Comparison of absolute and relative errors.

The aim of this paper is to investigate the solution of fractional-order partial differential equations and their coupled systems. A novel method is proposed, which effectively handles these problems under two-point non-local boundary conditions. The method is based on shifted Legendre polynomials, and some new operational matrices for these polynomials are constructed. In order to convert the partial differential equation together with its nonlocal boundary condition these matrices play important role. The matrices are used to convert the fractional-order derivatives and integrals, as well as the non-local boundary conditions to a system of algebraic equations. The convergence of the proposed method is rigorously analyzed and supported by a range of computational examples. The results obtained with the proposed method shows that the absolute and relative errors decreased for both the solutions X and Y as the parameter M increases. A significant reduction in both error types is observed, with the relative error |Xr| decreasing from approximately 10−1 to 10−8. We observed that the convergence rates lie in the range of 1.016 to 1.497 for |Xr|, and 0.985 to 1.451 for |Yr|. These results confirm the high precision and exponential convergence behavior of the proposed numerical method. All simulations are performed using MATLAB to validate the proposed approach. The algorithm is presented as pseudo code in the article. The MATLAB codes used for the simulation of the algorithm is presented as supplementary material.