) = 2 sin 1.

The exact solution is = (2 − 1)sin . The proposed method mentioned in section 4.1 is tested on this problem

where the domain [−1, 1] is divided into 10 equal subintervals. Numerical results for this problem are given in Table 3.

The maximum absolute error obtained by the proposed method is 5.304× 10−6.



-0.8 2.582482E-01 6.556511E-07

-0.6 3.613712E-01 7.152557E-07

-0.4 3.271114E-01 1.430511E-06

-0.2 1.907225E-01 1.713634E-06

0.0 -1.490116E-08 1.128331E-06

0.2 -1.907226E-01 3.129244E-06

0.4 -3.271114E-01 4.023314E-060.6 -3.613712E-01 4.798174E-06

0.8 -2.582482E-01 5.304813E-06

Example 4 Consider the nonlinear boundary value problem

(4) = 2 − 10 + 49 − 48 − 47 + 86 − 44 + 120− 48, 0 < < 1 (25)

subject to (0) = ′(0) = 0 , (1) = ′(1) = 1.

The exact solution is = 5 − 24 + 22. The nonlinear boundary value problem (25) is converted into a sequence

of linear boundary value problems generated by quasilinearization technique [6] as

(4)(+1) − [2()] (+1) = −10 + 49 − 48 − 47 + 86 − 44 + 120− 48− [()]

2, = 0, 1, 2, . . . (26)





subject to (+1)(0) = ′(+1)(0) = 0 , (+1)(1) = ′(+1)(1) = 1.

Here (+1) is the ( + 1)ℎ approximation for . The domain [0, 1] is divided into 10 equal subintervals and the

proposed method mentioned in section 4.1 is applied to the sequence of linear problems (26). Numerical results for this

problem are presented in Table 4. The maximum absolute error obtained by the proposed method is 7.092× 10−6.



0.1 1.981000E-02 4.302710E-07

0.2 7.712000E-02 1.810491E-06

0.3 1.662300E-01 3.576279E-06

0.4 2.790400E-01 4.202127E-06

0.5 4.062500E-01 4.947186E-06

0.6 5.385600E-01 7.092953E-06

0.7 6.678700E-01 6.556511E-06

0.8 7.884800E-01 3.874302E-06

0.9 8.982900E-01 2.861023E-06


(4) = sin + sin2 − [′′]2, 0 < < 1 (27)

subject to (0) = 0 , ′(0) = 1 , (1) = sin 1 , ′(1) = cos 1.

The exact solution is = sin . The nonlinear boundary value problem (27) is converted into a sequence of linear

boundary value problems generated by quasilinearization technique [6] as

(4)(+1) +

2′′()

′′(+1) = sin + sin2 +

′′()

2, = 0, 1, 2, . . . (28)

subject to (+1)(0) = 0 , ′(+1)(0) = 1 , (+1)(1) = sin 1 , ′(+1)(1) = cos 1.






0.1 9.983342E-02 9.685755E-07

0.2 1.986693E-01 4.082918E-06

0.3 2.955202E-01 7.957220E-06

0.4 3.894183E-01 1.019239E-05

0.5 4.794255E-01 1.180172E-05

0.6 5.646425E-01 1.358986E-05

0.7 6.442177E-01 1.221895E-050.8 7.173561E-01 7.748604E-06

0.9 7.833269E-01 4.529953E-06


(4) − 6−4 = −12(1 + )−4, 0 < < 1 (29)

subject to (0) = 0 , ′(0) = 1 , (1) = ln2 , ′(1) = 0.5.

The exact solution is = ln(1+ ). The nonlinear boundary value problem (29) is converted into a sequence of linear

boundary value problems generated by quasilinearization technique [6] as

(4)

(+1)

+ [24−4()] (+1) = −12(1 + )−4 + −4() [24() + 6], = 0, 1, 2, . . . (30)





subject to (+1)(0) = 0 , ′(+1)(0) = 1 , (+1)(1) = ln 2 , ′(+1)(1) = 0.5.






