Neutral Fractional Volterra Delay Integro‐Differential Equation With Variable Coefficients and Weakly‐Singular Kernel: Existence‐Uniqueness, Ulam‐Hyers Stability, Numerical Scheme

ABSTRACT This article comprehensively studies the Neutral Fractional Volterra Delay Integro‐Differential Equation (NFVDIDE) with variable coefficients and weakly‐singular kernel. It takes into account Caputo's fractional derivative of orders and for the current state and delayed state, respectively. To establish the existence results, we utilized the weighted norm concept and the Banach Contraction Principle. Furthermore, we established the Ulam‐Hyers stability result. Additionally, we proposed an interpolation‐based robust numerical algorithm to approximate the solutions of NFVDIDE and conducted a detailed error analysis. The proposed algorithm achieved a computational accuracy of . We analyzed several examples to demonstrate our theoretical findings and illustrated the efficacy of the obtained numerical scheme for various values of .