A Meshless Runge–Kutta Method for Some Nonlinear PDEs Arising in Physics.

This paper deals with the numerical solutions of a general class of one-dimensional nonlinear partial differential equations (PDEs) arising in different fields of science. The nonlinear equations contain, as special cases, several PDEs such as Burgers equation, nonlinear-Schrödinger equation (NLSE), Korteweg–De Vries (KDV) equation, and KdV–Schrödinger equations. Inspired by the method of lines, an RBF-FD approximation of the spatial derivatives in terms of local unknown function values, converts the nonlinear governing equations to a system of nonlinear ordinary differential equations(ODEs). Then, a fourth-order Runge–Kutta method is proposed to solve the resulting nonlinear system of first-order ODEs. For the RBF-FD approximation of derivatives, three kinds of different basis are investigated, and it is shown that the polynomial basis gives the highest accuracy. Solitary wave solutions form a special class of solutions of the model equations considered in this paper. The results reveal that the proposed method simulates this kind of solutions with high accuracy. Also, the method is able to simulate the collision of two soliton solutions, provided they are initially located far enough from each other. [ABSTRACT FROM AUTHOR]