A highly accurate Hermite polynomial-based least-squares approach for solving fractional Volterra-Fredholm integro-differential equations

This paper presents a comprehensive numerical study on the efficacy of a Hermite polynomial-based least-squares method for solving Volterra-Fredholm fractional integro-differential equations (V-FFIDEs). In our approach, we construct an approximate solution as a finite expansion of Hermite polynomials. This trial solution is systematically substituted into the governing V-FFIDE. Following the analytical evaluation of the fractional and integral operators, we formulate a residual function. The core of our method involves minimizing the squared norm of this residual over the problem domain, a process that transforms the original problem into a well-defined system of linear algebraic equations. To validate our methodology, we conducted a series of numerical experiments on a collection of representative examples. The results of our study, presented through detailed tables of numerical outcomes and comparative graphical illustrations, conclusively demonstrate the high accuracy, computational efficiency, and robust convergence of the proposed technique.