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Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection–diffusion equations

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Abstract

The present article develops a semi-discrete numerical scheme to solve the time-fractional stochastic advection–diffusion equations. This method, which is based on finite difference scheme and radial basis functions (RBFs) interpolation, is applied to convert the solution of time-fractional stochastic advection–diffusion equations to the solution of a linear system of algebraic equations. The mechanism of this method is such that time-fractional stochastic advection–diffusion equation is first transformed into elliptic stochastic differential equations by using finite difference scheme. Then meshfree method based on RBFs has been used to approximate the resulting equation. In other words, the approximate solution of time-fractional stochastic advection–diffusion equation is achieved with discrete the domain in the t-direction by finite difference method and approximating the unknown function in the x-direction by generalized inverse multiquadrics RBFs. In this method, the noise terms are directly simulated at the collocation points in each time step and it is the most important advantage of the suggested approach. Stability and convergence of the scheme are established. Finally, some test problems are included to confirm the accuracy and efficiency of the new approach.

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Acknowledgements

The authors would like to express our very great appreciation to editor and anonymous reviewers for their valuable comments and constructive suggestions which have helped to improve the quality and presentation of this paper.

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  1. Faculty of Mathematical Sciences and Statistics, Malayer University, Malayer, P. O. Box 65719-95863, Iran

    Farshid Mirzaee & Nasrin Samadyar

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  1. Farshid Mirzaee
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  2. Nasrin Samadyar
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Correspondence to Farshid Mirzaee.

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Mirzaee, F., Samadyar, N. Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection–diffusion equations. Engineering with Computers 36, 1673–1686 (2020). https://doi.org/10.1007/s00366-019-00789-y

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  • DOI: https://doi.org/10.1007/s00366-019-00789-y

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