Borel control and efficient numerical techniques to solve the Allen-Cahn equation governed by temporal multiplicative noise

• Published in: International journal of computer mathematics Vol. 101; no. 9-10; pp. 1152 - 1167
• Main Authors: Ahmed, Nauman, Macías-Díaz, Jorge E., Yasin, Muhammad W., Iqbal, Muhammad S.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02.10.2024; Taylor & Francis Ltd
• Subjects: Noise control; Nonlinear differential equations; numerical simulations; Partial differential equations; proposed schemes; stability and consistency analyses; Stochastic Allen-Cahn equation; Stochastic models; unique existence
• ISSN: 0020-7160; 1029-0265
• DOI: 10.1080/00207160.2024.2340694

In this manuscript, we investigate a stochastic version of the Allen-Cahn equation, which is a nonlinear partial differential equation from mathematical physics. The stochastic model considers temporal multiplicative noise in the form of the derivative of the standard Wiener process. The existence and the uniqueness of solutions for this stochastic model is established rigorously using the theory of distributions. As a corollary from these analytical results, some a priori optimal estimates for the solutions of this model are constructed. In this work, we develop reliable numerical schemes which possess similar features as those of the solutions for the analytical model. Moreover, we establish mathematically the von Neumann stability and the consistency for the schemes proposed in this paper. Both the analytical and numerical results derived in this work are computationally verified through some simulations.