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Applied Mathematics and Computation 290 (2016) 111–124

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Applied Mathematics and Computation

journal homepage: www.elsevier.com/locate/amc

An algorithm based on exponential modified cubic B-spline

differential quadrature method for nonlinear Burgers’

equation

Mohammad Tamsir a , Vineet K. Srivastava

b , c , Ram Jiwari d , ∗

a Department of Mathematics & Statistics, DDU Gorakhpur University, Gorakhpur 273009, India b Flight Dynamics Operations Division, ISRO Telemetry, Tracking and Command Network, Bangalore 560058, India c Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, India d Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

a r t i c l e i n f o

Keywords:

Expo-MCB-DQM

Burgers’ equation

SSP-RK54 method

Stability analysis

a b s t r a c t

In this paper, the authors developed a new differential quadrature method "exponential

modified cubic B-spline differential quadrature method (Expo-MCB-DQM)” by using expo-

nential modified cubic B-spline functions as test functions in the traditional differential

quadrature method [32]. The new method is tested on one and two dimensional nonlin-

ear Burgers’ equations. To check the efficiency and accuracy of the proposed method five

numerical problems have been considered. The numerical results of the method are com-

pared with some existing methods and found that the proposed numerical method pro-

duces more accurate results than existing methods. Stability analysis of the algorithm is

also done by using matrix stability analysis method.

© 2016 Elsevier Inc. All rights reserved.

1. Introduction

The structure of the Burgers’ equation is like a Navier–Stoke’s equation without the stress term. It is an easiest model for

understanding the various physical flows models, such as sound and shock wave theory, wave processes in thermo-elastic

medium, vorticity transportation, dispersion in porous media, mathematical modeling of turbulent fluid, hydrodynamic tur-

bulence etc. [2–6] .

In this work, the authors consider Burgers’ equations of the form

(a) One dimensional Burgers’ equation:

∂u

∂t + αu

∂u

∂x − υ

∂ 2 u

∂ x 2 = 0 ; a ≤ x ≤ b, t ∈ [0 , T ] , (1.1)

with initial conditions

u ( x, 0 ) = ψ ( x ) ; a ≤ x ≤ b (1.2)

and Dirichlet boundary conditions

u ( a, t ) = 0 = u ( b, t ) ; t ∈ [0 , T ] , (1.3)

∗ Corresponding author. Tel.: + 91 8191957249.

E-mail address: [email protected] (R. Jiwari).

http://dx.doi.org/10.1016/j.amc.2016.05.048

0 096-30 03/© 2016 Elsevier Inc. All rights reserved.


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112 M. Tamsir et al. / Applied Mathematics and Computation 290 (2016) 111–124

(b) Two dimensional coupled Burger’s equations

∂u

∂t + u

∂u

∂x + v

∂u

∂y = υ

(∂ 2 u

∂ x 2 +

∂ 2 u

∂ y 2

), (1.4)

∂v ∂t

+ u

∂v ∂x

+ v ∂v ∂y

= υ

(∂ 2 v ∂ x 2

+

∂ 2 v ∂ y 2

), (1.5)

with initial conditions

u ( x, y, 0 ) = ψ 1 ( x, y ) and v ( x, y, 0 ) = ψ 2 ( x, y ) , ( x, y ) ∈ � (1.6)

Dirichlet boundary conditions:

u ( x, y, t ) = ξ ( x, y, t ) and v ( x, y, t ) = ζ ( x, y, t ) , ( x, y ) ∈ ∂�, t > 0 (1.7)

where � = { ( x, y ) : a ≤ x ≤ b, c ≤ y ≤ d } is the computational square domain and ∂� is its boundary, u ( x, t ) is the

velocity component in one dimension, u ( x, y, t ) and v ( x, y, t ) are the velocity components in two dimension; ψ , ψ 1 ,

ψ 2 , ξ and ζ are known functions; ∂u ∂t

is unsteady term, u ∂u ∂x

is the nonlinear convection term, υ( ∂ 2 u

∂ x 2 +

∂ 2 u ∂ y 2

) is the

diffusion term and υ > 0 is viscosity coefficient and α is a positive constant. Bateman [1] was introduced this type of

equations and later a steady-state solution was proposed by Burger [2, 3] .

In the last decades, an enormous work has been done on Burgers’ equation. The Burgers’ equation is solved by various

analytical and numerical schemes such as Hopf–Cole transformation [4,8] , finite difference methods [7,10,12,14,16,17,41,42] ,

finite element method [7,13,15,19] , quadratic B-spline finite elements [14] , least-squares quadratic B-spline finite element

method [11] , B-spline collocation method [18] , Quartic B-spline collocation method [39] , reproducing kernel function

method [38] , sinc differential quadrature method [22] , polynomial differential quadrature method [20,21,44–47] , Quartic

B-spline differential quadrature method [22] , modified cubic B-splines collocation method [9] , Cubic B-spline differential

quadrature methods [21,23] , modified cubic B-spline differential quadrature method [31,43] , Galerkin quadratic B-spline

finite element method [48] , Galerkin method based on modified bi-quintic B-spline functions [49] and more recently Haar

wavelets [28–30] etc.

In 1972, Bellman et al. [32] was first introduced differential quadrature method (DQM) to solve differential equations.

After that the method was improved by Quan and Chang [33,34] . In DQM, several kinds of test functions have been used to

compute the weighting coefficients such as B-spline functions , cubic B-spline functions, sinc function, Lagrange interpolation

polynomials, Legendre polynomials, quartic B-spline functions, modified cubic B-spline functions [20–22,31,43] etc.

