Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochast
- PDF / 2,301,816 Bytes
- 14 Pages / 595.276 x 790.866 pts Page_size
- 56 Downloads / 195 Views
ORIGINAL ARTICLE
Combination of finite difference method and meshless method based on radial basis functions to solve fractional stochastic advection–diffusion equations Farshid Mirzaee1 · Nasrin Samadyar1 Received: 16 August 2018 / Accepted: 24 May 2019 © Springer-Verlag London Ltd., part of Springer Nature 2019
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. Keywords Finite difference method · Radial basis functions · Fractional partial differential equations · Stochastic partial differential equations · Fractional derivative · Stochastic analysis Mathematical Subject Classification 65M06 · 35R11 · 35R60 · 26A33 · 76M35
1 Introduction Many problems in the applied sciences and engineering are widely modeled via partial differential equations (PDEs). PDEs have been extensively studied by many researchers and various numerical methods have been presented to obtain their approximate solution. For instance meshless methods [1, 2], Sinc–Legendre collocation method [3], finite difference method [4], variational iteration method [5], Sumudu decomposition method [6], wavelet method [7], spectral method [8, 9], homotopy analysis method [10], * Farshid Mirzaee [email protected]; [email protected] Nasrin Samadyar [email protected] 1
Faculty of Mathematical Sciences and Statistics, Malayer University, Malayer, P. O. Box 65719‑95863, Iran
etc. One of the most applicable PDEs is advection–diffusion equation which is a combination of the diffusion and advection equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and advection. Since exact solution of these equations are rarely known, numerical analysis of them has been recently the subject of man
Data Loading...