0.1 9.531018E-02 2.235174E-08

0.2 1.823216E-01 7.599592E-07

0.3 2.623643E-01 2.026558E-06

0.4 3.364722E-01 2.413988E-06

0.5 4.054651E-01 3.129244E-06

0.6 4.700036E-01 4.917383E-06

0.7 5.306283E-01 4.887581E-06

0.8 5.877867E-01 3.099442E-06

0.9 6.418539E-01 2.324581E-06


(4) − = −4(2 cos + 3 sin ), 0 < < 1 (31)

subject to (0) = (1) = 0 , ′′(0) = 0 , ′′(1) = 2 sin 1 + 4 cos 1.

The exact solution is = (2 − 1)sin . The proposed method mentioned in section 4.2 is tested on this problem

where the domain [0, 1] is divided into 10 equal subintervals. Numerical results for this problem are givenn in Table 7.

The maximum absolute error obtained by the proposed method is 5.275× 10−6.



0.1 -9.883508E-02 2.168119E-06

0.2 -1.907226E-01 3.889203E-06

0.3 -2.689234E-01 5.215406E-06

0.4 -3.271114E-01 5.215406E-06

0.5 -3.595692E-01 5.215406E-06

0.6 -3.613712E-01 5.275011E-06

0.7 -3.285510E-01 4.589558E-06

0.8 -2.582482E-01 2.980232E-06

0.9 -1.488321E-01 1.385808E-06

7 Conclusions

In this paper, we have developed a Galerkin method with quintic B-splines as basis functions to solve a fourth or-

der boundary value problems with two different cases of boundary conditions. The quintic B-spline basis set has been

redefined into a new set of basis functions which vanish on the boundary where the Dirichlet boundary conditions are pre-

scribed. The proposed method is applied to solve a several number of linear and nonlinear problems to test the efficiency

of the method. The numerical results obtained by the proposed method are in good agreement with the exact solutions

available in the literature. The objective of this paper is to present a simple and accurate method to solve a fourth order

boundary value problem.





References

[1] O. A. Taiwo and O. M. Ogunlaran. Numerical solution of fourth order linear ordinary differential equations by cubic

spline collocation tau method. J. Math. Stat., 4(2008):264–268.

[2] S. S. Siddiqi and G. Akram. Quintic spline solutions of fourth order boundary value problems. Int. J. Numer. Anal.

Model., 5(2008):101–111.

[3] M. A. Noor and S. T. Mohyud-din. An efficient method for fourth order boundary value problems. Comput. Math.

Appl., 54(2007):1101–1111.

[4] O. O. Onyejekwe. A green element method for fourth order ordinary differential equations. Advances in Engineering

Software, 35(2004):517–525.

[5] M. E. Gamel and A. I. Zayed. Sinc -Galerkin method for solving nonlinear boundary value problems. Comput.

Math. Appl., 48(2004):1285–1298.

[6] R. E. Bellman and R. E. Kalaba. Quasilinearization and Nonlinear Boundary value problems. American Elsevier,

New York. 1965.

[7] L. Bers, F. John and M. Schecheter. Partial Differential Equations. John Wiley Inter Science, New York. 1964.

[8] J. L. Lions and E. Magenes. Non-Homogeneous Boundary Value Problems and Applications. Springer - Verlag,

Berlin. 1972.

[9] A. R. Mitchel and R. Wait. The Finite Element Method In Partial Differential Equations. John Wiley and Sons,

London. 1977.

[10] P. M. Prenter. Splines and Variational Methods. John-Wiley and Sons, New York. 1989.

[11] C. de Boor. A Practical Guide to Splines. Springer-Verlag. 1978.

[12] I. J. Schoenberg. On Spline Functions. MRC Report 625, University of Wisconsin. 1966.


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