A meshless numerical procedure for solving fractional reaction subdiffusion model via a new combination of alternating direction implicit (ADI) approach and interpolating element free Galerkin (EFG) method

Mostafa Abbaszadeh,Mehdi Dehghan

doi:10.1016/j.camwa.2015.09.011

Mostafa Abbaszadeh, Mehdi Dehghan

Open Access

https://doi.org/10.1016/j.camwa.2015.09.011

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Journal: Computers and Mathematics with Applications	Publication Date: Oct 9, 2015
Citations: 43	License type: elsevier-specific: oa user license

Affiliation: Amirkabir University of Technology

Abstract

A new numerical algorithm has been investigated for solving time fractional reaction–subdiffusion equation. The fractional derivative of the considered equation is described in the Riemann–Liouville sense. Firstly, we discrete the temporal dimension of the considered model using a finite difference scheme. A central difference scheme has been applied to discrete the first time derivative and then for discretizing the fractional integral term a difference scheme has been employed with convergence order O(τ1+γ). Moreover, to achieve a full discretization scheme a type of meshless method has been improved that is known as element free Galerkin (EFG) method. The EFG method for integration uses a background mesh. This method is based on the Galerkin weak form in which the test and trial functions are shape functions of moving least squares (MLS) approximation. Since the shape functions of traditional MLS lack the Kronecker δ property, essential boundary conditions of a boundary value problem can not be directly computed and other methods must be employed for this issue. To this end, a new class of MLS shape functions has been applied that is called shape functions of interpolating MLS. The new shape function has the mentioned property. In the EFG method, calculating the appeared two-dimensional integrals is a basic issue. In this research work, the alternating direction implicit approach is combined with the element free Galerkin method. Then, using the new proposed method, the two-dimensional integrals on rectangular domain will be changed to simple one-dimensional integrals. We prove that the new numerical algorithm is unconditionally stable and also we obtain an error bound for the new procedure using the energy method. Numerical examples are reported which demonstrate the theoretical results and the efficiency of proposed scheme.

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