Two meshless procedures: moving Kriging interpolation and element-free Galerkin for fractional PDEs

Abstract

In this paper, the element-free Galerkin (EFG) meshless method and moving Kriging collocation meshless technique are applied for finding the numerical solution of a class of two-dimensional (2D) nonlinear time fractional partial differential equations. The Klein–Gordon, sine-Gordon, diffusion wave and Cattaneo equations with Neumann boundary condition are studied. The time fractional derivative has been described in the Caputo’s sense. Firstly, we use a semi-implicit finite difference scheme of convergence order , and then for obtaining a full discrete scheme, the space derivative is discretized with the EFG and moving Kriging collocation techniques. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method, test and trial functions are moving least squares approximation (MLS) shape functions. Also, in the element-free Galekin method, we do not use any triangular, quadrangular, or other types of meshes. The EFG method is a global method while finite element method is a local one. The EFG method is not a truly meshless method and for integration uses a background mesh. We prove the unconditional stability and obtain an error bound for the EFG method using the energy method. Numerical examples are reported which support the theoretical results and the efficiency of the proposed scheme.

Keywords:

• Fractional diffusion wave (FDW) equation
• time fractional sine-Gordon and Klein Gordon equations
• time fractional Cattaneo equation
• element-free Galerkin (EFG)
• moving Kriging interpolation
• energy method and unconditional stability
• convergence and error estimate

AMS Subject Classifications:

• 65M12
• 65N12
• 65N30
• 65N40

Acknowledgements

The authors thank the anonymous reviewers for their constructive comments and suggestions.

Notes

No potential conflict of interest was reported by the authors.