In this paper, the authors proposed a new method “exponential modified cubic-B-spline differential quadrature method

(Expo-MCB-DQM)” and the method is applied for the numerical simulation of the Burgers’ equation. In this method, the

exponential modified cubic-B-spline basis functions are used in traditional DQM to determine the weighting coefficients.

The method transforms the Burgers’ equation into a system of the first order nonlinear ordinary differential equations. The

resulting system is solved by employing an optimal five-stage, order four strong stability-preserving time-stepping Runge–

Kutta method. The purpose of preferring SSP-RK54 scheme is to reduce storage space. The efficiency and adaptability of the

method is confirmed by taking five test problems.

2. Exponential modified cubic B-spline differential quadrature method

Let us assume that the one dimensional domain [ a , b ] is discretized into N grid points a = x 1 < x 2 · · · < x N = b uniformly

with step size x = x i +1 − x i . By the basic fundamental of DQM, the r th order spatial partial derivatives of the unknown

u ( x, t ) with respect to x are approximated at x i , i = 1 , 2 , . . . , N as follows

∂ r u ( x i , t )

∂ x r =

N ∑

j=1

a ( r )

i j u

(x j , t

), i = 1 , 2 , . . . , N, (2.1)

where a (r) i j

are the weighting coefficients of the r th order spatial partial derivatives with respect to x .

For two dimensional domain [ a, b ] × [ c, d ] , it is assumed that N and M grid points a = x 1 < x 2 · · · < x N = b and c = y 1 <

y 2 · · · < y M

= d are uniformly distributed with step size x = x i +1 − x i and y = y j+1 − y j in x and y directions, respectively.

In the same manner, the r th order spatial partial derivatives of u ( x, y, t ) with respect to x (keeping y j as fixed) and with

respect to y (keeping x i as fixed), approximated at the point ( x i , y j ) as follows

∂ r u

(x i , y j , t

)∂ x r

=

N ∑

k =1

a ( r ) ik

u

(x k , y j , t

), i = 1 , 2 , . . . , N, j = 1 , 2 , . . . , M (2.2)

∂ r u

(x i , y j , t

)∂ y r

=

M ∑

k =1

b ( r ) jk

u ( x i , y k , t ) , i = 1 , 2 , . . . , N, j = 1 , 2 , . . . , M. (2.3)

where a (r) i j

and b (r) i j

are the weighting coefficients of the r th order spatial partial derivatives with respect to x and y .

In this work, exponential cubic B-spline basis functions are used to find the weighting coefficients of one and two di-

mensional problems.

M. Tamsir et al. / Applied Mathematics and Computation 290 (2016) 111–124 113

Table 1

Coefficients of the exponential cubic B-spline functions E i and

its derivatives at the node x i .

x i −2 x i −1 x i x i +1 x i +2

E i (x ) 0 s −ph 2( phc−s )

1 s −ph 2( phc−s )

0

E ′ i (x ) 0 p(c−1)

2( phc−s ) 0 − p(c−1)

2( phc−s ) 0

E ′′ i (x ) 0 p 2 s

2( phc−s ) − p 2 s

phc−s p 2 s

2( phc−s ) 0

2.1. Exponential cubic B-spline basis functions

The exponential cubic B-spline basis functions are defined as

E i ( x ) =

1

h

3

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

b 2 (( x i −2 − x ) − 1

p ( sinh ( p ( x i −2 − x ) ) ) ), x ∈ [ x i −2 , x i −1 )

a 1 + b 1 ( x i − x ) + c 1 exp ( p ( x i − x ) ) + d 1 exp ( −p ( x i − x ) ) , x ∈ [ x i −1 , x i )

a 1 + b 1 ( x − x i ) + c 1 exp ( p ( x − x i ) ) + d 1 exp ( −p ( x − x i ) ) , x ∈ [ x i , x i +1 )

b 2 (( x − x i +2 ) − 1

p ( sinh ( p ( x − x i +2 ) ) ) ), x ∈ [ x i +1 , x i +2 )

0 , otherwise

(2.4)

where

a 1 =

phc

phc − s , b 1 =

p

2

(c ( c − 1 ) + s 2

( phc − s ) ( 1 − c )

), c 1 =

1

4

(exp ( −ph ) ( 1 − c ) + s ( exp ( −ph ) − 1 )

( phc − s ) ( 1 − c )

)

d 1 =

1

4

(exp ( ph ) ( c − 1 ) + s ( exp ( ph ) − 1 )

( phc − s ) ( 1 − c )

), b 2 =

p

2 ( phc − s ) , c = cosh (ph ) , s = sinh (ph ) .

In Eq. (2.4) , the free parameter p is used to obtain different forms of exponential cubic B-spline functions. The set

{ E 0 , E 1 , . . . , E N , E N+1 } is chosen in such a way that it forms a basis over the domain a ≤ x ≤ b. The values of exponential

cubic B-splines and its derivatives at the nodal points are depicted in Table 1.

The basis exponential cubic B-spline basis functions are modified in such way that the resulting matrix system of equa-

tions is diagonally dominant. The exponential cubic B-spline basis functions are modified as

φ1 ( x ) = E 1 ( x ) + 2 E 0 ( x )

φ2 ( x ) = E 2 ( x ) − E 0 ( x )

φm

( x ) = E m

( x ) f or m = 3 , . . . , N − 2

φN−1 ( x ) = E N−1 ( x ) − E N+1 ( x )

φN ( x ) = E N ( x ) + 2 E N+1 ( x )

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎭

, (2.5)

where { φ1 , φ2 , . . . , φN } forms a basis in the region a ≤ x ≤ b.

