Abstract
One of the most powerful tools for solving partial differential equations is approximation using the radial basis functions (RBFs). This method can be very spectrally accurate because it is meshfree. This paper presents a semi-discretization numerical scheme to solve the stochastic coupled nonlinear time fractional sine-Gordon equations, which are obtained by replacing integer time derivative with the Caputo fractional time derivative of order \(\alpha \) (\(1<\alpha \le 2\)) and adding stochastic factors. Using this method, the stochastic coupled nonlinear time fractional sine-Gordon equation is transformed to a system of nonlinear algebraic equations that can be solved by a suitable numerical method. This method is a combination of finite difference (FD) method and RBFs. First, time is overcome by forward FD method, then in the direction of space using the meshless method based on RBFs, the unknown function is approximated. This method is very practical, accurate and appropriate. Finally, two examples show the accuracy and efficiency of the proposed method.
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The authors would like to express our very great appreciation to reviewers for their valuable comments and suggestions which have helped to improve the quality and presentation of this paper.
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Communicated by Eduardo Souza de Cursi.
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Mirzaee, F., Rezaei, S. & Samadyar, N. Application of combination schemes based on radial basis functions and finite difference to solve stochastic coupled nonlinear time fractional sine-Gordon equations. Comp. Appl. Math. 41, 10 (2022). https://doi.org/10.1007/s40314-021-01725-x
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DOI: https://doi.org/10.1007/s40314-021-01725-x
Keywords
- Fractional stochastic sine-Gordon equation
- Stochastic partial differential equations
- Caputo fractional derivative
- Finite difference method
- Brownian motion process