An Approach with Haar Wavelet Collocation Method for Numerical Simulations of Modified KdV and Modified Burgers Equations

S.Saha Rayand A.K.Gupta

1　Introduction

Generalized modified KdV equation[Wazwaz(2009)]is a nonlinear partial differential equation of the form

whereqandrare parameters.

Generalized modified Burgers’equation[Irk(2009)]is a nonlinear partial differential equation of the form

wherepandνare parameters.

The modified Korteweg-de Vries(mKdV)equations are most popular soliton equations and have been extensively investigated.The modified KdV equation is of important significance in many branches of nonlinear science field.The mKdV equa-tion appears in many fields such as Alfvén waves in a collisionless plasma,acoustic waves in certain anharmonic lattices,models of traffic congestion,transmission lines in Schottky barrier,ion acoustic soliton,elastic media etc.[Yan(2008)]

Similarly,the modified Burgers’equation[Bratsos(2011)]has the strong nonlinear aspects of the governing equation in many practical transport problems such as nonlinear waves in medium with low frequency absorption,wave processes in thermoelastic medium,turbulence transport,ion reflection at quasi perpendicular shocks,transport and dispersion of pollutants in rivers and sediment transport etc.Various mathematical methods such as Petrov-Galerkin method[Roshan and Bhamra(2011)],Quintic spline method[Ramadan and El-Danaf(2005)],Sextic B-spline collocation method[Irk(2009)],local discontinuous Galerkin method[Zhang,Yu and Zhao(2013)],and Lattice Boltzmann model[Duan,Liu and Jiang(2008)]have been used in attempting to solve modified Burgers’equations.Dehghan et al.have applied mixed finite difference and Galerkin methods for solving the Burgers [Dehghan, Saray and Lakestani(2014)]and Burgers-Huxley[Dehghan,Saray and Lakestani(2012)]equations.Generalized Benjamin–Bona–Mahony–Burgers equation[Dehghan,Abbaszadeh and Mohebbi(2014)]and KdV equations[Dehghan and Shokri(2007)]have also been investigated by Dehghan et al.via the method of radial basis functions.In 2008,Alipanah and Dehghan have solved the population balance equations by applying rationalized Haar functions.

Zhi-Zhong et al.(2008)improve a numerical method based on two types of wavelets viz.the Haar wavelet and biorthogonal wavelet to compute the band structures of 2D phononic crystals consisting of general anisotropic materials.In 2011,Zhou et al.proposed an efficient wavelet-based algorithm for solving a class of fractional vibration,diffusion and wave equations with strong nonlinearities.Yi and Chen(2012)and Wang et al.(2013)applied Haar wavelet operational matrix method to solve a class of fractional partial differential equations.Using the Haar wavelet operational matrix of fractional order differentiation,the fractional partial differential equations have been reduced to Sylvester equation.Wei et al.(2012)present a computational method for solving space-time fractional convection-diffusion equations with variable coefficients which is based on the Haar wavelets operational matrix of fractional order differentiation.They also exhibit error analysis in order to show the efficiency of the method.Saha Ray and Gupta(2013)proposed Haar wavelet collocation method for solving generalized Burger-Huxley and Huxley equations.

The Haar wavelet method consists of reducing the problem to a set of algebraic equation by expanding the term,which has maximum derivative.Our aim in the present work is to implement the Haar wavelet method to stress its power in handling nonlinear equations,so that one can execute it to various types of strong nonlinear equations.

This paper is systematized as follows:in Section 1,introduction to modified KdV and modified Burgers’equation is discussed.In Section 2,the mathematical preliminaries of Haar wavelet are presented.Sections 3 and 5 define the mathematical models of modified KdV and modified Burgers’equation respectively.The Haar wavelet method has been applied to solve modified KdV and modified Burgers’equation in Sections 4 and 6 respectively.The convergence of Haar wavelet method is discussed in Section 7.The numerical results and discussions are discussed in Section 8 and Section 9 concludes the paper.

