Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method

Applied Mathematics and Computation 224 (2013) 166–177

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Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method Geeta Arora, Brajesh Kumar Singh ⇑ Department of Mathematics, School of Allied Sciences, Graphic Era Hill University, Dehradun 248002, Uttarakhand, India

a r t i c l e

i n f o

Keywords: Burgers’ equation Cubic B-spline Modified cubic B-spline Differential quadrature method (DQM) Thomas algorithm SSP-RK43 scheme

a b s t r a c t In this paper, a new numerical method, ‘‘modified cubic-B-spline differential quadrature method (MCB-DQM)’’ is proposed to find the approximate solution of the Burgers’ equation. The modified cubic-B-spline basis functions are used in differential quadrature to determine the weighting coefficients. The MCB-DQM is used in space, and the optimal four-stage, order three strong stability-preserving time-stepping Runge–Kutta (SSP-RK43) scheme is used in time for solving the resulting system of ordinary differential equations. To check the efficiency and accuracy of the method, four examples of Burgers’ equation are included with their numerical solutions, L2 and L1 errors and comparisons are done with the results given in the literature. The proposed method produces better results as compared to the results obtained by almost all the schemes available in the literature, and approaching to the exact solutions. The presented method is seen to be easy, powerful, efficient and economical to implement as compared to the existing techniques for finding the numerical solutions for various kinds of linear/nonlinear physical models. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction We consider the well known one dimensional nonlinear Burgers’ equation

@u @u @2u þ au  m 2 ¼ 0; @t @x @x

ðx; tÞ 2 X  ½0; T;

ð1:1Þ

where X ¼ ða; bÞ, with the initial condition

uðx; 0Þ ¼ f ðxÞ;

x 2 ½a; b;

ð1:2Þ

and the boundary conditions

uða; tÞ ¼ 0 and uðb; tÞ ¼ 0; t 2 ½0; T;

ð1:3Þ

where m > 0 is a small parameter known as the coefficient of kinematic viscosity and a is some positive constant. Such type of equations was first introduced by Bateman [5]. Also, he proposed the steady-state solution of the problem. Burgers [6,7] has introduced this equation to capture some features of turbulent fluid in a channel caused by the interaction of the opposite effects of convection and diffusion, and hence Eq. (1.1) is referred to as ‘‘Burgers’ equation’’. The structure of Burgers’ equation is similar to the one dimensional Navier–Stoke’s equation without the stress term. It is the simplest model of nonlinear partial differential equation for diffusive waves in fluid dynamics. This model arises in many physical problems ⇑ Corresponding author. E-mail addresses: [email protected] (G. Arora), [email protected] (B.K. Singh). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.08.071

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including one-dimensional sound/shock waves in a viscous medium, waves in fluid filled viscous elastic tubes, magnetohydrodynamic waves in a medium with finite electrical conductivity, mathematical modeling of turbulent fluid, and in continuous stochastic processes. In the last years, a great deal of effort to compute efficiently the numerical solutions of the Burgers’ equation for small and large both values of the kinematic viscosity has been expanded. The Burgers’ equation is solved for both the infinite and the finite domain [9]. The various numerical techniques to compute the numerical solutions of the Burgers’ equation are: automatic differentiation method [1], finite elements method [32,34], Galerkin finite element method [13], cubic B-splines collocation method [12], cubic B-spline quasi-interpolation [15], modified cubic B-spline collocation method [31], spectral collocation method [27], sinc differential quadrature method [22], polynomial based differential quadrature method [23], cubic B-spline differential quadrature method [20], quartic B-splines differential quadrature method [24], quartic B-splines collocation method [38], quadratic B-splines finite difference element method [4,33], fourth-order finite difference method [14], factorized diagonal Padé approximation [2], non-polynomial spline approach [37], a novel numerical scheme [45], explicit and exact-explicit finite difference methods [25], Hopf–Cole transformation [11,25], least-squares quadratic B-splines finite element method [26], reproducing kernel function method [41], implicit fourth-order compact finite difference [29], weighted average differential quadrature method [16], variational method [3], parameter-uniform implicit difference scheme [17], one dimensional fourier expansion [30], etc. The differential quadrature method (DQM), is an efficient technique to solve partial differential equations (PDEs), was first introduced by Bellman et al. [8]. It was further improved by Quan and Chang [35,36] to solve the weighting coefficients. Various kinds of test functions such as spline functions, Lagrange interpolation polynomials, sinc function [35,36,22], etc. are used to determine the weighting coefficients, for further details on DQM we refer to [42,43,40,36]. B-splines are a set of special spline functions that can be used to construct piece-wise polynomial by computing the appropriate linear combination. These functions have their computational advantage from the fact that any B-spline basis function of order m is nonzero over at most m adjacent intervals and zero otherwise. Due to smoothness and capability to handle local phenomena, B-spline basis functions offer distinct advantages in comparison to other basis functions. Cubic B-spline functions have already been used as basis functions to solve many physical models. Recently, Korkmaz and Dag˘ [21] used cubic B-spline functions with DQM to solve advection–diffusion equation. Mittal and Jain [31] solved Burgers’ equation by modified cubic B-spline collocation method. In this paper, a new numerical method, ‘‘modified cubic-B-spline differential quadrature method (MCB-DQM)’’ is proposed to find the approximate solution of the Burgers’ equation. In this method, the modified cubic-B-spline basis functions are used in DQM to determine the weighting coefficients (i.e., spatial derivatives) which produces the system of first order ordinary differential equations (ODEs). The resulting system of ODEs is solved by the optimal four-stage, order three strong stability-preserving time-stepping Runge–Kutta (SSP-RK43) scheme [44]. The SSP-RK43 scheme needs less storage space that causes less accumulation of numerical errors. This is why we preferred SSP-RK43 scheme. The MCB-DQM solutions to the Burgers’ equation have been computed without transforming the equation and without using the linearization. The comparison of the MCB-DQM numerical solutions with analytical solutions are presented to illustrate the efficiency and adaptability of the method. The L2 and L1 errors are also evaluated and compared with results given in the literature. This paper is organized as follows. In Section 2, the description of the modified cubic B-spline differential quadrature method is given. In Sections 3, procedure for implementation of method is described. Numerical examples are given to establish the applicability and accuracy of the proposed method in Section 4. The conclusion is given in Section 5 that briefly summarizes the numerical outcomes. 2. Description of the method The differential quadrature method (DQM) is an approximation to derivatives of a function using the weighted sum of the functional values at certain discrete points. Since the weighting coefficients are dependent on the spatial grid spacing only, one can assume uniformly distributed N nodes/knots: a ¼ x1 < x2 ; . . . ; xN1 < xN ¼ b such that xiþ1  xi ¼ h on the real axis. The first and second order spatial derivatives of the uðx; tÞ at any time on the knot xi for i ¼ 1; . . . ; N are given by

