A reliable strategy for a category of third-kind nonlinear fractional integro-differential equations

This work develops a novel numerical approach for solving a class of third-kind fractional integro-differential equations, incorporating the Caputo–Hadamard derivative. To simplify the solution process, a set of piecewise basis functions, known as the piecewise shifted Jacobi polynomials, is introduced. Additionally, two closed-form expressions for the classical and Hadamard fractional integrals are derived. Furthermore, two corresponding operational matrices are constructed to enable efficient numerical computation of these integrals. In the proposed method, the fractional term of the problem is first expressed as a finite expansion in terms of the generated piecewise basis functions. Employing the derived operational matrices and exploiting the linear independence of the basis, the original equation is transformed into a system of algebraic equations, which is then solved to obtain the numerical solution. A rigorous convergence analysis is conducted, combining theoretical and numerical investigations to validate the reliability of the proposed scheme. The method is implemented on three illustrative examples, demonstrating its exceptional accuracy and computational efficiency in solving this class of fractional integro-differential equations.