SPECTRAL OPTIMAL CONTROL SOLUTION FOR NONLINEAR WEAKLY SINGULAR SYSTEM OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

This research introduces an innovative numerical method for addressing a specific category of nonlinear weakly singular fractional integro-differential equations (NWSFIDEs). The proposed approach primarily involves utilizing the Riemann–Liouville fractional-order integral definition to avoid the issue of singularity in the core problem. Then, Genocchi operational integration matrix of fractional order is presented to fully approximate the fractional-order integral. To determine the fractional-order derivative of the unknown function, we employed a series of partially wavelet-based fractional-order Mittag-Leffler functions (PWBFMLFs). To develop the proposed numerical method, the core problem is reformulated as an equivalent optimal control problem involving a specific experimental function. Theorems for error and convergence analysis of the proposed techniques are examined. Theoretical predictions are validated through numerical examples, which show that the proposed method, requiring fewer degrees of freedom, surpasses existing methods.