Solving Time-Fractional Nonlinear Variable-Order Delay PDEs Using Feedforward Neural Networks

Abstract

This study presents an innovative application of Feedforward Neural Networks ‘FNNs’ to solve Variable-Order Fractional Partial Differential Equations ‘VO-FPDEs’ with time delays. Utilizing the Caputo definition, the variable-order fractional derivatives are approximated in terms of integer-order derivatives. The problem is reformulated as a system of partial differential equations with delay terms, which is then addressed using ‘FNNs’ to achieve explicit approximate solutions. Comprehensive error and convergence analyses validate the method’s precision and reliability. The effectiveness of the proposed approach is highlighted through numerical examples, with graphical and tabular representations showcasing minimal absolute errors and robust convergence. These results demonstrate the proposed method’s efficiency and simplicity, establishing it as a powerful tool for addressing complex fractional delay problems. © 2025 Elsevier B.V., All rights reserved.