2.2. To determine the weighting coefficients

Taking r = 1 in Eq. (2.1) and substituting the values of φm

(x ) , m = 1 , 2 , . . . , N, we get a system of linear equations

φ′ m

( x i ) =

N ∑

j=1

a (1) i j

φm

(x j

), for i, m = 1 , 2 , . . . , N. (2.6)

With the help of Eq. (2.5) and Table 1, Eq. (2.6) reduces into a tri-diagonal system of equations

A

� a ( 1 ) [ i ] =

� R [ i ] , f or i = 1 , 2 , ..., M, (2.7)

where A = [ φi j ] is the coefficient matrix of order N given by:

A =

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

phc−ph phc−s

s −ph 2 ( phc−s )

0 1



1


. . . . . .

. . . s −ph

2 ( phc−s ) 1



1 0


phc−ph phc−s

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦


� a (1) [ i ] = [ a (1)

i 1 , a (1)

i 2 , . . . , a (1)

iN ] T is the weighting coefficient vector corresponding to knot point x i , and the coefficient vector

� R [ i ] = [ φ′

1 ,i , φ′ 2 ,i , . . . , φ

′ N−1 ,i , φ

′ N,i ]

T corresponding to knot point x i , i = 1 , 2 , . . . , N are evaluated as

� R [ 1 ] =

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

− p ( c−1 ) phc−s

p ( c−1 ) phc−s

0

0

. . . 0

0

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, � R [ 2 ] =

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

− p ( c−1 ) 2 ( phc−s )

0

p ( c−1 ) 2 ( phc−s )

0

. . . 0

0

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, · · · , � R [ N − 1 ] =

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

0

. . . 0

− p ( c−1 ) 2 ( phc−s )

0

p ( c−1 ) 2 ( phc−s )

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, � R [ N ] =

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

0

. . . 0

0

− p ( c−1 ) phc−s

p ( c−1 ) phc−s

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

We note that the coefficient matrix A is invertible. The tri-diagonal system of equations is solved for each knot

point x i (i = 1 , 2 , . . . , N) using the Thomas algorithm, which gives the weighting coefficients a (1) i 1

, a (1) i 2

, . . . , a (1) iN−1

, a (1) iN

(i =1 , 2 , . . . , N) of the first order partial derivative.

The weighting coefficients a (2) i j

, 1 ≤ i, j ≤ N for the second order and higher order partial derivatives are determined by

the formula [35] ⎧ ⎪ ⎨

⎪ ⎩

a ( r )

i j = r

(a (

1 ) i j

a ( r−1 )

ii − a (

r−1 ) i j

x i −x j

), f or i � = j and i = 1 , 2 , 3 , . . . , N; r = 2 , 3 , . . . , N − 1

a ( r )

ii = −

N ∑

j =1 , j � = i a (

r ) i j

, for i = j, (2.8)

where a ( r−1 ) i j

and a (r) i j

are the weighting coefficients of the ( r − 1 ) th and r th order partial derivatives with respect to x .

In the same manner, the weighting coefficients b (1) i j

of the first order partial derivatives with respect to y and weighting

coefficients b (2) i j

, 1 ≤ i, j ≤ N for the second derivatives can also be computed from the formula ⎧ ⎪ ⎨

⎪ ⎩

b ( r )

i j = r

(b (

1 ) i j

b ( r−1 )

ii − b (

r−1 ) i j

x i −x j

), f or i � = j and i = 1 , 2 , 3 , . . . , N; r = 2 , 3 , . . . , N − 1

b ( r )

ii = −

N ∑

j =1 , j � = i b (

r ) i j

, for i = j,

where b ( r−1 ) i j

and b (r) i j

are the weighting coefficients of the ( r − 1 ) th and r th order partial derivatives with respect to y.

3. Exponential modified cubic B-spline differential quadrature algorithm for Burgers’ equations

First we discretize the spatial derivatives of Burgers’ Eq. (1.1) by exponential modified cubic B-spline differential quadra-

ture method then, we get the following system of nonlinear ordinary differential equation

du ( x i , t )

dt = −α u ( x i , t )

N ∑

j=1

a (1) i j

u

(x j , t

)− υ

N ∑

j=1

a (2) i j

u

(x j , t

), a ≤ x i ≤ b, t > 0 , i = 1 , 2 , . . . , N, (3.1)

Eq. (3.1) can be written as

du ( x i , t )

dt = L ( u ( x i , t ) ) , i = 1 , 2 , . . . , N. (3.2)

with the initial and boundary conditions ( 1.2 ) and ( 1.3 ).