ux ðxi ; tÞ ¼

N X

aij uðxj ; tÞ;

for j ¼ 1; . . . ; N;

ð2:1Þ

j¼1

uxx ðxi ; tÞ ¼

N X bij uðxj ; tÞ;

for j ¼ 1; . . . ; N;

ð2:2Þ

j¼1

where aij and bij are weighting coefficients of the first and second order derivatives with respect to x, respectively [8]. The cubic B-spline basis functions at the knots are defined as follows

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8 > ðx  xj2 Þ3 > > > > 3 3 > > ðx  xj2 Þ  4ðx  xj1 Þ 1< uj ðxÞ ¼ 3 ðxjþ2  xÞ3  4ðxjþ1  xÞ3 > h > > > ðx  xÞ3 > jþ2 > > : 0

x 2 ðxj2 ; xj1 Þ x 2 ðxj1 ; xj Þ ð2:3Þ

x 2 ðxj ; xjþ1 Þ x 2 ðxjþ1 ; xjþ2 Þ otherwise;

where fu0 ; u1 ; . . . ; uN ; uNþ1 g forms a basis over the region ½a; b. The values of cubic B-splines and its derivatives at the nodal points are tabulated in Table 0.1. The cubic B-spline basis functions are modified in such way that the resulting matrix system of equations is diagonally dominant. The modified cubic B-spline basis functions at the knots are defined as follows [31]

/1 ðxÞ ¼ u1 ðxÞ þ 2u0 ðxÞ /2 ðxÞ ¼ u2 ðxÞ  u0 ðxÞ /j ðxÞ ¼ uj ðxÞ for j ¼ 3; . . . ; N  2 /N1 ðxÞ ¼ uN1 ðxÞ  uNþ1 ðxÞ /N ðxÞ ¼ uN ðxÞ þ 2uNþ1 ðxÞ

9 > > > > > > =

ð2:4Þ

> > > > > > ;

where f/1 ; /2 ; . . . ; /N g forms a basis over the region ½a; b. 2.1. To determine the weighting coefficients The first order derivative approximation is given by

/0k ðxi Þ ¼

N X aij /k ðxj Þ;

for i ¼ 1; . . . ; N; k ¼ 1; . . . ; N

ð2:5Þ

j¼1

For the first knot point x1 , the approximation can be given as

/0k ðx1 Þ ¼

N X a1j /k ðxj Þ;

for k ¼ 1; . . . ; N

ð2:6Þ

j¼1

which gives a tridiagonal system of equation as

3 3 2 3 2 6=h 7 a11 60 4 1 7 7 6 6 76 7 6 6=h 7 6 76 a12 7 6 7 6 1 4 1 76 7 7 6 6 .. .. .. 76 .. 7 6 0 7 6 76 . 7 ¼ 6 . 7 6 . . . 76 7 6 .. 7 6 76 7 7 6 6 1 4 1 74 a1N1 5 6 7 6 7 6 4 0 5 5 4 1 4 0 a1N 0 1 6 2

