Existence and uniqueness of solutions for a generalized time-integro-fractional reaction–diffusion equation via the separation method of semi-fixed variables

Abstract

This paper investigates the existence, uniqueness, and analytical solutions of a generalized time-integro-fractional reaction--diffusion equation, which serves as an effective mathematical model for describing diffusion phenomena with nonlocal memory 
effects encountered in physics, biology, engineering, and related disciplines. The governing equation is formulated in terms of a generalized fractional derivative together with an integral memory operator to capture hereditary characteristics of the 
underlying processes. First, the integro-fractional problem is transformed into an equivalent Volterra-type integral equation. Sufficient conditions for the existence and uniqueness of solutions are established by employing the Banach contraction principle and suitable fixed-point techniques under appropriate Lipschitz continuity assumptions. Subsequently, the separation method of semi-fixed variables is developed to obtain exact analytical solutions of the proposed generalized time-integro-fractional 
reaction--diffusion equation. The method effectively reduces the original integro-fractional partial differential equation into a system of solvable fractional ordinary differential equations, leading to closed-form solutions expressed in terms of generalized Mittag--Leffler functions. Several illustrative examples are presented to demonstrate the applicability, accuracy, and computational efficiency of the proposed approach. The obtained results reveal that the separation method of semi-fixed 
variables provides a simple, systematic, and reliable analytical framework for solving a broad class of generalized integro-fractional reaction--diffusion equations with memory effects, thereby extending existing methodologies for fractional differential 
equations