CAS wavelet quasi-linearization technique for the generalized Burger–Fisher equation

In this article, we propose a method for the solution of the generalized Burger–Fisher equation. The method is developed using CAS wavelets in conjunction with quasi-linearization technique. The operational matrices for the CAS wavelets are derived and constructed. Error analysis and procedure of implementation of the method are provided. We compare the results produce by present method with some well known results and show that the present method is more accurate, efficient, and applicable.


Introduction
The Burger-Fisher equation has important applications in various fields of financial mathematics, gas dynamic, traffic flow, applied mathematics, and physics. This equation shows a prototypical model for describing the interaction between the reaction mechanism, convection effect, and diffusion transport [1]. Consider the generalized Burger-Fisher equation: subject to the initial and boundary conditions: uð0; tÞ ¼ EðtÞ : uð1; tÞ ¼ FðtÞ : The exact solution is given in Chen and Zhang [2]: where a; b; and c are non-zero parameters. Wavelet analysis is a new development in the area of applied mathematics. Wavelets are a special kind of functions which exhibits oscillatory behavior for a short period of time and then die out. In wavelets, we use a single function and its dilations and translations to generate a set of orthonormal basis functions to represent a function. We define wavelet (mother wavelet) by Radunovic [3]: where a and b are called scaling and translation parameter, respectively. If jaj\1, the wavelet (3) is the compressed version (smaller support in time domain) of the mother wavelet and corresponds to mainly higher frequencies. On the other hand, when jaj [ 1, the wavelet (3) has larger support in time domain and corresponds to lower frequencies.
Discretizing the parameters via a ¼ 2 Àk and b ¼ n2 Àk , we get the discrete wavelets transform as: w k;n ðxÞ ¼ 2 k 2 wð2 k x À nÞ: These wavelets for all integers k and n produce an orthogonal basis of L 2 ðRÞ. It is somewhat surprising that  [4][5][6][7] in which CAS method is used for the solution of integro-differential equations, and CAS wavelets is not implemented for the solution of nonlinear Lane Emdentype equation. In Yi et al. [4], CAS wavelet method is utilized for the solution of integro-differential equations with a weakly singular kernel of fractional order. In addition, error analysis of the CAS wavelets is provided. The CAS wavelets operational matrices are implemented for the numerical solution of nonlinear Volterra integro-differential equations of arbitrary order in Saeedi et al. [5]. CAS wavelet approximation method is presented for the solution of Fredholm integral equations in Yousefi and Banifatemi [6]. The operational matrices are utilized to convert the Fredholm integral equation into a system of algebraic equations. In Shamooshaky et al. [7], authors presented a CAS wavelet method for solving boundary integral equations with logarithmic singular kernels which occur as reformulations of a boundary value problem for Laplace's equation.
The purpose of this article is to propose a numerical method for solving the generalized Burger-Fisher equation using CAS wavelets in conjunction with quasi-linearization technique. The properties of quasi-linearization technique are used to discretize the nonlinear partial differential equation and then utilize the properties of CAS wavelets to convert the obtained discrete partial differential equation into a Sylvester system. The solution of the obtained system provides the values of CAS wavelets coefficients which lead to the solution of the generalized Burger-Fisher equation.

Function approximations
We can expand any function yðxÞ 2 L 2 ½0; 1Þ into truncated CAS wavelet series as: . . .; w 2 k À1;M ðxÞ T : Any function of two variables uðx; tÞ 2 L 2 ½0; 1Þ Â ½0; 1Þ can be approximated as: The collocation points for the CAS wavelets are taken as . . .;m. The CAS wavelets matrix Wm ;m is given as: ; . . .; W 2m À 1 2m In particular, we fix k ¼ 2; M ¼ 1, we have n ¼ 0; 1; 2; 3 ; m ¼ À1; 0; 1 and i ¼ 1; 2; . . .; 12, the CAS wavelets matrix is given as: The CAS wavelets operational matrix of integration For simplicity, we write (5) as: where c l ¼ c m;n , w l ¼ w m;n ðxÞ. The index l is determined by the equation l ¼ Mð2n þ 1Þ þ ðn þ m þ 1Þ andm ¼ 2 k ð2M þ 1Þ. In addition, C ¼ ½c 1 ; c 2 ; . . .; cm T , WðxÞ ¼ ½w 1 ðxÞ; w 2 ðxÞ; . . .; wmðxÞ T : Equation (6) can be written as: where C ism Âm 0 coefficient matrix and its entries are c l;p ¼ \w l ðxÞ; \uðx; tÞ; w p ðtÞ [ [ : The index l and p are determined by the equations An arbitrary function uðx; tÞ 2 L 2 ½0; 1Þ Â ½0; 1Þ, can be expanded into block-pulse functions [8] as: where a i;j are the coefficients of the block-pulse functions b i and b j . The CAS wavelets can be expanded intom-set of block-pulse functions as: The qth integral of block-pulse function can be written as: where q [ 0 and F q mÂm is given in Kilicman and Al Zhour [8] with The CAS wavelets operational matrix of integration P q mÂm of integer order q are utilize for solving differential equations.
In particular, for k ¼ 2, M ¼ 1, q ¼ 2, the CAS wavelet operational matrix of integration P 2 12Â12 is given by: This phenomena makes calculations fast, because the operational matrices Wm Âm and P q mÂm contains many zero entries.