6 1

We apply well known ‘‘Thomas algorithm’’ to solve the resulting tridiagonal system of equations whose solution provides the coefficients a11 ; a12 ; . . . ; a1N . Similarly, for the second knot point x2 , the approximation can be given as

/0k ðx2 Þ ¼

N X a2j /k ðxj Þ;

for k ¼ 1; . . . ; N

ð2:7Þ

j¼1

which again gives a tridiagonal system of equations as

Table 0.1 The coefficients of cubic B-splines and its derivatives at knots xj . xj2

uj ðxÞ u0j ðxÞ u00j ðxÞ

xj1

xj

xjþ1

xjþ2 0

0

1

4

1

0

3=h

0

3=h

0

6=h

2

12=h

2

6=h

0 2

0

G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177

169

3 3 2 3 2 3=h 7 a21 60 4 1 7 7 6 6 76 7 6 0 7 6 76 a22 7 6 7 6 1 4 1 76 7 7 6 6 .. .. .. 76 .. 7 6 3=h 7 6 ¼ 7 7 7 6 6 6 . . . . 0 76 7 7 6 6 7 7 7 6 6 6 1 4 1 74 a2N1 5 6 .. 7 6 7 6 4 . 5 4 1 4 0 5 a2N 0 1 6 2

6 1

whose solution provides the coefficients a21 ; a22 ; . . . ; a2N . Proceeding in the similar manner, up to the second last knot point xN1 the coefficients ak1 ; ak2 ; . . . ; akN for k ¼ 3 . . . N  1 are determined. At the knot point xN1 , the tridiagonal system of equations is given as

3 3 2 3 2 0 7 aN11 60 4 1 7 6 6 .. 7 76 7 7 6 6 76 aN12 7 6 . 7 6 1 4 1 76 7 7 6 6 .. .. .. 76 7 6 0 7 6 .. 76 7 7¼6 6 . . . . 76 7 6 3=h 7 6 7 7 7 6 6 6 1 4 1 74 aN1N1 5 6 7 6 7 6 4 0 5 4 1 4 0 5 aN1N 3=h 1 6 2

6 1

Now, the matrix system of equations corresponding to the last knot point xN is given as

3 3 2 2 3 7 aN1 60 4 1 0 7 6 76 7 6 . 7 6 76 aN2 7 6 .. 7 6 1 4 1 76 7 6 7 .. .. .. 76 .. 7 6 6 7; 76 . 7 ¼ 6 6 . 0 . . 7 6 76 7 6 6 7 7 7 6 6 4 6=h 5 1 4 1 74 aNN1 5 6 7 6 4 6=h 1 4 0 5 aNN 1 6 2

6 1

which provides the coefficients aN1 ; aN2 ; . . . ; aNN . Thus, we have evaluated the weighting coefficient aij i ¼ 1; 2; . . . ; N; j ¼ 1; 2; . . . ; N. Using these coefficients, the weighting coefficient bij for i ¼ 1; 2; . . . ; N; j ¼ 1; 2; . . . ; N is evaluated as follows [40]

 bij ¼ 2aij aij 

 1 ; xi  xj

for i – j;

and bii ¼ 

N X

for

bij :

i¼1;i – j

3. Implementation of method On substituting the first and second order approximation of the spatial derivatives, obtained by using MCB-DQM, the Burgers’ Eq. (1.1) can be rewritten as N N X X @ui ¼ m bij uðxj Þ  auðxi Þ aij uðxj Þ; i ¼ 1; . . . ; N: @t j¼1 j¼1

ð3:1Þ

Thus, Eq. (3.1) is reduced into a set of ordinary differential equations in time, that is, for i ¼ 1; . . . ; N, we have

dui ¼ Lðui Þ; dt

ð3:2Þ

where L denotes a spatial nonlinear differential operator. There are various methods to solve this system of ODE. We preferred the optimal four-stage, order three strong stability-preserving time-stepping Runge–Kutta (SSP-RK43) scheme [44] to solve the system of ODE. In this scheme the Eq. (3.2) is integrated from time t 0 to t0 þ Mt through the following operations

Mt Lðum Þ 2 Mt uð2Þ ¼ uð1Þ þ Lðuð1Þ Þ 2 2 m uð2Þ Mt ð3Þ u ¼ u þ þ Lðuð2Þ Þ 3 6 3 Mt mþ1 ð3Þ ð3Þ ¼ u þ Lðu Þ; u 2 uð1Þ ¼ um þ

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and consequently the solution uðx; tÞ at a particular time level is completely known. 4. Numerical experiments and discussion In this section, the numerical solutions by the proposed method (MCB-DQM) are evaluated for some examples of Burgers’ equation. Existence of analytical solutions help to measure the accuracy of numerical methods. In the present study, the accuracy and the efficiency of this method is measured for various numerical examples by evaluating the discrete L2 and L1 error norms which are defined as follows

