Fuzzy Mixed Volterra-Fredholm Integral Equation with Delay: An Analytical and Numerical Solution using the Adomian Decomposition Method

Resumen

This study addresses a class of fuzzy mixed Volterra–Fredholm integral equations with delay, incorporating both analytical and numerical techniques based on the Adomian Decomposition Method (ADM). The formulation captures uncertainty by introducing fuzziness in the limits of both the Volterra and Fredholm integrals using triangular fuzzy numbers. To ensure mathematical rigor, the existence and uniqueness of the fuzzy solution are established using fixed-point theorems within a fuzzy Banach space. The ADM is systematically extended to handle delayed fuzzy kernels and construct approximate solutions iteratively. Numerical results at various values of the independent variable demonstrate excellent agreement between exact and ADM-based approximate fuzzy solutions, with minimal absolute error across α-cut levels. Graphical comparisons further validate the method’s accuracy and convergence. This approach contributes to the growing body of fuzzy integral equation theory, offering an effective tool for solving hybrid fuzzy systems arising in engineering and uncertain environments.