Advancing Solutions for Fractional Volterra–Fredholm Integro-Differential Equations: A Comprehensive Review of Recent Numerical Methods

Fractional Volterra–Fredholm integro differential equations (FVFIDEs) combine fractional derivatives with nonlocal Volterra and Fredholm operators, making them a central tool for modeling systems with memory and spatial interactions. Analytical solutions are rare, and numerical methods have become essential for practical applications. This review synthesizes recent advances in numerical approaches, including predictor–corrector schemes, spectral methods, iterative algorithms, and hybrid techniques. Emphasis is placed on convergence analysis, computational efficiency, and comparative performance. By consolidating developments from 2020–2025, the article provides a structured overview of current solution strategies and highlights open challenges, offering guidance for researchers seeking effective tools to advance the study of FVFIDEs.

Fractional calculus; Volterra–Fredholm equations; Integro differential equations; Numerical methods; Spectral techniques; Predictor–corrector schemes; Convergence analysis; Applied mathematics

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