L2 ¼

h

N h X

uexact j



uj

i2

!1=2 ;

j¼1

  N   and L1 ¼ maxuexact  uj ; j j¼1

where uj represent the numerical solution at node j. Example 1. The Burgers’ Eq. (1.1) with a ¼ 1 is solved over the region ½0; 1:2 and the initial and boundary conditions are as given in [1,4,23]

uðx; 1Þ ¼

1 þ exp

x 1

4m

; ðx2  14Þ

and uð0; tÞ ¼ 0;

uð1:2; tÞ ¼ 0;

for t > 1:

In this problem the initial condition is taken at t ¼ 1. The exact solution of the problem is

uðx; tÞ ¼ 1þ



t t0

x t 1=2

exp

 x2  ;

for t P 1;

where t 0 ¼ exp



 1 : 8m

4mt

In this example, the parameter values are taken as m ¼ 0:005; h ¼ 0:01; Mt ¼ 0:01. The comparison of the numerical solutions obtained by MCB-DQM, at different time levels are presented with the solutions obtained by Mittal and Jain [31], Shu et al. [41] and the exact solutions in Table 1.1. In Table 1.2, L2 and L1 errors at different time levels t 6 3:50 are compared with the errors obtained by several earlier schemes. Further, the L2 and L1 errors at t ¼ 3:60 are compared with errors obtained by the three methods recently proposed by Korkmaz–Dag˘ [20] and are reported in Table 1.3. It is found that our results are much better than all the three methods. It is evident from Table 1.1, 1.2 and 1.3 that our method produces better approximate numeric solutions than almost all the earlier schemes and approaching towards the exact solutions. The absolute errors at t ¼ 3:5 are plotted in Fig. 1. It is evident from Fig. 1 that the absolute errors are very small as compared to that given by Mittal and Jain [[31] Fig. 10]. The absolute errors for different time levels are also plotted in Fig. 2. It is found that the errors are decreasing with increasing time and the maximum error is shifting towards the boundary x ¼ 1:2 only. The physical behavior of the numerical solutions at m ¼ 0:005 for different time levels t 6 3:5 is depicted in Fig. 3. Example 2. In this example, we take the particular solution of the Burgers’ Eq. (1.1), for a ¼ 1, over the region ½0; 2 as considered in [32]:

Table 1.1 Comparison of the MCB-DQM numerical solutions of Example 1 with exact solutions, for x

0.20

0.40

0.6

0.8

t

1.7 2.5 3.0 3.5 1.7 2.5 3.0 3.5 1.7 2.5 3.0 3.5 1.7 2.5 3.0 3.5

Shu et al. [41] with h ¼ 104 ; Mt ¼ :01

m ¼ 0:005.

Mittal & Jain [31]

MCB-DQM

b¼1

b ¼ 0:5

h ¼ 0:005; Mt ¼ 10

h ¼ 0:01; Mt ¼ 0:01

0.1176565 0.0800527 0.0667147 0.0571820 0.2332111 0.1591735 0.1328314 0.1139606 0.2940048 0.2347876 0.1973222 0.1697753 0.0008917 0.1103866 0.2088346 0.2119293

0.1174841 0.0798389 0.0665176 0.0570060 0.2348504 0.1596608 0.1330273 0.1140077 0.2961269 0.2376699 0.1990478 0.1708231 0.0006640 0.1036067 0.2093735 0.2143409

0.1176452 0.0799990 0.0666658 0.0571422 0.2351690 0.1599771 0.1333211 0.1142780 0.2958570 0.2381299 0.1994839 0.1712257 0.0006381 0.1021325 0.2088032 0.2145938

0.1176450 0.0799989 0.0666658 0.0571422 0.2351680 0.1599770 0.1333210 0.1142780 0.2959160 0.2381200 0.1994800 0.1712240 0.0006464 0.1020930 0.2088380 0.2145870

3

Exact value

0.1176452 0.0799990 0.0666658 0.0571422 0.2351677 0.1599769 0.1333209 0.1142779 0.2959097 0.2381207 0.1994805 0.1712242 0.0006465 0.1020957 0.2088359 0.2145869

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G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177 Table 1.2 Comparison of L2 and L1 errors in the MCB-DQM solutions of Example 1 for schemes. Methods

N

t ¼ 1:7

Mt

m ¼ 0:005 at different time levels t 6 3:50 with the errors obtained by earlier

t ¼ 2:4 3

L2  10

L1  10

3

t ¼ 3:1 3

L2  10

L1  10

3

MCB-DQM QRTDQ [24] BS.FEM [10] C.S.C.[39] Galerkin [46] QBCM1[28] QBCM2 [28] PDQ [18] CBCDQ [19]

121 101 50 50 200 200 200 200 101

0.01 0.001 0.100 0.100 0.010 0.010 0.010 0.010 0.001

0.00191 0.109 0.857 0.857 0.857 0.017 0.358 0.015

0.00777 0.434 2.576 2.576 2.576 0.061 1.211 0.056

0.00086 0.100 0.423 0.423 0.423 0.012 0.251 0.011 0.210 t ¼ 2:5

0.00308 0.339 1.242 1.242 1.242 0.058 0.807 0.064 0.680

QBCM [12] CBCM [12] QRKM [12]

