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# Meshless Local Petrov-Galerkin and RBFs Collocation Methods for Solving 2D Fractional Klein-Kramers Dynamics Equation on Irregular Domains

M. Dehghan1, M. Abbaszadeh2, A. Mohebbi3

Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave.,15914, Tehran, Iran, Corresponding author. E-mail addresses: mdehghan@aut.ac.ir, mdehghan.aut@gmail.com, (M. Dehghan)
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave.,15914, Tehran, Iran, E-mail addresses: m.abbaszadeh@aut.ac.ir (M. Abbaszadeh)
Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran. E-mail addresses: a_mohebbi@kashanu.ac.ir (A. Mohebbi).

Computer Modeling in Engineering & Sciences 2015, 107(6), 481-516. https://doi.org/10.3970/cmes.2015.107.481

## Abstract

In the current paper the two-dimensional time fractional Klein-Kramers equation which describes the subdiffusion in the presence of an external force field in phase space has been considered. The numerical solution of fractional Klein-Kramers equation is investigated. The proposed method is based on using finite difference scheme in time variable for obtaining a semi-discrete scheme. Also, to achieve a full discretization scheme, the Kansa's approach and meshless local Petrov-Galerkin technique are used to approximate the spatial derivatives. The meshless method has already proved successful in solving classic and fractional differential equations as well as for several other engineering and physical problems. The fractional derivative of equation is described in the Riemann-Liouville sense. In this paper we use a finite difference scheme to discretize the time fractional derivative of mentioned equation as the obtained scheme is of convergence order O(τ1+γ) for 0 < γ < 1. Also, we solve the mentioned equation on non-rectangular domains. The aim of this paper is to show that the meshless methods based on the strong form i.e. the radial basis functions collocation approach and local weak form i.e. meshless local Petrov-Galerkin idea are also suitable for the treatment of the fractional Klein-Kramers equation. Numerical examples confirm the high accuracy and acceptable results of proposed schemes.

## Keywords

Dehghan, M., Abbaszadeh, M., Mohebbi, A. (2015). Meshless Local Petrov-Galerkin and RBFs Collocation Methods for Solving 2D Fractional Klein-Kramers Dynamics Equation on Irregular Domains. CMES-Computer Modeling in Engineering & Sciences, 107(6), 481–516. https://doi.org/10.3970/cmes.2015.107.481

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