Solution of time‐fractional stochastic nonlinear sine‐Gordon equation via finite difference and meshfree techniques

In this article, we introduce a numerical procedure to solve time‐fractional stochastic sine‐Gordon equation. The suggested technique is based on finite difference method and radial basis functions interpolation. By using this algorithm, first time‐fractional stochastic nonlinear sine‐Gordon equatio...

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Published in:	Mathematical methods in the applied sciences Vol. 45; no. 7; pp. 3426 - 3438
Main Authors:	Mirzaee, Farshid, Rezaei, Shadi, Samadyar, Nasrin
Format:	Journal Article
Language:	English
Published:	Freiburg Wiley Subscription Services, Inc 15.05.2022
Subjects:	Algorithms Approximation Brownian motion process Differential equations Elliptic functions Finite difference method Interpolation Mathematical analysis Meshless methods Nonlinear systems Radial basis function radial basis functions stochastic partial differential equations time‐fractional stochastic sine‐Gordon equation Trigonometric functions
ISSN:	0170-4214, 1099-1476
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Abstract	In this article, we introduce a numerical procedure to solve time‐fractional stochastic sine‐Gordon equation. The suggested technique is based on finite difference method and radial basis functions interpolation. By using this algorithm, first time‐fractional stochastic nonlinear sine‐Gordon equation is converted to elliptic stochastic differential equations. Then, the meshfree method based on radial basis functions (RBFs) is used to approximate the obtained equation. In fact, the finite difference method is used to approximate the unknown function in the time direction and generalized Gaussian RBF is applied to estimate the obtained equation in the space direction. The most important advantage of this method is that the noise terms are simulated directly at the collocation points at each time step. By employing this method, the equation decreased to a nonlinear system of algebraic equations which can be solved simply. The obtained results of solving three examples confirm the validity and capability of the proposed solution.
AbstractList	In this article, we introduce a numerical procedure to solve time‐fractional stochastic sine‐Gordon equation. The suggested technique is based on finite difference method and radial basis functions interpolation. By using this algorithm, first time‐fractional stochastic nonlinear sine‐Gordon equation is converted to elliptic stochastic differential equations. Then, the meshfree method based on radial basis functions (RBFs) is used to approximate the obtained equation. In fact, the finite difference method is used to approximate the unknown function in the time direction and generalized Gaussian RBF is applied to estimate the obtained equation in the space direction. The most important advantage of this method is that the noise terms are simulated directly at the collocation points at each time step. By employing this method, the equation decreased to a nonlinear system of algebraic equations which can be solved simply. The obtained results of solving three examples confirm the validity and capability of the proposed solution.
Author	Mirzaee, Farshid Rezaei, Shadi Samadyar, Nasrin
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SubjectTerms	Algorithms Approximation Brownian motion process Differential equations Elliptic functions Finite difference method Interpolation Mathematical analysis Meshless methods Nonlinear systems Radial basis function radial basis functions stochastic partial differential equations time‐fractional stochastic sine‐Gordon equation Trigonometric functions
Title	Solution of time‐fractional stochastic nonlinear sine‐Gordon equation via finite difference and meshfree techniques
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