200 200 200

0.010 0.010 0.010

0.072 2.466 0.026

0.311 27.577 0.091

0.051 2.111 0.031

0.189 25.15 0.115

MCB-DQM MCB-CM [31] [41] (b ¼ 0:5) [41](b ¼ 1) MCB-DQM

121 241 12001 12001 121

0.010 0.010 0.010 0.010 0.010

0.00191 0.0252 0.38421 3.08966 0.00191

0.00777 0.0994 1.34728 10.4040 0.00777

0.00778 0.0151 0.49135 2.72048 0.00778

0.00275 0.0549 1.55470 8.29747 0.00275

t ¼ 3:25 3

L1  10

L2  103

L1  103

0.00065 0.091 0.230 0.235 0.235 0.601 0.630 0.584 0.190

0.00331 0.266 0.680 0.688 0.688 4.434 4.790 4.301 0.530

0.001341

0.00918

1.129 1.925 1.111 t ¼ 3:50

8.983 21.084 8.000

0.006177 0.0117 0.525855 2.12110 0.006177

0.04335 0.0486 1.52196 5.94321 0.04335

L2  10

3

t ¼ 3:00 0.00056 0.0118 0.51508 2.39922 0.00056

0.0017 0.0414 1.5529 6.9880 0.0017

Table 1.3 Comparison of L2 and L1 errors in the MCB-DQM solutions of Example 1 for m ¼ 0:005 at t ¼ 3:6 with the errors obtained in [20]. Korkmaz & Dag˘ [20]

MCB-DQM

Method I

Method II

Method III

L2  10

0:01

0:18

0:16

0:14

L1  103

0:07

0:46

0:52

0:54

3

4.5

x 10

−5

4

Absolute Error

3.5 3 2.5 2 1.5 1 0.5 0

0

0.2

0.4

0.6

x

0.8

Fig. 1. Absolute errors in the MCB-DQM numeric solutions of Example 1 for

uðx; tÞ ¼ 2pm

sinðpxÞexpðp2 m2 tÞ þ 4 sinð2pxÞexpð4p2 m2 tÞ ; 4 þ cosðpxÞexpðp2 m2 tÞ þ 2 cosð2pxÞexpð4p2 m2 tÞ

1

1.2

m ¼ 0:005 at t ¼ 3:5 with h ¼ 0:01; Mt ¼ 0:01.

for x 2 ð0; 2Þ and t P 0;

ð4:1Þ

where the initial condition is evaluated from (4.1), and the boundary conditions are taken to be uð0; tÞ ¼ 0 and uð2; tÞ ¼ 0. In this example, we have computed L1 and L2 errors at t ¼ 0:1; 1:0 with the parameter h ¼ 0:01; Mt ¼ 0:01, and at the different values of m. The comparison of the computed errors with the errors obtained by Mittal and Jain [[31] Table 5.1] are reported in Table 2.1. It is evident that as the value of m decreases the absolute error decreases rapidly. Also, for a given value of m, the computed errors are less than that obtained by Mittal and Jain [31], and hence, the numerical solutions produced by

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G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177 x 10

4.5

−5

t=1.7 t=2.5 t=3.0 t=3.5

4

Absolute Error

3.5 3 2.5 2 1.5 1 0.5 0

0

0.2

0.4

0.6

0.8

x

Fig. 2. Absolute errors in the MCB-DQM numeric solutions of Example 1 for

1

1.2

m ¼ 0:005 at t 6 3:5 with h ¼ 0:01; Mt ¼ 0:01.

0.45 0.4

t=0.0 t=1.7 t=2.5 t=3.0 t=3.5

0.35

u(x,t)

0.3 0.25 0.2 0.15 0.1 0.05 0 0

0.2

0.4

0.6

x

0.8

Fig. 3. Physical behavior of the MCB-DQM numeric solutions of Example 1 for

1

1.2

1.4

m ¼ 0:005 at t 6 3:5 with h ¼ 0:01; Mt ¼ 0:01.

Table 2.1 Comparison of L1 and L2 in the MCB-DQM numeric solutions of Example 2 with the errors obtained in [31].

m

t ¼ 0:1

t ¼ 1:0

Mittal & Jain [31] h ¼ 0:025; Mt ¼ 103

MCB-DQM h ¼ 0:1; Mt ¼ 0:01

Mittal & Jain [31] h ¼ 0:025; Mt ¼ 103

MCB-DQM h ¼ 0:1; Mt ¼ 0:01

L1

L2

L1

L2

L1

L2

L1

L2

102

4:41E  03

3:55E  03

3:89E  03

3:41E  03

3:13E  02

2:66E  02

2:92E  02

2:63E  02

103

4:60E  05

3:72E  05

4:09E  05

3:55E  05

4:45E  04

3:59E  04

3:93E  04

3:45E  04

104

4:62E  07

3:74E  07

4:11E  07

3:56E  07

4:61E  06

3:72E  06

4:09E  06

3:55E  06

105

4:62E  09

3:74E  09

4:11E  09

3:56E  09

4:62E  08

3:74E  08

4:11E  08

3:56E  08

106

4:62E  11

3:74E  11

4:11E  11

3:56E  11

4:62E  10

3:74E  10

4:11E  10

3:56E  10

our method are accurate than [31], for the current problem. Also, the absolute errors at t ¼ 1 taking v ¼ 104 ; 105 ; 106 have been shown in Fig. 4. The physical behavior of the numerical solutions at m ¼ 0:01 for different time levels are depicted in Fig. 5.

