Meshfree Methods For The Variable-Order Fractional Advection-Diffusion Equation

The fractional advection-diffusion equation originates from the traditional advectiondiffusion equation and can model some complex physical phenomena with behaviors of time heredity and space global dependence more accurately such as anomalous diffusion.It takes advantages over its integer-order counterpart of accurate describing of complex phenomena such as convection diffusion,heat conduction,viscoelasticity,turbulence,seepage and abnormal diffusion.The variable-order fractional advection-diffusion equations have strong flexibility in simulating complex physical systems with memory dependency.But the analytical solutions of most fractional differential equations are quite difficult to be obtained because they contain weakly singular integral.The non-symmetric radial basis function collocation method is used to solve the variable-order fractional advection-diffusion equations.This method requires no domain discretization,which greatly reduces the amount of computational cost,and it of very high accuracy.Firstly,the basic preliminaries of non-symmetric radial basis function collocation method and fractional derivatives are introduced,containing Wendland’s C6 function and its derivatives,the relationship between two types of fractional derivatives.Then,nonsymmetric radial basis function collocation method is used to solve 1D and 2D time-space linear/nonlinear time-space variable-order fractional advection-diffusion equations.The time fractional derivatives are approximated by full-implicit difference scheme and the space fractional derivatives are discretized via Kansa’s method combined with Wendland’s C6 compactly supported radial basis function.The Gauss-Jacobi quadrature is utilized to evaluate the weakly singular integration during the computation of the space fractional derivative of radial basis function.The nonlinear terms are approximated by directly linearized method.Finally,some numerical experiments are conducted to verify the accuracy and efficiency of the proposed method.