A Jacobi Dual-Petrov Galerkin-Jacobi Collocation Method for Solving Korteweg-de Vries Equations
چکیده
and Applied Analysis 3 nonlinear term is treated with the Chebyshev collocation method. The time discretization is a classical Crank-Nicholson-leap-frog scheme. Yuan and Wu 43 extended the Legendre dual-Petrov-Galerkin method proposed by Shen 44 , further developed by Yuan et al. 45 to general fifth-order KdV-type equations with various nonlinear terms. The main aim of this paper is to propose a suitable way to approximate the thirdorder differential equations in space, by dual-Petrov Galerkin method-based on Jacobi polynomials such that it can be implemented efficiently and at the same time has a good convergence property. Moreover, we introduce the Jacobi dual Petrov Galerkin-Jacobi collocation JDPG-JC method for solving third-order differential equations with nonlinear term. The method is basically formulated in the Jacobi spectral form with general indexes α, β > −1 but the nonlinear term being treated by the Jacobi collocation method with other two general indexes θ, θ > −1 so that the schemes can be implemented at JacobiGauss-Lobatto points efficiently. Therefore, we can generalize Legendre Petrov-Galerkin and Chebyshev collocation method to Jacobi Petrov-Galerkin and Jacobi collocation method. Some other cases can be obtained directly as special cases from our proposed JDPG-JC approximations. We, therefore, motivated our interest in JDPG-JC approximations. Finally, numerical results are presented in which the usual exponential convergence behaviour of spectral approximations is exhibited. The layout of the paper is as follows. In Section 2, we give an overview of Jacobi polynomials and their relevant properties needed hereafter. Section 3 is devoted to the theoretical derivation of the Jacobi dual-Petrov Galerkin JDPG method for linear thirdorder differential equations subject to homogeneous boundary conditions. Section 4 gives the corresponding results for those obtained in Section 3 but for the KDV equation using JDPG-JCmethod. In Section 5, we present some numerical results exhibiting the accuracy and efficiency of our numerical algorithms. Some concluding remarks are given in the final section. 2. Preliminaries Let SN I be the space of polynomials of degree at most N on the interval I −1, 1 , we set WN { u ∈ SN : u ±1 u′ 1 0 } , W∗ N { u ∈ SN : u ±1 u′ −1 0 } . 2.1 And let P α,β n x n 0, 1, 2, . . . be the Jacobi polynomials orthogonal with the weight functions w x 1 − x α 1 x , where α, β > −1. Let x α,β N,j , 0 ≤ j ≤ N, be the zeros of 1 − x2 ∂xP α,β N . Denote by α,β N,j , 0 ≤ j ≤ N, the weights of the corresponding Gauss-Lobatto quadrature formula. They are arranged in decreasing order. We define the discrete inner product and norm as follows: