Numerical approximations of stochastic Gray-Scott model with two novel schemes

Xiaoming Wang; Muhammad W. Yasin; Nauman Ahmed; Muhammad Rafiq; Muhammad Abbas; Xiaoming Wang; Muhammad W. Yasin; Nauman Ahmed; Muhammad Rafiq; Muhammad Abbas

doi:10.3934/math.2023257

AIMS Mathematics

2023, Volume 8, Issue 3: 5124-5147. doi: 10.3934/math.2023257

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Numerical approximations of stochastic Gray-Scott model with two novel schemes

1.
School of Mathematics & Computer Science, Shangrao Normal University, 344001 Shangrao, China
2.
Department of Mathematics, University of Narowal, Narowal, Pakistan
3.
Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
4.
Department of Mathematics, Faculty of Science & Technology, University of Central Punjab, Lahore, Pakistan
5.
Department of Mathematics, Mathematics Research Center, Near East University, Near East Boulevard, 99138 Nicosia/Mersin, Turkey
6.
Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan

Received: 20 August 2022 Revised: 16 October 2022 Accepted: 20 October 2022 Published: 13 December 2022
MSC : 35R60, 62L20

This article deals with coupled nonlinear stochastic partial differential equations. It is a reaction-diffusion system, known as the stochastic Gray-Scott model. The numerical approximation of the stochastic Gray-Scott model is discussed with the proposed stochastic forward Euler (SFE) scheme and the proposed stochastic non-standard finite difference (NSFD) scheme. Both schemes are consistent with the given system of equations. The linear stability analysis is discussed. The proposed SFE scheme is conditionally stable and the proposed stochastic NSFD is unconditionally stable. The convergence of the schemes is also discussed in the mean square sense. The simulations of the numerical solution have been obtained by using the MATLAB package for the various values of the parameters. The effects of randomness are discussed. Regarding the graphical behavior of the stochastic Gray-Scott model, self-replicating behavior is observed.
- stochastic Gray-Scot model,
- SFE and stochastic NSFD schemes,
- analysis of schemes,
- graphical behavior
Citation: Xiaoming Wang, Muhammad W. Yasin, Nauman Ahmed, Muhammad Rafiq, Muhammad Abbas. Numerical approximations of stochastic Gray-Scott model with two novel schemes[J]. AIMS Mathematics, 2023, 8(3): 5124-5147. doi: 10.3934/math.2023257

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Abstract

This article deals with coupled nonlinear stochastic partial differential equations. It is a reaction-diffusion system, known as the stochastic Gray-Scott model. The numerical approximation of the stochastic Gray-Scott model is discussed with the proposed stochastic forward Euler (SFE) scheme and the proposed stochastic non-standard finite difference (NSFD) scheme. Both schemes are consistent with the given system of equations. The linear stability analysis is discussed. The proposed SFE scheme is conditionally stable and the proposed stochastic NSFD is unconditionally stable. The convergence of the schemes is also discussed in the mean square sense. The simulations of the numerical solution have been obtained by using the MATLAB package for the various values of the parameters. The effects of randomness are discussed. Regarding the graphical behavior of the stochastic Gray-Scott model, self-replicating behavior is observed.

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