Analysis of Random Fractional Functional Volterra–Fredholm Integro-Differential Equations with Infinite Delay and ANN-Based Approximation

In this paper, we study a class of random fractional functional Volterra–Fredholm integro-differential equations with infinite delay with the Caputo fractional derivative. This model under consideration is a model with hereditary memory, nonlocal Volterra and Fredholm integral interactions, and bounded random perturbations. Sufficient conditions for the existence and uniqueness of a random integral solution of the problem are derived by transforming the problem into an equivalent random fractional integral equation, by means of the Banach fixed-point theorem and Krasnosel’skii’s fixed-point theorem under appropriate assumptions of continuity, boundedness and Lipschitz. An illustrative example is given to establish the validity of the obtained theoretical results. By graphical visualization the qualitative behavior of the random fractional solution is analyzed, with smooth, bounded, and stable dynamics of the solution. Furthermore, a computational framework based on an artificial neural network is proposed to determine a family of random fractional solutions, corresponding to various realizations in a bounded random parameter. The results from both the analysis and the computation validate the proposed framework for modeling random hereditary fractional dynamical systems with infinite-delay effects.