2　Haar wavelets and the operational matrices

The Haar wavelet family forx∈[0,1)is defined as follows[Debnath(2002);Lepik(2007);Saha Ray(2012)]

In these formulae integerm=2j,j=0,1,2,...,Jindicates the level of the wavelet;k=0,1,2,...,m?1 is the translation parameter.Maximum level of resolution isJ.The indexiis calculated from the formulai=m+k+1;in the case of minimal valuesm=1,k=0 we havei=2.The maximum possible value ofi=2M=2J+1.It is assumed that the valuei=1 corresponds to the scaling function for which

It is expedient to introduce the 2M×2MmatricesH,P,QandRwith the elementsH(i,l)=hi(xl),P(i,l)=pi(xl),Q(i,l)=qi(xl)andR(i,l)=ri(xl).

3　Generalized modified KdV equation

Consider the generalized modified KdV equation[Kaya(2005);Wazwaz(2004)]

4　Application of Haar wavelet method for solving modified KdV equation

Haar wavelet solution ofu(x,t)is sought by assuming thatcan be expanded in terms of Haar wavelets as

Integrating eq.(12)with respect totfromtstotand thrice with respect toxfrom 0 tox,the following equations are obtained

By using the boundary condition atx=1,eq.(15)becomes

It is obtained from eq.(6)that,

Substituting the above equations in eq.(8),we have

From the above equation,the wavelet coefficientsas(i)can be successively calculated.This process starts with

5　Modified Burgers’equation

Consider the generalized modified Burgers’equation[Roshan and Bhamra(2011)]

wherepis a positive constant andν(>0)can be interpreted as viscosity.

To show the effectiveness and accuracy of proposed scheme,we consider two test examples takingp=2.The numerical solutions thus obtained are compared with the analytical solutions as well as available numerical results.

The initial condition associated with eq.(21)will be

with boundary conditions

6　Haar wavelet based scheme for modified Burgers’equation

Now,integrating eq.(22)with respect totfromtstotand twice with respect toxfrom 0 toxthe following equations are obtained

Substituting equation(27)and(28)in eqs.(24),(25)and(26),we have

Discretising the above results by assumingwe obtain

Substituting equations(31),(32),(33)and(34)in eq.(21),we have

From eq.(35),the wavelet coefficientsas(i)can be successively calculated.This process starts with

Example 1.Consider modified Burgers’equation with the following initial and boundary conditions[Roshan and Bhamra(2011);Ramadan and El-Danaf(2005)]

The exact solution of eq.(21)is given by[Roshan and Bhamra(2011);Ramadan and El-Danaf(2005)]

Example 2.In this example,we consider modified Burgers’equation with initial and boundary conditions in the following form

In case of example 1,the Haar wavelet numerical solutions have been compared with the results obtained by Ramadan et al.(2005),using the collocation method with quintic splines and in case of example 2,the solutions have been compared with the results obtained by Duan et al.(2008),using 2-bit lattice Boltzmann method(LBM).Tables 1 and 2 cite the comparison of Haar wavelet solution with LBM and quintic splines numerical solutions att=0.4 andt=2,and the numerical solutions at different time stages are exhibited in Fig.6.

Table 1:Comparison of Haar wavelet solutions with the LBM solutions and 5-Splines solution of modified Burgers’equation(example 2)att=0.4 and ν =0.01.

Table 2:Comparison of Haar wavelet solutions with the LBM solutions and 5-Splines solution of modified Burgers’equation(example 2)att=2.0 and ν =0.01.

7　Error of collocation method

From eq.(3),the Haar wavelet family foris defined as follows

Hence,usingE(x)as least square of the error on ?,we have

8　Numerical Results and discussions

The error function is given by

The errors for modified Burgers’equation are measured using two different norms,namelyL2andL∞,defined by

The following table exhibits theL2andL∞error norm for modified Burgers’equation takingp=2 andν=0.001 and different values oft.In tables 3,Jis taken as 5 i.e.M=32 and?tis taken as 0.001.

