Existence and Uniqueness of the Solution of a Stochastic Partial Functional Differential Equation of a Special Form and Methods of its Computer Modeling

• Date: 2025
• Main Authors: Юрченко, Ігор, Ясинський, Володимир
• Format: Article
• Language: Ukrainian
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2025
• Online Access: http://mcm-math.kpnu.edu.ua/article/view/332502
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

This article investigates the Cauchy problem for a stochastic partial differential-functional equation (SPDFE) of a special form, which models dynamic processes with memory under the influence of random perturbations. In particular, we examine the mathematical conditions ensuring the existence and uniqueness of the solution. The theoretical results are based on modern tools of stochastic analysis and functional differential calculus. Since analytical solutions to such equations are generally unavailable, the paper explores approaches for approximate numerical solutions. We describe the discretization of space and time, and techniques for approximating the functional (memory-dependent) terms using a historical buffer. Stochastic perturbations are modeled as space-time noise based on the Wiener process. We developed a Python-based software implementation using the Euler–Maruyama method to validate the theoretical results and provide a practical illustration of memory-driven dynamics. The considered equation models the evolution of a system under spatial diffusion, damping, memory effects (via past state dependence), and stochastic noise. For the numerical solution, we use discretized approximations of the memory integral as an average over a buffer of previous values. A graphical visualization of the spatiotemporal evolution is constructed, including a heatmap that shows how the system state u(t, x) «spreads» and fluctuates under the combined influence of memory and noise. The results are promising for further applications in the theory and practice of computational modeling of complex dynamical systems with memory and stochasticity. In particular, the developed approaches may be applied in modeling processes in physics (heat conduction with delay, diffusion in memory-structured media), biology and ecology (population dynamics or epidemic spread with incubation delays), financial mathematics (volatility depending on past states), and technical or information control systems with stochastic effects and signal delays.