On substituting the approximated values of the spatial derivatives by Expo-MCB-DQM, Eqs. (1.4) and (1.5) become

∂u

(x i , y j , t

)∂t

= −u

(x i , y j

) N ∑

k =1

a (1) ik

u

(x k , y j

)− v

(x i , y j

) M ∑

k =1

b (1) jk

u ( x i , y k )

+ υ

[

N ∑

k =1

a (2) ik

u

(x k , y j

)+

M ∑

k =1

b (2) jk

u ( x i , y k )

]

, (x i , y j

)∈ �, t > 0 , i = 1 , 2 , . . . , N, j = 1 , 2 , . . . , M. (3.3)

∂v (x i , y j , t

)∂t

= −u

(x i , y j

) N ∑

k =1

a (1) ik

v (x k , y j

)− v

(x i , y j

) M ∑

k =1

b (1) jk

v ( x i , y k )

+ υ

[

N ∑

k =1

a (2) ik

v (x k , y j

)+

M ∑

k =1

b (2) jk

v ( x i , y k )

]

, (x i , y j

)∈ �, t > 0 , i = 1 , 2 , . . . , N, j = 1 , 2 , . . . , M. (3.4)


Eqs. (3.3) and ( 3.4 ) can be written as the following system of nonlinear first order ordinary differential equations:

du

(x i , y j , t

)dt

= F 1 (u

(x i , y j , t

)), i = 1 , 2 , . . . , N and j = 1 , 2 , . . . , M. (3.5)

dv (x i , y j , t

)dt

= F 2 (u

(x i , y j , t

)), i = 1 , 2 , . . . , N and j = 1 , 2 , . . . , M. (3.6)

where L, F 1 and F 2 denotes spatial nonlinear differential operator.with the initial and boundary conditions ( 1.6 ) and ( 1.7 ).

The above system of nonlinear first order ordinary differential equations with the initial conditions and boundary condi-

tions cannot be solved directly by Runge–Kutta methods. So, first we have applied boundary conditions on the systems ( 3.1 ),

( 3.5 ) and ( 3.6 ), then we have got a system of nonlinear first order ordinary differential equations with initial conditions only.

There are various methods in literature to solve a system of nonlinear first order ordinary differential equations. We pre-

ferred the optimal five-stage, order four strong stability-preserving time-stepping Runge–Kutta (SSP-RK54) method [36,37] to

solve the system of nonlinear first order ordinary differential equations. The purpose of preferring SSP-RK54 scheme is to

reduce storage space The SSP-RK54 scheme is defined through the following steps [36] :

u

(1) = u

m + 0 . 391752226571890 tL ( u

m )

u

(2) = 0 . 4 4 4370493651235 u

m + 0 . 555629506348765 u

(1) + 0 . 368410593050371 tL ( u

(1) )

u

(3) = 0 . 620101851488403 u

m + 0 . 379898148511597 u

(2) + 0 . 251891774271694 tL ( u

(2) )

u

(4) = 0 . 178079954393132 u

m + 0 . 821920045606 86 8 u

(3) + 0 . 544974750228521 tL ( u

(3) )

u

(m +1) = 0 . 517231671970585 u

(2) + 0 . 096059710526147 u

(3) + 0 . 0 636924686 6 6290 tL ( u

(3) )

+0 . 386708617503269 u

(4) + 0 . 226007483236906 tL ( u

(4) )

4. Stability analysis of the algorithm

After discretization via DQM and linearization of the non-linear term u u x , u u x + v u y and u v x + v v y by assuming u and vlocally constant [40] , Eq. (3.1) is reduced into a set of ordinary differential equations in time as

d { U } dt

= P { U } + { E } . (4.1)

Eqs. (3.3) and ( 3.4 ) are reduced into set of ordinary differential equations in time as

d { W } dt

=

[A O

O B

]{ W } + { K } , (4.2)

where,

(i) { U} = ( u 2 , u 3 , . . . , u N−1 ) is an unknown vector of the functional values at the interior grid points.

(ii) { E} is a vector containing non-homogeneous part and boundary conditions.

(iii) P = −αU i j A 1 + υA 2 .

(iv) O

′ s are null matrices.

(v) { K} = ( F , G ) T is a vector containing non-homogeneous part and boundary conditions.

(vi) { W } = ( U, V ) T , where U and V are unknown vectors of the functional values at the interior grid points:

U = ( u 22 , u 23 , . . . , u 2(M−1) , u 32 , u 33 , . . . u 3(M−1) , . . . u (M−1)2 , u (M−1)3 . . . , u (M−1)(M−1) ) ,

V = ( v 22 , v 23 , . . . , v 2(M−1) , v 32 , v 33 , . . . v 3(M−1) , . . . v (M−1)2 , v (M−1)3 , . . . , v (M−1)(N−1) ) .

(vii)

A = −U i j A 1 − V i j B 1 + υA 2 + υB 2 ,

B = −U i j A

′ 1 − V i j B

′ 1 + υA

′ 2 + υB

′ 2 ,

where A r and B r are square block diagonal matrices (N − 2) × (M − 2) of the weighting coefficients a (r) i j

, b (r) i j

(r = 1 , 2) re-

spectively as given below

A r =

⎡

⎢ ⎢ ⎢ ⎣

a (r) 22

I a (r) 23

I . . . a (r) 2(N−1)

I

a (r) 32

I a (r) 33

I . . . a (r) 3 ,N−1

I

. . . . . .

. . . . . .

a (r) (N−1)2

I a (r) (N−1)3

I . . . a (r) (N−1)(N−1)

I

⎤

⎥ ⎥ ⎥ ⎦


Fig. 1. Eigen values of A 1 (left) and A 2 (right) for different grid sizes.

Fig. 2. Eigen values of B 1 (left) and B 2 (right) for different grid sizes.

B r =

⎡

⎢ ⎢ ⎣

M r O . . . O

O M r . . . O

. . . . . .

. . . . . .

