An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions

A H Bhrawy,M A Alghamdi,Eman S Alaidarous

doi:10.1155/2014/295936

Abstract

One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs) as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.

Highlights

For several decades, numerical methods have been developed to obtain more accurate solutions of differential and integral equations
A basic property of the Jacobi polynomials is that they are the eigenfunctions to a singular Sturm-Liouville problem: (1 − x2) φ󸀠󸀠 (x) + [θ − θ + (θ + θ + 2) x] φ󸀠 (x) (1)
We propose an efficient numerical algorithm to solve the coupled nonlinear hyperbolic types equations in the following form: Dt2u (y, t) = γu (y, t) V (y, t)

Summary

Introduction

Numerical methods have been developed to obtain more accurate solutions of differential and integral equations. In [41], Dehghan and Shokri used the radial basis functions to solve a two-dimensional SineGordon equation; in [42] they developed numerical scheme to solve the one-dimensional nonlinear KleinGordon equation with quadratic and cubic nonlinearity using collocation points and approximating the solution using Thin Plate Splines and RBFs. There are no results on Jacobi-Gauss-Lobatto collocation (J-GL-C) method for solving nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary and nonlocal conditions. The nonlocal conservation conditions are efficiently treated by Jacobi-Gauss-Lobatto quadrature rule at (N + 1) nodes to obtain a system of ODEs in time and proper initial value software can be applied to solve this system of ODEs. Several illustrative problems with various kinds of exact solutions such as triangular, soliton, and exponentialtriangular solutions are presented for demonstrating the high accuracy of this scheme.

Jacobi Polynomials

The Problem and the Numerical Algorithm

Test Problems

Conclusion