A new approach for numerical solution of the modified Kawahara equation

In this paper, numerical solution of the modified Kawahara equation is obtained by using the mesh-free method based on the collocation with radial basis functions(RBFs). Three different types of RBFs are used for this purpose. Stability analysis of the method is discussed. Accuracy of the method is tested in terms of L2 and L∞ error norms.


Introduction
In this article a mesh-free method has been used for the numerical solution of the modified Kawahara equation.It is a nonlinear partial differential equation of the form: with initial condition and boundry conditions where D = {x : a < x < b} and ∂ D is its boundary and p, q are nonzero real constant.Modified Kawahara equation (1.1) was given first by Kawahara [1], as an important divergent equation.Equation (1.1) has wide applications in physics such as plasma waves, capillary-gravity water waves, water waves with surface tension, shallow water waves and so on [1]- [4].There are several methods for finding the analytical and numerical solution of modified Kawahara equation like, tanh-function method, extended tanh-function method, sine-cosine method, Jacobi function method, direct algebraic method, Adomian decomposition method [4]- [12].
In 1982, Frank [13] by applying interpolation methods for scattered data availabel at that time, observed radial basis function is better than all other methods regarding efficiency, stability and ease of implementations.Hardy in 1971 [14] introduced a general scattered data approximation method, called multiquadric (MQ) method.Kansa [15] in 1990 used modified MQ scheme to solve partial differential equations.The existence and uniqueness of the method was discussed by Madych et al. [16] and Michelli [17].In this paper we represent a mesh-free collocation method based on RBFs such as: to solve equation (1.1).The layout of the paper is as follows.First, in Section 2, the method is discussed for modified Kawahara equation.Section 3, is devoted to stability analysis of the method.Numerical results are given in Section 4 to illustrate the efficiency and accuracy of the presented method.
2 Mesh-free interpolation method by RBFs Let us consider the modified Kawahara equation with the initial condition and boundary condition Now, we apply the Crank-Nicolson scheme to equation (2.4) as follows: where u n+1 = u(t n+1 ), t n+1 = t n + δt and δt is the time step.The equation (2.7) is nonlinear.To linearize the nonlinear term (u 2 u x ) n+1 in equation (2.7) we can use following formula which obtained by applying the Taylor expansions, From equation (2.8), we can write: (2.9) Substituting equation (2.9) in equation (2.7) we get where u n is the nth iteration of the approximate solution.We assume u n approximated as follows: where x i = a + iδ x are collocation points in [a, b], φ : R n → R is a radial basis function and r i j = ∥x i − x j ∥ is the Euclidean norm between x i and x j , where x j = a + jδ x are centers in [a,b], j = 0(1)N.The parameters λ n j in equation (2.11) are unknowns to be determined by the collocation method.Now, by putting equation (2.11) in equation (2.10) for x i , i = 1(1)N − 1 we get the following results: where φ Also, from equations (2.6) and (2.11) we obtain following equations for the boundary points, (2.13) The system (2.12) and (2.13) contains (N + 1) equations and (N + 1) unknowns λ n+1 j which can be find by Gaussian elimination method.Using first value of u from initial condition, we can obtain the value of λ n j from equation (2.11).We can simplify the Eqs.(2.12) and (2.13) in the matrix form:

.14)
where International Scientific Publications and Consulting Services and The notation " * " denotes the Hadamard matrix multiplication.Equation (2.14) can be written as where From equation (2.11) we get Using equation (2.15) and (2.16) we can write: (2.17) From Eq. (2.17), we can obtain the solution at any time level n.For distinct collocation points, A 1 is always invertible [16].Invertibility of matrix M can not be provide [15] but in case of parameter-dependent RBFs, invertibility of M depends on shape parameters c.Optimal value of c calculate numerically in any problem.Therefore, we have the following algorithm for our purpose.
Algorithm of the method 1. choose N collocation point from the domain set [a,b].

Stability analysis
In this section, the stability of the approximation (2.17) is investigated by using the matrix method.To apply this method, we have linearized the non-linear term u 2 u x by assuming u as a constant.The error e n at the nth time level is given by: International Scientific Publications and Consulting Services where u n exact and u n app are the exact and approximate solutions at the nth time level respectively.The error equation for equation (1.1) is as follows: where ]. Now we can write equation (3.19) as: Numerical scheme is stable if ∥P∥ 2 ≤ 1, which is equivalent to ρ(P) ≤ 1, where ρ(P) denotes the spectral radius of the matrix P. By attention to above subjects, stability is assured if maximum eigenvalue of P satisfied in the following condition: where λ H and λ K , are eigenvalues of the matrices H and K, respectively.For real eigenvalues, the inequality (3.21) hold true if −1 ≤ λ H and λ K ≥ λ H −1 δt .This shows that the scheme (2.17), is stable if The equation (3.23) is satisfied if: then, inequality (3.23) hold true and the scheme is stable.The stability of the scheme (2.17) and conditioning of the component matrices H, K of the matrix P depend on the minimum distance between any two collocation points δ x, in the domain set [a, b], and the local shape parameter c.

Numerical solution
In this section we consider examples that solved by presented method in previous section for finding solution of modified Kawahara equations.In order to illustrate the accuracy of the method, we used the error norm L 2 and L ∞ which are defined as follows: where δ x is spatial step.,30] and δ x = 1.We used MQ, IMQ and GA radial basis function with shape parameter respectively 2.9, 0.0001 and 0.43.The L 2 and L ∞ in numerical solutions of Example 4.1 are tabulated in Table 1 for t = 0, 0.1, 0.5, 1.         3 for N = 4, 6, 8, 10.Also, for RBFs GA with c = 2.5 and IMQ with c = 0.36, are shown in Table 4 and Table 5, respectively.We compare our results with the Chebyshev spectral collocation method [18] applied to same example.We consider the same parameter values for the modified Kawahara equation ( 4.32) as considered in [18], namely; t = 0.1, 0.15, 0.2.Table 6 shows the compared results (δt = 10 −3 , [a, b] = [−10, 10], c(MQ) = 10.5).Also, Table 7 shows the comparison results of the approximate solutions obtained by present method and variational iteration method [19]    In this work, we have applied mesh-free method for solution of modified Kawahara equation based on radial basis functions.The numerical results, tables and figures, show that errors are very small and this scheme is accurate and efficient approach for the solution of such type of nonlinear partial differential equations.

Table 1 :
Numerical results for Example 4.1

Table 3 :
L ∞ and L 2 norm with MQ-RBF for Example 4.2.

Table 5 :
L ∞ and L 2 norm with IMQ-RBF for Example 4.2.

Table 7 :
Numerical solutions for Example 4.2.