Example 3. The initial and the boundary conditions of the Burgers’ Eq. (1.1) over the region ½0; 1 with a ¼ 1, are considered as in [26,34,16]:

uðx; 0Þ ¼ 4xð1  xÞ and uð0; tÞ ¼ uð1; tÞ ¼ 0: The description of the numerical solutions of this example for

m ¼ 0:1 and 0:01 is given below:

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G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177

8

−4

x 10

v=1.0 E−06 v=1.0 E−05 v=1.0 E−04

6 4

Errors

2 0 −2 −4 −6 −8 0

0.2

0.4

0.6

0.8

1

x

1.2

Fig. 4. Errors in the MCB-DQM numeric solutions of Example 2 at t ¼ 1 for

1.4

1.6

1.8

2

m ¼ 104 ; 105 and 106 taking h ¼ 0:01; Mt ¼ 0:01.

Table 3.1 Comparison of the computed results form Example 3 to Jiwari et al. [16], Kutluay et al. [26] for

m ¼ 0:1, and the exact solutions.

x

T

[26] Mt ¼ 0:0001 h ¼ 0:0125

[16] Mt ¼ 0:0001 h ¼ 0:04

MCB-DQM Mt ¼ 0:001 h ¼ 0:025

Exact

0.25

0.4 0.8 1.0 3.0 0.4 0.8 1.0 3.0 0.4 0.8 1.0 3.0

0.32091 0.20211 0.16782 0.02828 0.58788 0.37111 0.30183 0.04185 0.65054 0.39068 0.30057 0.03106

0.31744 0.19952 0.16557 0.02775 0.58443 0.36733 0.29830 0.04106 0.64556 0.38526 0.29582 0.03043

0.317526 0.199558 0.165601 0.027761 0.584541 0.367406 0.298352 0.041069 0.645641 0.385369 0.295885 0.030443

0.31752 0.19956 0.16560 0.02775 0.58454 0.36740 0.29834 0.04106 0.64562 0.38534 0.29586 0.03044

0.50

0.75

Table 3.2 Comparison of the MCB-DQM numerical solutions of Example 3 with the numeric solutions due to Mittal & Jain [31] for m ¼ 0:01. x

t

[31] h ¼ 0:025 Mt ¼ 0:001

MCB-DQM h ¼ 0:025 Mt ¼ 0:001

Exact

0.25

0.4 0.6 0.8 1.0 3.0

0.36225 0.28202 0.23044 0.19468 0.07613

0.36226 0.28204 0.23045 0.19469 0.07613

0.36226 0.28204 0.23045 0.19469 0.07613

0.50

0.4 0.6 0.8 1.0 3.0

0.68368 0.54832 0.45371 0.38567 0.15218

0.68368 0.54832 0.45371 0.38568 0.15218

0.68368 0.54832 0.45371 0.38568 0.15218

0.75

0.4 0.6 0.8 1.0 3.0

0.92052 0.78300 0.66272 0.56932 0.22782

0.92049 0.78297 0.66271 0.56932 0.22775

0.92050 0.78299 0.66272 0.56932 0.22774

(a) In Table 3.1, we have computed the numerical solutions with parameter values m ¼ 0:1; h ¼ 0:025; Mt ¼ 0:001 at different time levels. The comparison of our results with the results obtained in [16,26] are repored in Table 3.1. Also, the numerical solutions obtained by Jiwari et al. [[16]Table 3] are better than the solutions obtained in [26,34]. Thus, we found that our solutions are more accurate than the solutions obtained in [16,26,34] and approaching to exact solutions.

G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177

1

1

0.8

0.8

0.6

0.6

u(x,t)

u(x,t)

174

0.4

0.2

0

0.2 0.2 0 0

0.6

0.4

0.6

0.8

x

0 0.2

0 0

0.4 0.2

0.4

0.2

0.8 1

0.4 0.4

0.6 0.6

0.8

t

1

x

Fig. 6. Physical behavior of the MCB-DQM numeric solutions of Example 3 at

0.8 1

t

1

m ¼ 0:01 (left) and at m ¼ 0:1 (right) for t 6 1 with h ¼ 0:025; Mt ¼ 0:001.

0.1

u(x,t)

0.05

0

−0.05

−0.1 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

x

Fig. 5. Physical behavior of the MCB-DQM numeric solutions of Example 2 for

0.4

0.6

0.8

1

t

m ¼ 0:01 at t 6 1 with m ¼ 0:01; h ¼ 0:01; Mt ¼ 0:01.

