Numerical Solution of Nonlinear Time-Fractional Telegraph Equation by Radial Basis Function Collocation Method

Abstract

A $$\theta$$ -weighted method and radial basis functions are applied for finding the solution of time-fractional nonlinear telegraph equation in Caputo sense. The Caputo derivatives are approximated by the difference formulas introduced in Cao et al. (Fract Calc Appl Anal 18(3):735–761, 2015) and Li et al. (Comput Math Appl 62:855–875, 2011). By choosing the centers of radial basis functions as collocation points, in each time step a nonlinear system of algebraic equations is obtained. A fixed point iteration method for solving the system is presented. By the fixed point method the computations of the nonlinear system are reduced to some linear systems of algebraic equations. The QR decomposition method is proposed for solving these linear systems. Several numerical examples in one- and two-dimensional cases are included to demonstrate the efficiency and the accuracy of the our method.