CAS wavelets operational matrix of integration for boundary value problems
We need another operational matrix of fractional integration while solving boundary value problems. In this subsection, we drive an operational matrix of integration for dealing with the boundary conditions while solving boundary value problem. Let gðxÞ 2 L 2 ½0; 1 be a given function, then gðxÞI q x¼1 w n;m ðxÞ ¼ gðxÞ CðqÞ Since the CAS wavelets are supported on the intervals where Q q n;m ¼ 2 Expand the Eq. (13) at the collocation points, x i ¼ 2iÀ1 2m , where i ¼ 1; 2; :::;m, to obtain where G 1Âm ¼ ½gðx 1 Þ; gðx 2 Þ; :::; gðxmÞ, In particular, for k ¼ 2; M ¼ 1; q ¼ 2; and gðxÞ ¼ x 2 sinðxÞ, we have Quasi-linearization [9] The quasi-linearization approach is a generalized Newton-Raphson technique for functional equations. It converges quadratically to the exact solution, if there is convergence at all, and it has monotone convergence.
Quasi-linearization for the nonlinear partial differential equations is as follows. Given the problem of the form: with the initial condition uðx; 0Þ ¼ hðxÞ; and boundary conditions of the form: where g is the nonlinear function of u and u x . Quasi-linearization approach for Eq. (15) implies: with the initial and boundary conditions of the form: Starting with an initial approximation u 0 ðx; tÞ, we have a linear equation for each u rþ1 ; r ! 0:

Procedure of implementation
In this section, the procedure of implementing the method for nonlinear partial differential equation is explained. The procedure begins with the conversion of nonlinear partial differential equation into discretize form by quasi-linearization technique, explained in Sect. 3. Next the discretized nonlinear partial differential equation is solved by CAS wavelet operational matrix method.
Consider the following discretized nonlinear partial differential equation: Approximate the highest order term by CAS wavelet quasilinearization method as: Applying the integral operator on above equation, we have where p(t) and q(t) are pðtÞ ¼ h 2 ðtÞ À h 1 ðtÞ À ðI 2 x¼1 W T ðxÞÞC rþ1 WðtÞ qðtÞ ¼ h 1 ðtÞ: By putting the values of p(t) and q(t) in u rþ1 ðx; tÞ, we get Equation (17) implies that We make substitution as: G ¼ f ðx; tÞ À dðxÞh 1 ðtÞ À dðxÞ ðh 2 ðtÞ À h 1 ðtÞÞx À bðxÞðh 2 ðtÞ À h 1 ðtÞÞ and G ¼ W T ðxÞMWðtÞ for simplification and get Apply second-order integral on above equation to get Now, by equating Eqs. (18) and (19) and simplification, it is WðtÞ þ g 2 ðxÞt þ g 1 ðxÞ À ðh 2 ðtÞ À h 1 ðtÞÞx À h 1 ðtÞÞðWðtÞ À1 Þ: For simplification let kðx; tÞ ¼ g 2 ðxÞt þ g 1 ðxÞ À ðh 2 ðtÞ À h 1 ðtÞÞx À h 1 ðtÞ above equation at collocation points which can be written in matrix form as: After simplification, we obtain the sylvester equation: where x mÂm 0 þ KÞðW À1 Þ, and, A, B and D are diagonal matrices, which are given by: The matrix K is defined as From Eq. (21), we get C rþ1 which is used in Eq. (18) to get the solution u rþ1 at the collocation points.

