Exact Solutions and Convergence Analysis of Linear Fractional Integro‐Differential Equations With Caputo Derivatives

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Abstract
Modeling systems with memory and nonlocal effects is significantly influenced by fractional integro‐differential equations; however, precise analytical solutions are still hard to find. This article establishes a consistent deductive mathematical framework for the emergence of closed‐form solutions to linear fractional Fredholm and Volterra integro‐differential equations that are formulated in the Caputo sense. The Banach fixed‐point theorem is used to establish the uniqueness of explicit solutions, which are derived using the Riemann–Liouville fractional integral and Mittag–Leffler functions in conjunction with each other. As the fractional order approaches unity, the proposed solutions naturally recover classical integer‐order models, thereby bridging the gap between classical and fractional dynamics. Comparative analysis with the Adomian Decomposition and Variational Iteration Methods demonstrates the Modified Adomian Decomposition Method’s superior computational efficiency and convergence. Error analysis and numerical simulations verify the theoretical discoveries. In addition to contributing to the analytical advancement of fractional integro‐differential modeling, these findings offer benchmark solutions for validating numerical schemes.

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