Fractional Stochastic Partial Differential Equations: Numerical Advances and Practical Applications—A State of the Art Review

Abstract

This article aims to provide a comprehensive review of the latest advancements in numerical methods and practical implementations in the field of fractional stochastic partial differential equations (FSPDEs). This type of equation integrates fractional calculus, stochastic processes, and differential equations to model complex dynamical systems characterized by memory and randomness. It introduces the foundational concepts and definitions essential for understanding FSPDEs, followed by a comprehensive review of the diverse numerical methods and analytical techniques developed to tackle these equations. Then, this article highlights the significant expansion in numerical methods, such as spectral and finite element methods, aimed at solving FSPDEs, underscoring their potential for innovative applications across various disciplines.