Meshfree methods for the nonlinear variable-order fractional advection–diffusion equation

Abstract

The nonlinear variable-order fractional advection–diffusion equation is applied to simulate complex engineering problem that exhibits time memory and space global dependence, for example, anomalous diffusion. In this paper, Kansa’s method is used to solve the nonlinear time–space variable-order fractional advection–diffusion equation, in which the variable order of space fractional derivative relies on both time and space while the variable order of the time fractional derivative is determined by space. Moreover, the convection coefficients, diffusion coefficients and source term are nonlinear functions that depend on exact solution. The finite difference method is contracted for the time fractional derivative and Wendland’s C6 compactly supported radial basis functions are used to substitute the space fractional derivative in the problem. The nonlinear terms are approximated by using an explicit difference scheme. The Gauss-Jacobi quadrature rule is used to approximate the weakly singular integral of space fractional derivative after radial basis function approximation. To illustrate the effectiveness and accuracy of the method, three numerical examples are solved, in which 2D experiments include the cases of regular and irregular regions. The results show that Kansa’s method has obvious advantages for variable-order fractional advection–diffusion equation.