Analytical Representation and Applications of Solutions to a Loaded Fractional Integro-Differential Equation

We study the Cauchy problem for a loaded fractional integro-differential equation with a time-dependent diffusion coefficient. By reducing the problem to an equivalent Volterra integral equation of the second kind, we derive explicit analytical representations of solutions under appropriate regularity assumptions. The construction of the associated resolvent kernel allows us to establish existence and uniqueness results and to investigate the role of the fractional order and the loading term in the solution structure. Two illustrative examples are presented to demonstrate the applicability of the proposed approach.