Numerical solution of multi-term time fractional wave diffusion equation using transform based local meshless method and quadrature

Abstract

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles. In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The present papers deals with the approximation of one and two dimensional multi-term time fractional wave diffusion equations. In this work a numerical method which combines Laplace transform with local radial basis functions method is presented. The Laplace transform eliminates the time variable with which the classical time stepping procedure is avoided, because in time stepping methods the accuracy is achieved at a very small step size, and these methods face sever stability restrictions. For spatial discretization the local meshless method is employed to circumvent the issue of shape parameter sensitivity and ill-conditioning of collocation matrices in global meshless methods. The bounds of the stability for the differentiation matrix of our numerical scheme are derived. The method is tested and validated against 1D and 2D wave diffusion equations. The 2D equations are solved over rectangular, circular and complex domains. The computational results insures the stability, accuracy, and efficiency of the method.