Table 4.1 Comparison of the MCB-DQM numeric solutions of Example 4 for m ¼ 1:0 with the numercal solutions obtained by Dag˘ et al. [12], Korkmaz [22], Mittal & Jain [31], and exact solutions. x

t

[12] h ¼ 0:0125

[22] h ¼ 0:025 Mt ¼ 0:000125

MCB-DQM h ¼ 0:025 Mt ¼ 0:00025

Exact

Mt ¼ 104

[31] h ¼ 0:025 Mt ¼ 0:00025

0.25

0.4 0.6 0.8 1.0 3.0

0.01357 0.00189 0.00026 0.00004 0.00000

0.01354 0.00188 0.00026 0.00004 0.00000

0.01363 0.00190 0.00026 0.00003 0.00000

0.0135710 0.0018888 0.0002624 0.0000365 0.0000000

0.01357 0.00189 0.00026 0.00004 0.00000

0.50

0.4 0.6 0.8 1.0 3.0

0.01923 0.00267 0.00037 0.00005 0.00000

0.01920 0.00266 0.00037 0.00005 0.00000

0.01932 0.00269 0.00037 0.00005 0.00000

0.0192336 0.0026719 0.0003712 0.0000516 0.0000000

0.01923 0.00267 0.00037 0.00005 0.00000

0.75

0.4 0.6 0.8 1.0 3.0

0.01362 0.00189 0.00026 0.00004 0.00000

0.01360 0.00188 0.00026 0.00004 0.00000

0.01369 0.00190 0.00026 0.00003 0.00000

0.0136298 0.0018899 0.0002625 0.0000365 0.0000000

0.01363 0.00189 0.00026 0.00004 0.00000

(b) In Table 3.2, we have computed the numerical solutions with parameter values m ¼ 0:01; h ¼ 0:025; Mt ¼ 0:001. Since the numerical solutions obtained by Mittal and Jain [[31] Table 5.1] are better than the solutions obtained in [1,2,45], hence the comparison of the obtained solutions with the exact solutions as well as with the solutions obtained in [31] are reported in Table 3.2. It is observed that our solutions are accurate than the results obtained in [1,2,31,45].

175

G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177 Table 4.2 Comparison of the MCB-DQM numeric solutions of Example 4 for x

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

m ¼ 1:0 with the solutions obtained by Dag˘ et al. [12].

[12] h ¼ 0:0125

MCB-DQM h ¼ 0:025

[12] h ¼ 0:00625

MCB-DQM h ¼ 0:0125

Mt ¼ 105

Mt ¼ 104

Mt ¼ 105

Mt ¼ 104

0.10952 0.20975 0.29184 0.34785 0.37149 0.35896 0.30983 0.22776 0.12065

0.109530 0.209771 0.291860 0.347874 0.371517 0.358981 0.309845 0.227773 0.120666

0.10953 0.20977 0.29186 0.34788 0.37153 0.35900 0.30986 0.22778 0.12067

0.109526 0.209766 0.291855 0.347869 0.371512 0.358975 0.309839 0.227766 0.120659

Exact

0.10954 0.20979 0.29190 0.34792 0.37158 0.35905 0.30991 0.22782 0.12069

Table 4.3 Comparison of the MCB-DQM numeric solutions of Example 4 for m ¼ 0:1 with the numercal solutions obtained by Dag˘ et al. [12], Korkmaz [22], Mittal & Jain [31], and exact solutions. x

t

[12] h ¼ 0:0125

[22] h ¼ 0:025 Mt ¼ 0:00125

MCB-DQM h ¼ 0:025 Mt ¼ 0:004

Exact

Mt ¼ 104

[31] h ¼ 0:025 Mt ¼ 0:0025

0.25

0.4 0.6 0.8 1.0 3.0

0.30890 0.24075 0.19569 0.16258 0.02720

0.30892 0.24077 0.19572 0.16261 0.02718

0.30910 0.24093 0.19586 0.16274 0.02720

0.3089280 0.2407550 0.1956840 0.1625700 0.0272047

0.30889 0.24074 0.19568 0.16256 0.02720

0.50

0.4 0.6 0.8 1.0 3.0

0.56965 0.44723 0.35925 0.29192 0.04019

0.56970 0.44729 0.35930 0.29195 0.04016

0.56973 0.44736 0.35943 0.29213 0.04032

0.5696530 0.4472170 0.3592450 0.2919250 0.0402085

0.56963 0.44721 0.35924 0.29192 0.04021

0.75

0.4 0.6 0.8 1.0 3.0

0.62538 0.48715 0.37385 0.28741 0.02976

0.62520 0.48694 0.37365 0.28724 0.02974

0.62573 0.48760 0.37434 0.28788 0.29881

0.6253490 0.4872040 0.3739350 0.2874930 0.0297753

0.62544 0.48721 0.37392 0.28747 0.02977

1 1

0.8

u(x,t)

u(x,t)

0.8 0.6 0.4

0.4 0.2

0

0.2 0 0

0.6

0.2 0.4 0.2

0.4

0.6 0.6

0.8

x

0.8 1

1

t

Fig. 7. Physical behavior of MCB-DQM numeric solutions at

0 0

0 0.2

0.2

0.4

0.4

0.6

0.6

x

0.8

0.8 1

1

t

m ¼ 0:1 (left) and at m ¼ 1:0 (right) of Example 4 for t 6 1 with h ¼ 0:025; Mt ¼ 0:001.