Error analysis
Lemma If the CAS wavelets expansion of a continuous function u rþ1 ðx; tÞ converges uniformly, then the CAS wavelets expansion converges to the function u rþ1 ðx; tÞ.
Multiply both sides of Eq. (22) by w p;q ðtÞ and w r;s ðxÞ , then integrating from 0 to 1 with respect to x as well as t, we obtain (23) using orthonormality of CAS wavelet: Thus, c rþ1 pq;rs ¼ hhv rþ1 ðx; tÞ; w p;q ðxÞi; w r;s ðtÞi for p; r N, q; s Z. This implies that u rþ1 ðx; tÞ ¼ v rþ1 ðx; tÞ.
Theorem Assume that u rþ1 ðx; tÞL 2 ð½0; 1 Â ½0; 1Þ is a differentiable function with bounded partial derivative on ð½0; 1 Â ½0; 1Þ that is 9 c [ 0; 8 ðx; tÞ ð½0; 1 Â ½0; 1Þ : o 2 xo 2 t j c: The function u rþ1 ðx; tÞ is expanded as an infinite sum of the CAS wavelets and the series converges uniformly to u rþ1 ðx; tÞ, that is u rþ1 ðx; tÞ ¼ CAS j ð2 k 0 t À n þ 1Þdxdt: Let 2 k x À n þ 1 ¼ p and 2 k 0 t À i þ 1 ¼ q then we have c rþ1 nm;ij ¼ Use integration with respect to p to get Now, applying integration with respect to q, we obtain Again, integrating with respect to p and q, we obtain ðÀcosð2jpqÞ À sinð2jpqÞÞ j j 2 dp 2 dq 2 : Hence, the series P 1 n¼0 P mZ P 1 i¼0 P jZ c rþ1 nm;ij is absolutely convergent. In addition, we can obtain ðx; tÞ converges to u rþ1 ðx; tÞ as k; k 0 ; M; M 0 À! 1. Since u rþ1 ðx; tÞ is obtained at ðr þ 1Þth iteration of quasi-linearization technique so according to the convergence analysis of quasi-linearization technique [9] which states that u rþ1 ðx; tÞ converges to u(x, t) as r À! 1, if there is convergence at all. This suggest that solution by CAS wavelet quasi-linearization technique u k;k 0 ;M;M 0 rþ1 ðx; tÞ converges to u(x, t) when k, k 0 , M, M 0 and r À! 1.

Applications of CAS wavelet quasilinearization technique
Consider the generalized Burgers-Fisher equation: subject to the initial and boundary conditions: Implement the CAS wavelet quasi-linearization technique on Eq. (25), as described in Sect. 4, we get the following results as given in Tables 1, 2, and 3, and Fig. 1. We consider the three different forms of Eq. (25) using different values of a; b and c: E RTDM ; E VIM ; E ADM and E CAS represents the absolute error by reduced differential transform method, variational iteration method, Adomian decomposition method, and present method, respectively.
Solution of generalize Burger-Fisher equation for a ¼ 0:001; b ¼ 0:001 and c ¼ 1 by present method at M ¼ M 0 ¼ 5; k ¼ k 0 ¼ 4 and r ¼ 4 is given in Table 1. The obtained results are compared with the results obtained from reduced differential transform method (RDTM) [10] and variational iteration method (VIM) [10]. Table 2 is used to list the results of generalized Burger-Fisher equation at a ¼ 0:001; b ¼ 0:001 and c ¼ 2. We implement the proposed method at M ¼ M 0 ¼ 7, k ¼ k 0 ¼ 5 and r ¼ 3. We compared our results with the results obtained from reduced differential transform method (RDTM) [10]. Present method at k ¼ k 0 ¼ 5; M ¼ M 0 ¼ 7; r ¼ 4 is implemented on generalized Burger-Fisher equation with a ¼ 0:001; b ¼ 0:001 and c ¼ 1. The obtained results are listed in Table 3.

Conclusion
We have derived and constructed the CAS wavelets matrix, Wm Âm , and the CAS wavelets operational matrix of q th order integration, P q mÂm , and CAS wavelets operational matrix of integration for boundary value problems, W g;q mÂm . These matrices are successfully utilized to solve the generalized Burger-Fisher equation.
According to Tables 1, 2, and 3, our results are more accurate as compared to reduced differential transform method, variational iteration method and Adomian decomposition method. Figure 1 shows that our results converge to the exact solution while increasing k; k 0 ; M and M 0 , when r ¼ 4.
It is shown that present method gives excellent results when applied to generalized Burger-Fisher equation. The different types of non-linearities can easily be handled by the present method.