Using the spectral meshless radial basis functions method for solving time fractional Burgers' equation

Abstract

‎This study presents a local meshless method based on radial basis functions for the numerical solution of the non-linear time-fractional Burgers' equation (TFBE) involving the fractional derivatives of Caputo (CFD)‎, ‎and Caputo-Fabrizio (CFFD)‎. ‎Time is discretized using the implicit finite difference scheme with $\theta= 1$‎, ‎while radial basis functions (RBFs)‎, ‎which do not require a mesh to approximate the solution‎, ‎are used for spatial discretization‎. ‎Rubin–Graves technique is used to linearize non-linear terms‎. ‎Approximation of existing spatial derivatives is done using central and finite difference methods and the temporal derivatives are approximated by using the definition of the Caputo and Caputo-Fabrizio derivatives‎. ‎With the mentioned techniques‎, ‎a system of algebraic equations is established‎. ‎By comparing the results obtained from solving this system with the previous results‎, ‎it is clear that the presented technique provides accurate‎, ‎stable‎, ‎and convergent results‎.