Solvability and Stability of Solutions of (q, τ)-Fractional Integro-Differential Models

In this paper, we investigate a set of nonlinear (q, τ)-fractional Fredholm integrodifferential equations that involve memory-type integral kernels and generalized fractional derivatives. Using a Galerkin technique based on (q, τ)-Legendre polynomials, we designed an approximation solution and provided a numerical scheme for calculating the integral terms and fractional derivative. Through comparison with benchmark functions and residual analysis, the approximation’s convergence is confirmed. In addition, we propose adequate conditions under which perturbed solutions stay close to the real solution in order to study the Ulam–Hyers stability of the generalized equation. The existence, uniqueness, and stability of the solution under Lipschitz-type conditions on the kernel function and the nonlinearity are demonstrated using a fixed-point theorem. The numerical tests show the method’s robustness and validate the theoretical results. These findings offer a solid foundation for the study and modeling of nonlinear systems in the (q, τ)-setting that are controlled by nonlocal fractional models.