TY - JOUR
T1 - Borel control and efficient numerical techniques to solve the Allen–Cahn equation governed by temporal multiplicative noise
AU - Ahmed, Nauman
AU - Macías-Díaz, Jorge E.
AU - Yasin, Muhammad W.
AU - Iqbal, Muhammad S.
N1 - Publisher Copyright:
© 2024 Informa UK Limited, trading as Taylor & Francis Group.
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
KW - numerical simulations
KW - proposed schemes
KW - stability and consistency analyses
KW - Stochastic Allen–Cahn equation
KW - unique existence
UR - http://www.scopus.com/inward/record.url?scp=85189980991&partnerID=8YFLogxK
U2 - 10.1080/00207160.2024.2340694
DO - 10.1080/00207160.2024.2340694
M3 - Article
AN - SCOPUS:85189980991
SN - 0020-7160
JO - International Journal of Computer Mathematics
JF - International Journal of Computer Mathematics
ER -