O O . . . M r

⎤

⎥ ⎥ ⎦

; where M r =

⎡

⎢ ⎢ ⎢ ⎣

b (r) 22

b (r) 23

. . . b (r) 2(M−1)

b (r) 32

b (r) 33

. . . b (r) 3(M−1)

. . . . . .

. . . . . .

b (r) (M−1)2

b (r) (M−1)3

. . . b (r) (M−1)(M−1)

⎤

⎥ ⎥ ⎥ ⎦

I and O are the matrices of order (N − 2) × (M − 2) .

Similarly, A

′ r and B ′ r are square block diagonal matrices each of which having order (N − 2) × (M − 2) of the weighting

coefficients a (r) i j

and b (r) i j

(r = 1 , 2) respectively.

Stability of the proposed scheme for the solution of nonlinear viscous Burgers’ equation directly depends upon the sta-

bility of the system of ordinary differential Eq. (4.2) for one dimension and (4.4) for two dimensions. Stability of ( 4.2 ) and

(4.4) depends on the eigen values of the coefficient matrices P , A and B . The system ( 4.2 ) and (4.4) will be stable if the real

part of each eigen value of P and Q are either negative or zero.

The Figs. 1 and 2 show that the real part of the eigen values of the matrices A 1 , A 2 , B 1 , and B 2 are either negative or

zero for different values of grid points. The real parts of the eigen values of the matrices P , A and B are either negative or

zero since these matrices depend upon the matrices A 1 , A 2 , B 1 , and B 2 . This shows that the developed algorithm is stable.

5. Numerical experiments and discussion

In this section, five numerical problems are considered to show the accuracy and efficiency of the proposed algorithm.

The error norms L 2 and L ∞

are calculated by using the following definitions

L 2 := || u exact − u computed | | 2 =

√

h

n ∑

j=1

| u j exact − u j

computed | 2 L ∞

:= || u exact − u computed | | ∞

= max j

| u j exact − u j

computed |

⎫ ⎪ ⎬

⎪ ⎭

,


Table 1.1

Comparison of L 2 and L ∞ errors of Expo-MCB-DQM of Problem 1 for υ = 0 . 005 with the existing methods at different

time levels.

Methods N t t = 1 . 7 t = 2 . 4 t = 3 . 1

L 2 × 10 3 L ∞ × 10 3 L 2 × 10 3 L ∞ × 10 3 L 2 × 10 3 L ∞ × 10 3

Present (p = 1) 121 0 .01 0 .00173 0 .00680 0 .0 0 0799 0 .00288 0 .0 0 0657 0 .00354

Present ( p = 0.015) 121 0 .01 0 .00173 0 .00679 0 .0 0 0799 0 .00288 0 .0 0 0657 0 .00354

MCB-DQM [31] 121 0 .01 0 .00191 0 .00777 0 .0 0 086 0 .00308 0 .0 0 065 0 .00331

QRTDQ [22] 101 0 .001 0 .109 0 .434 0 .100 0 .339 0 .091 0 .266

BS.FEM [24] 50 0 .1 0 .857 2 .576 0 .423 1 .242 0 .230 0 .680

C.S.C. [25] 50 0 .01 0 .857 2 .576 0 .423 1 .242 0 .235 0 .688

QBCM1 [26] 200 0 .01 0 .017 0 .061 0 .012 0 .058 0 .601 4 .434

QBCM2 [26] 200 0 .001 0 .358 1 .211 0 .251 0 .807 0 .630 4 .790

Galerkin [27] 200 0 .01 0 .857 2 .576 0 .423 1 .242 0 .235 0 .688

t = 2 . 5

QBCM [18] 200 0 .01 0 .0721 0 .31153 0 .0510 0 .18902

CBCM [18] 200 0 .01 2 .4664 27 .577 2 .1118 25 .1517

t = 3 . 5

MCB-CM [9] 241 0 .01 0 .0252 0 .0994 0 .0151 0 .0549 0 .0117 0 .0486

β = 0 . 5 [38] 121 0 .01 0 .38421 1 .34728 0 .49135 1 .55470 0 .525855 1 .52196

β = 1 [38] 121 0 .01 3 .08966 10 .4040 2 .72048 8 .29747 2 .12110 5 .94321

MCB-DQM [31] 121 0 .01 0 .00191 0 .00777 0 .00778 0 .00275 0 .006177 0 .04335

Present ( p = 1) 121 0 .01 0 .00173 0 .00680 0 .0 0 0729 0 .00256 0 .006152 0 .0431

Present ( p = 0.015) 121 0 .01 0 .00173 0 .00679 0 .0 0 0729 0 .00256 0 .006152 0 .0431

where u exact and u computed represent the exact and computed solutions at the node x j , respectively.

Problem 1. In this problem, the Burgers’ Eq. (1.1) with α = 1 considered over the domain [0, 1.2] with the following initial

and boundary conditions [ 19 , 21 , 31 ]

u ( x, 1 ) =

x

1 + exp

(1

4 υ

(x 2 − 1

4

)) , and u (0 , t) = 0 , u (1 . 2 , t) = 0 , for t > 1 .

The exact solution of the problem is given by

u ( x, t ) =

x t

1 +

(t t 0

)1 / 2 exp

(x 2

4 υt

) , where t 0 = exp

(1

8 υ

), for t ≥ 1 .