Table 3:L2and L∞ error norm for modified Burgers’equation(example 1)at different values of t with ν =0.001 and ?t=ts+1?ts=0.001.

The following tables show the comparisons of the exact solutions with the approximate solutions of modified KdV equation takingq=6,r=?0.001 and different values oft.In tables 4-7,Jis taken as 3 i.e.M=8 and?tis taken as 0.0001.

In case ofr=?0.001,theR.M.S.errorbetween the numerical solutions and the exact solutions of modified KdV equations fort=0.2,0.5,0.8 and 1 are 0.000017137,0.0000433416,0.0000695581 and 0.0000870423 respectively and forr=?0.1andt=0.2,0.5,0.8and1theR.M.S.erroris found to be0.00209359,0.00624177,0.011631 and 0.0159099 respectively.In the following tables[8-11]alsoJhas been taken as 3 i.e.M=8 and?tis taken as 0.0001.

Figures 1-4 represent the comparison graphically between the numerical and exact solutions of modified Burgers’equation for different values oftandν=0.001.The behaviour of numerical solutions of modified Burgers’equation is cited in figure 5 and 6.Similarly,in case of modified KdV equation,the Figures 7-11 demonstrate

the comparison graphically between the numerical and exact solutions for different values oftandr.

Table 4:The absolute errors for modified KdV equation at various collocation points of x with t=0.2 and r=?0.001.

Table 5:The absolute errors for modified KdV equation at various collocation points of x with t=0.5 and r=?0.001.

Table 6:The absolute errors for modified KdV equation at various collocation points of x with t=0.8 and r=?0.001.

Table 7:The absolute errors for modified KdV equation at various collocation points of x with t=1 and r=?0.001.

Table 8:The absolute errors for modified KdV equation at various collocation points of x with t=0.2 and r=?0.1.

Table 9:The absolute errors for modified KdV equation at various collocation points of x with t=0.5 and r=?0.1.

Table 11:The absolute errors for modified KdV equation at various collocation points of x with t=1 and r=?0.1.

Figure 1:Comparison of Numerical solution and exact solution of modified Burger’s equation(example 1)when t=2 and ν =0.001.

Figure 2:Comparison of Numerical solution and exact solution of modified Burger’s equation(example 1)when t=4 and ν =0.001.

Figure 3:Comparison of Numerical solution and exact solution of modified Burger’s equation(example 1)when t=6 and ν =0.001.

Figure 5:Behaviour of numerical solutions for modified Burgers’equation(example 1)when ν =0.001 and ?t=0.001 at times t=2,4,6 and 8.

Figure 6:Behaviour of numerical solutions for modified Burgers’equation(example 2)when ν =0.01 and ?t=0.001 at times t=0.4,0.8,2 and 3.

Figure 7:Comparison of Numerical solution and exact solution of modified KdV equation when t=0.2 and r=?0.001.

Figure 8:Comparison of Numerical solution and exact solution of modified KdV equation when t=0.5 and r=?0.001.

Figure 9:Comparison of Numerical solution and exact solution of modified KdV equation when t=0.8 and r=?0.001.

Figure 10:Comparison of Numerical solution and exact solution of modified KdV equation when t=1.0 and r=?0.001.

Figure 11:Comparison of Numerical solution and exact solution of modified KdV equation when t=0.2 and r=?0.1.

9　Conclusions

In this paper,the modified KdV equation and modified Burgers’equation have been solved by Haar wavelet method.The results thus found are then compared with the exact solutions as well as solutions available in open literature.These have been reported in tables and also have been shown in the graphs.These results demonstrated in Tables justify the accuracy and efficiency of the proposed schemes based on Haar wavelet.The numerical schemes are reliable and convenient for solving modified KdV and modified Burgers’equations.The main advantages of the scheme are its simplicity and applicability.Also it has less computational errors.Moreover,the errors may be reduced significantly if we increase level of resolution which prompts more number of collocation points.

Acknowledgement:This research work was financially supported by DST,Government of India under Grant No.SR/S4/MS.:722/11.

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