A numerical approach to solve the stochastic Allen-Cahn equation of fractional order

Document Type : Original Article

Authors

Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.

Abstract

In this paper, we employ a collocation method based on Legendre polynomials (LPs) to solve the time-fractional stochastic Allen-Cahn equation. This method is applied to convert the solution of this stochastic equation to the solution of a nonlinear system of algebraic equations. The numerical approach is completely described. Finally, a test example is implemented to validate the robustness of the proposed scheme.

Keywords


[1] R. Aboulaich, A. Darouichi, I. Elmouki, A. Jrai , A Stochastic Optimal Control Model for BCG
Immunotherapy in Super cial Bladder Cancer, Math. Model. Nat. Phenom, 12(5) 2017, 99-119.
[2] N. Abdollahi, D. Rostamy, Identifying an unknown time-dependent boundary source in time-fractional
diffusion equation with a non-local boundary condition, Journal of Computational and Applied Math-
ematics, 355 2019, 36-50.
[3] A. Babaei, S. Banihashemi, A stable numerical approach to solve a time-fractional inverse heat con-
duction problem, Iran J. Sci. Technol. Trans. A, 42(4) 2018, 2225-2236.
[4] A. Babaei, H. Jafari, M. Ahmadi, A fractional order HIV/AIDS model based on the effect of screening
of unaware infectives, Mathematical Methods in the Applied Sciences, 42(7) 2019, 2334-2343.
[5] A. Babaei, H. Jafari, S. Banihashemi, A collocation approach for solving time-fractional stochastic
heat equation driven by an additive noise, Symmetry, 12(6) 2020, 904.
[6] A. Babaei, H. Jafari, S. Banihashemi, A numerical scheme to solve a class of two-dimensional nonlinear
time-fractional diffusion equations of distributed order, Engineering with Computers, 2020, 1-13.
[7] D. Baleanu, A. Jajarmi, H. Mohammadi, H. Rezapour, A new study on the mathematical modelling of
human liver with Caputo-Fabrizio fractional derivative, Chaos, Solitons & Fractals, 134 2020, 109705.
[8] N. Bellomo, Z. Brzezniak, L.M.D. Socio, Nonlinear stochastic evolution problems in applied sciences,
Kluwer Academic Publishers, Springer: Dordrecht, The Netherlands, 1992.
[9] F. Biagini, Y. Hu, B. Oksendal, and T. Zhang, Stochastic calculus for fractional Brownian motion
and applications, Springer Science & Business Media, London, 2008.
[10] C. Canuto, M. Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: Fundamentals in Single
Domains, Springer-Verlag, 2006.
[11] A.S. Chaves, A fractional diffusion equation to describe Levy 
ights, Phys. Lett. A, 239(1-2) 1998,
13-16.
[12] Q. Du, T. Zhang, Numerical approximation of some linear stochastic partial differential equations
driven by special additive noises, SIAM J. Numer. Anal., 40(4) 2002, 1421-1445.
[13] M. Hairer, M.D. Ryser, H. Weber, Triviality of the 2D stochastic Allen-Cahn equation, Electronic
Journal of Probability, 17 2012, 1-14.
[14] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations,
North-Holland Mathematics Studies, Elsevier: Amsterdam, The Netherlands, 2006.
[15] M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, Walter de Gruyter GmbH
& Co KG, Berlin, 2019.
[16] B.P. Moghaddam, L. Zhang, A.M. Lopes, J.A.T. Machado, Z.S. Mostaghim, Computational scheme
for solving nonlinear fractional stochastic differential equations with delay, Stochastic Analysis and
Applications, 37(6) 2019, 893-908.
[17] B. Oksendal, Stochastic differential equations: an introduction with applications, 5th edition,
Springer-Velarge, Berlin, Germany, 1998.
[18] I. Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional
differential equations, to methods of their solution and some of their applications, Academic Press:
New York, NY, USA, 1999.
[19] D. Rostamy, S.Qasemi, Stochastic Adsorption-Desorption With a New Discontinuous Galerkin
Method, Journal of Numerical Mathematics and Stochastics, 11(1) 2019, 68-81.
[20] C. Roth, A combination of  nite difference and Wong-Zakai methods for hyperbolic stochastic partial
differential equations, Stoch. Anal. Appl, 24(1) 2006, 221-240.
[21] D.N. Tien, Fractional stochastic differential equations with applications to  nance, J. Math. Anal.
Appl, 397(1) 2013, 334-348.
[22] J. Yang, Y. Tan, R.A. Cheke, Thresholds for extinction and proliferation in a stochastic tumour-
immune model with pulsed comprehensive therapy, Commun Nonlinear Sci Numer Simulat, 73 2019,
363-378.
Volume 2, Issue 4
December 2021
Pages 1-10
  • Receive Date: 24 September 2021
  • Revise Date: 30 November 2021
  • Accept Date: 30 November 2021
  • First Publish Date: 30 November 2021