The numerical solutions of this example are computed with the parameter values υ = 0 . 005 , h = 0 . 01 and t = 0 . 01

at different time levels. Table 1.1 shows the comparison of the proposed algorithm in term of L 2 and L ∞

errors with the

existing methods MCB-DQM [31] , MCB-CM [9] , QBCM [18] , CBCM [18] , QRTDQ [22] , BS.FEM [24] , C.S.C. [25] , QBCM1 [26] ,

QBCM2 [26] , Galerkin [27] , β = 0 . 5 [38] , β = 1 [38] . The physical behavior of the problem for υ = 0 . 005 at different time

levels with h = 0 . 01 , t = 0 . 01 is shown in Fig. 3. The absolute errors for different time levels are also depicted in Fig. 4 . It

is observed that the absolute errors much better than those given in [31] .

Example 2. Considered the Burger’s Eq. (1.1) , for α = 1 , over the region [0, 1] with initial condition [ 20 , 31 ]

u ( x, 0 ) = sin (πx ) ,

and the boundary conditions

u ( 0 , t ) = u (1 , t) = 0 .

The exact solution of this problem is given by Cole [4] in terms of an infinite series as

u ( x, t ) =

4 πυ∑ ∞

j=1 j I j (

1 2 πυ

)sin ( jπx ) exp (− j 2 π2 υt)

I 0 (

1 2 πυ

)+ 2

∑ ∞

j=1 I j (

1 2 πυ

)cos ( jπx ) exp (− j 2 π2 υt)

,

where I j are the modified Bessel’s functions.

Table 2.1 shows the numerical solution of the problem with parameter p = 1 , υ = 1 . 0 , h = 0 . 0125 , h = 0 . 025 and t =10 −4 at t = 0 . 1 , and a comparison is made with [ 18 , 31 ]. It is found that Expo-MCB-DQM produces the results similar to

those given in [18] at the half of the grid points and better than those given in [31] . Also, the numerical solution of the

problem with parameter value p = 1 , υ = 0 . 1 , h = 0 . 025 and t = 0 . 004 at different time levels reported in Table 2.2 . It

is clear that our results are much better than obtained in [ 9 , 18 , 20 , 31 ]. The physical behavior of solutions are depicted in

Fig. 5 at υ = 0 . 1 and υ = 1 . 0 for t ≤ 1 with h = 0 . 025 and t = 0 . 0 0 01 .


Fig. 3. Physical behavior of Expo-MCB-DQM solutions of Problem 1 for υ = 0 . 005 at different time levels with h = 0 . 01 , t = 0 . 01 .

Fig. 4. Absolute errors in the Expo-MCB-DQM numeric solutions of Problem 1 for υ = 0 . 005 at different time levels with h = 0 . 01 , t = 0 . 01 .

Table 2.1

Comparison of Expo-MCB-DQM solutions of Problem 2 for υ = 1 . 0 with the existing solutions and exact solutions at t = 1.0.

x [18] h = 0 . 0125 [18] h = 0 . 00625 MCB-DQM [31]

h = 0 . 025

Present (p = 1)

h = 0 . 025

MCB-DQM [31]

h = 0 . 0125

Present (p = 1)

h = 0 . 0125

Exact

t = 10 −5 t = 10 −5 t = 10 −4 t = 10 −4 t = 10 −4 t = 10 −4

0 .1 0 .10952 0 .10953 0 .109530 0 .109541 0 .109526 0 .109538 0 .10954

0 .2 0 .20975 0 .20977 0 .209771 0 .209795 0 .209766 0 .209792 0 .20979

0 .3 0 .29184 0 .29186 0 .291860 0 .291899 0 .291855 0 .291896 0 .29190

0 .4 0 .34785 0 .34788 0 .347874 0 .347927 0 .347869 0 .347923 0 .34792

0 .5 0 .37149 0 .37153 0 .371517 0 .371581 0 .371512 0 .371577 0 .37158

0 .6 0 .35896 0 .35900 0 .358981 0 .359049 0 .358975 0 .359045 0 .35905

0 .7 0 .30983 0 .30986 0 .309845 0 .309909 0 .309839 0 .309904 0 .30991

0 .8 0 .22776 0 .22778 0 .227773 0 .227822 0 .227766 0 .2278217 0 .22782

0 .9 0 .12065 0 .12067 0 .120666 0 .120691 0 .120659 0 .120686 0 .12069


Table 2.2

Comparison of Expo-MCB-DQM solutions of Problem 2 for υ = 0 . 1 with the existing and exact solutions.

x t Dag et al. [18]

h = 0 . 0125

Mittal and Jain [9]

h = 0 . 025

Korkmaz [20] h = 0 . 025 Arora and Singh

[31] h = 0 . 025

Present (p = 1)