The physical behavior of the current problem for different time levels t 6 1, is depicted in Fig. 6. The similar figures are also depicted in [1,31]. Example 4. In this example, the initial condition for the Burgers’ Eq. (1.1), for a ¼ 1, over the region ½0; 1 is considered as given in [22]:

176

G. Arora, B.K. Singh / Applied Mathematics and Computation 224 (2013) 166–177

uðx; 0Þ ¼ sinðpxÞ;

ð4:2Þ

and the boundary conditions

uð0; tÞ ¼ uð1; tÞ ¼ 0:

ð4:3Þ

The analytical solution of this problem is given by Cole [11] in terms of an infinite series as

uðx; tÞ ¼

P 2 2 1 4pm 1 j¼1 jIj ð2pmÞ sinðjpxÞexpðj p mtÞ ; P 2 2 1 I0 ð2p1 mÞ þ 2 1 j¼1 I j ð2pmÞ cosðjpxÞexpðj p mtÞ

where Ij are the modified Bessel’s functions. The numerical solutions of this example for different values of

ð4:4Þ

m are given below:

(a) In Table 4.1, we have computed the numerical solutions of this example at different time levels with parameter values m ¼ 1:0; h ¼ 0:025 and Mt ¼ 0:00025. The comparison of our results with the exact solutions as well as the solutions obtained in [12,31,22] are reported in Table 4.1. It is found that our method produces comparable results as obtained in [12,31,22]. Further, the numerical solutions are computed at t ¼ 0:1 with parameter values h ¼ 0:025; 0:0125 m ¼ 1:0; Mt ¼ 0:0001, and compared with the solutions obtained in Dag˘ et al. [[12] Table 1]. The results are reported in Table 4.2. It is found that we require half of the grid points to produces the results similar to [12]. (b) Also, the numerical solutions of this example are computed at different time levels with parameter values m ¼ 0:1; h ¼ 0:025 and Mt ¼ 0:004. The comparison of our solutions with the exact solutions as well as the solutions obtained in [12,31,22] are reported in Table 4.3. It is evident that the MCB-DQM numerical solutions are better as compared to the results obtained in [12,31,22]. The physical behavior of this example for

m ¼ 0:1 and m ¼ 1:0 are depicted in Fig. 7.

5. Conclusion In this paper, we have developed a method (MCB-DQM) to solve nonlinear partial differential equations. In this method, the modified cubic B-splines are used in differential quadrature method as basis function to evaluate the weighting coefficients, and hence the derivatives. In this way, we find a system of ordinary differential equations (ODEs) which is solved by SSP-RK43 scheme. To check the efficiency and accuracy of the method, four examples of Burgers’ equation are included with their numerical solutions, L2 and L1 errors and done the comparisons with the results given in the literature. It is evident that our method produces better results as compared to the results obtained by almost all the schemes available in the literature, and approaching to the exact solutions. The cubic B-spline basis functions are modified in such a way that it reduces matrix size and complexity when applied with differential quadrature method. In this method we require less number of grid points as compared to the earlier given methods. This method is hence easy to implement and economical in terms of data complexity, which results in less errors and so, the easiness of the implementation of MCB-DQM, and low memory storage can be counted as advantages of this method. Also, this method can be easily implemented to solve two-dimensional nonlinear partial differential equations. Acknowledgement The authors thank the anonymous referees for their time, effort, and extensive comments which improve the quality of the presentation of the paper. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.amc.2013.08.071. References [1] A. Asaithambi, Numerical solution of the Burgers’ equation by automatic differentiation, Appl. Math. Comput. 216 (2010) 2700–2708. [2] K. Altparmak, Numerical solution of Burgers’ equation with factorized diagonal Padè approximation, Int. J. Numer. Methods Heat Fluid Flow 21 (3) (2011) 310–319. [3] E.N. Aksan, A. Ozdes, A numerical solution of Burgers’ equation, Appl. Math. Comput. 156 (2004) 395–402. [4] E.N. Aksan, Quadratic B-spline finite element method for numerical solution of the Burgers’ equation, Appl. Math. Comput. 174 (2006) 884–896. [5] H. Bateman, Some recent researches on the motion of fluids, Mon. Weather Rev. 43 (1915) 163–170. [6] J.M. Burgers, Mathematical example illustrating relations occurring in the theory of turbulent fluid motion, Trans. Roy. Neth. Acad. Sci. Amsterdam 17 (1939) 1–53. [7] J.M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., vol. I, Academic Press, New York, 1948, pp. 171–199.

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