Solving Systems of Fractional-Order Differential Equations Using a Reproducing Kernel-Based Approach

This paper introduces a new technique utilizing the reproducing kernel method (RKM) to solve both linear and nonlinear systems of fractional-order differential equations (SFDEs). The technique carefully integrates essential elements, including the solution space, basis functions, strategic point selection, and a suitable inner product. While solving SFDEs can be notoriously complex, leading to issues such as extended computation times, elevated matrix condition numbers, reduced accuracy compared to alternative methods, and discrepancies between theoretical predictions and numerical outcomes, our approach effectively mitigates these challenges. We achieve this by eliminating the need for Gram–Schmidt orthogonalization and incorporating efficient linear algebra techniques. By strategically selecting the solution space and evaluation points, we develop an RKM-based approach that combines computational efficiency, high accuracy, and straightforward implementation.