h = 0 . 025

Exact

t = 10 −4 t = 0 . 0025 t = 0 . 00125 t = 0 . 004 t = 0 . 004

0 .25 0 .4 0 .30890 0 .30892 0 .30910 0 .3089280 0 .308893 0 .30889

0 .6 0 .24075 0 .24077 0 .24093 0 .2407550 0 .240738 0 .24074

0 .8 0 .19569 0 .19572 0 .19586 0 .1956840 0 .195674 0 .19568

1 .0 0 .16258 0 .16261 0 .16274 0 .1625700 0 .162563 0 .16256

3 .0 0 .02720 0 .02718 0 .02720 0 .0272047 0 .0272005 0 .02720

0 .50 0 .4 0 .56965 0 .56970 0 .56973 0 .5696530 0 .569631 0 .56963

0 .6 0 .44723 0 .44729 0 .44736 0 .4472170 0 .447202 0 .44721

0 .8 0 .35925 0 .35930 0 .35943 0 .3592450 0 .359231 0 .35924

1 .0 0 .29192 0 .29195 0 .29213 0 .2919250 0 .291910 0 .29192

3 .0 0 .04019 0 .04016 0 .04032 0 .0402085 0 .0402023 0 .04021

0 .75 0 .4 0 .62538 0 .62520 0 .62573 0 .6253490 0 .625424 0 .62544

0 .6 0 .48715 0 .48694 0 .48760 0 .4872040 0 .487194 0 .48721

0 .8 0 .37385 0 .37365 0 .37434 0 .3739350 0 .373901 0 .37392

1 .0 0 .28741 0 .28724 0 .28788 0 .2874930 0 .287456 0 .28747

3 .0 0 .02976 0 .02974 0 .029881 0 .0297753 0 .0297697 0 .02977

Fig. 5. Physical behavior of Expo-MCB-DQM of Problem 2 at υ = 0 . 1 (left) and at υ = 1 (right) for t ≤ 1 with h = 0 . 025 and t = 0 . 0 0 01 .

Table 3.1

Comparison of L 2 and L ∞ errors of Expo-MCB-DQM of Problem 3 with the existing methods.

N Mittal and Jain [9] Kaysar et al. [46] Jiwari et al. [47] Present scheme

L ∞ L 2 L ∞ L 2 L ∞ L 2 L ∞ L 2

10 4 .62E −07 3 .28E −07 4 .881E −07 3 .455E −07 4 .708E −08 6 .459E −08 1 .467E −07 6 .330E −08

20 1 .16E −07 8 .19E −08 1 .431E −07 1 .012E −07 1 .091E −08 4 .465E −09 3 .029E −08 1 .014E −08

40 2 .907E −08 2 .047E −08 5 .668E −08 4 .003E −08 1 .980E −09 2 .786E −10 3 .956E −09 1 .207E −09

80 7 .271E −09 5 .119E −09 3 .499E −08 4 .002E −08 7 .18E −09 2 .665E −10 8 .861E −11 1 .322E −10

Problem 3. In this problem, the Burgers’ Eq. (1.1) with α = 1 is considered with the following exact solution [9,47]

u ( x, t ) = 2 πυsin (πx ) exp (−π2 υt)

σ + cos (πx ) exp (−π2 υt) , for x ∈ ( 0 , 1 ) and t ≥ 0 ,

where, the parameter σ > 1 .

The initial and boundary conditions are taken from the exact solution. The comparison of L ∞

and L 2 errors with parame-

ters υ = 0 . 005 , σ = 100 , p = 0 . 1 , t = 0 . 01 at t = 1 for different grid sizes, is reported in Table 3.1 . It is evident that present

method results are much better than the results obtained in [9,46] and are in good agreement with the results obtained in

[47] . Absolute error for parameters σ = 2 , υ = 0 . 001 at t = 0 . 5 at different t is depicted in Fig. 6.


Fig. 6. Absolute error of Problem 3 for σ = 2 , υ = 0 . 001 at t = 0 . 5 for different time step t.

Table 4.1

Errors and rate of convergence for u component for υ = 10 −2 , t = 0.0 0 01 at t = 1 . 0 .

Grid L 2 L ∞

Srivastava et al. [42] Shukla et al. [43] Expo-MCB-DQM Srivastava et al. [42] Shukla et al. [43] Expo-MCB-DQM

p = 10 ROC p = 10 ROC

4 × 4 8 .5708E −02 1 .6388E −02 1 .5865E −02 – 9 .7046E −02 2 .8788E −03 2 .3325E −03 –

8 × 8 4 .9429E −02 1 .9286E −03 1 .8037E −03 3 .137 4 .6886E −02 1 .9572E −04 1 .6816E −04 3 .794

16 × 16 1 .9192E −02 3 .9474E −04 3 .8329E −04 2 .234 2 .0467E −02 2 .0486E −05 1 .9610E −05 3 .100

32 × 32 8 .6812E −03 8 .1181E −05 8 .0461E −05 2 .252 9 .0744E −03 2 .2202E −06 2 .1967E −06 3 .158

64 × 64 – 1 .5322E −05 1 .5355E −05 2 .387 – 2 .1838E −07 2 .1795E −07 3 .333

Problem 4. In this problem, the two dimensional Burgers’ Eqs. (1.4) and ( 1.5 ) are considered over the domain D ={ ( x, y ) : 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } with an exact solution generated by using the Hopf–Cole transformation [8]

u ( x, y, t ) =

3

4

− 1

4

(1 + e

Re (4 y −4 x −t) 32

)

v ( x, y, t ) =

3

4

+

1

4

(1 + e

Re (4 y −4 x −t) 32

)The initial and boundary conditions are taken from the exact solutions. In this problem, numerical solutions are com-

puted with the parameters p = 10 , υ = 10 −2 , t = 0.0 0 01 at t = 1 . 0 for different grid sizes and reported in Tables 4.1 and

4.2 in the form of errors and the rate of convergence for u and v , respectively. It is found that that the Expo-MCB-DQM

performs much better than [42,43] and gives more than quadratic rate of convergence (see Figs. 7 and 8 ).

