An implicit RBF meshless approach for time fractional diffusion equations

Abstract

This paper aims to develop an implicit meshless approach based on the radial basis function (RBF) for numerical simulation of time fractional diffusion equations. The meshless RBF interpolation is firstly briefed. The discrete equations for two-dimensional time fractional diffusion equation (FDE) are obtained by using the meshless RBF shape functions and the strong-forms of the time FDE. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approach. It has proven that the present meshless formulation is very effective for modeling and simulation of fractional differential equations.

An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations

“…Also, they have discussed the stability and convergence of their method. Authors of [50] presented an implicit meshless method based on the radial basis functions for the numerical simulation of time-fractional diffusion equation. Authors of [95] presented an implicit meshless approach based on the moving least squares (MLS) approximation for the numerical simulation of fractional advectiondiffusion equation.…”

The method of approximate particular solutions for the time-fractional diffusion equation with a non-local boundary condition

“…Because of the non-local properties of fractional operators, obtaining the analytical solutions of the fractional differential equations (FDEs) is more challenging or sometimes even impossible. Hence the proposal, development, and analysis of numerical methods to solve FDEs are at present a quite active field of research, and many methods have been considered, for instance, finite difference method [2][3][4], finite element method [5,6], spectral method [7][8][9], meshless method [10][11][12][13][14], and so on.…”

Error analysis of a high-order compact ADI method for two-dimensional fractional convection-subdiffusion equations

“…The finite difference method is one of the most popular numerical methods used for solving time and/or space FDEs (see [16][17][18][19][20][21][22][23][24][25][26][27]). There are also a few other interesting studies by the finite element method [28][29][30], the spectral method [31][32][33], the implicit meshless method [34] and the radial basis function approximation method [35].…”