Problem 5. In this problem, the Eqs. (1.4) and ( 1.5 ) are solved in the computational domain 0 ≤ x ≤ 0 . 5 , 0 ≤ y ≤ 0 . 5 as taken

in [43] with initial conditions

u ( x, y, 0 ) = sin (πx ) + cos (πy ) , v ( x, y, 0 ) = x + y,

}; 0 ≤ x ≤ 0 . 5 , 0 ≤ y ≤ 0 . 5 ,


Fig. 7. Comparison of (a) numerical and (b) exact solution for component u and v of Problem 4 for Re = 100 , h = 0 . 05 , t = 0 . 0 0 01 at t = 1 .

Fig. 8. Comparison of (a) numerical and (b) exact solution for component u and v of Problem 4 for Re = 200 , h = 0 . 05 , t = 0 . 0 0 01 at t = 1 .


Table 4.2

Errors and rate of convergence for v component for υ = 10 −2 , t = 0.0 0 01 at t = 1 . 0 .

Grid L 2 L ∞

Srivastava et al. [42] Shukla et al. [43] Expo-MCB-DQM Srivastava et al . [42] Shukla et al. [43] Expo-MCB-DQM

p = 10 ROC p = 10 ROC

4 × 4 8 .5708E −02 1 .6388E −02 1 .5865E −02 – 9 .7046E −02 2 .8788E −03 2 .3325E −03 –

8 × 8 4 .9431E −02 1 .9286E −03 1 .8037E −03 3 .137 4 .6887E −02 1 .9573E −04 1 .6816E −04 3 .794

16 × 16 1 .9196E −02 3 .9474E −04 3 .8329E −04 2 .234 2 .0471E −02 2 .0486E −05 1 .9610E −04 3 .100

32 × 32 8 .6878E −03 8 .1181E −05 8 .0461E −05 2 .252 9 .0813E −03 2 .2202E −06 2 .1967E −06 3 .158

64 × 64 – 1 .5322E −05 1 .5355E −05 2 .387 – 2 .1838E −07 2 .1795E −07 3 .333

Table 5.1

Comparison of the results of Expo-MCB-DQM for Re = 50 with grid size 20 × 20 and t = 0.0 0 01 at t = 0.625.

Grid ( x, y ) u ( x, t ) v ( x, t )

I-LFDM [42] MCB-DQM [43] Present I-LFDM [42] MCB-DQM [43] Present

(0 .1, 0.1) 0 .97146 0 .97056 0 .970558 0 .09869 0 .09842 0 .098419

(0 .3, 0.1) 1 .15280 1 .15152 1 .15152 0 .14158 0 .14107 0 .141070

(0 .2, 0.2) 0 .86308 0 .86244 0 .862434 0 .16754 0 .16732 0 .167317

(0 .4, 0.2) 0 .97985 0 .98078 0 .980779 0 .17111 0 .17223 0 .172228

(0 .1, 0.3) 0 .66316 0 .66336 0 .663354 0 .26378 0 .26380 0 .263801

(0 .3, 0.3) 0 .77233 0 .77226 0 .772256 0 .22655 0 .22653 0 .226526

(0 .2, 0.4) 0 .58181 0 .58273 0 .582728 0 .32851 0 .32935 0 .329347

(0 .4, 0.4) 0 .75862 0 .76179 0 .761787 0 .32502 0 .32884 0 .328842

Fig. 9. Physical behavior the numerical solution of Problem 5 for Re = 100 , h = 0 . 05 , t = 0 . 0 0 01 at t = 1 .

and boundary conditions

u ( 0 , y, t ) = cos (πy ) , u ( 0 . 5 , y, t ) = 1 + cos (πy ) , v ( 0 , y, t ) = y, v ( 0 . 5 , y, t ) = 0 . 5 + y,

⎫ ⎪ ⎬

⎪ ⎭

; 0 ≤ y ≤ 0 . 5 , t ≥ 0 ,

u ( x, 0 , t ) = 1 + sin (πx ) , u ( x, 0 . 5 , t ) = sin (πx ) , v ( x, 0 , t ) = x,

v ( x, 0 . 5 , t ) = x + 0 . 5

⎫ ⎪ ⎬

⎪ ⎭

; 0 ≤ x ≤ 0 . 5 , t ≥ 0 .

Table 5.1 shows the comparison of numerical solution for the component u ( x, t ) and v ( x, t ) for Re = 50, grid size 20 × 20 ,

t = 0 . 0 0 01 at t = 0.625 with [ 42 , 43 ]. The table concludes that the present results are good in agreement with the results

[42,43] (see Figs. 9–11 ).

6. Conclusions

In this paper, the authors developed a new differential quadrature method “Expo-MCB-DQM” to solve nonlinear par-

tial differential equations. The proposed method is tested on well known nonlinear Burgers’ equations. Finally, the authors

summarize the outcomes of this analysis as follows:




(i) A different technique based on exponential modified cubic-B-spline functions is proposed to find the weighting coef-

ficients of differential quadrature method than the traditional technique of Lagrange interpolation [32] .

(ii) To the best knowledge of the authors, this is new differential quadrature technique for solving differential equations.

(iii) The new proposed technique gives better results than the results discussed in [9,18,22,24–27,31,38,42,43,47] and good

accuracy for small number of grid points.

(iv) The present method with some modifications can be easily extended to solve model equations in two or higher

dimensional problems including mechanical, physical or biophysical effects, such as nonlinear convection, reaction,

linear diffusion and dispersion.

(v) The low memory storage, ease of the implementation and good accuracy for small number of grid points are the

advantage of the proposed method.

Acknowledgment

The authors are very thankful to the reviewers for their valuable suggestions to improve the quality of the paper .

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