A Novel Orthogonal TU¯ Chebyshev Polynomial Basis for High-Accuracy Solution of Fractional Integro-Differential Equations

The main question addressed in this study is whether a newly constructed orthogonal basis, which is a combination of first- and second-kind Chebyshev polynomials, can provide a more accurate and efficient numerical method for solving fractional integro-differential equations. To this end, a new family of normalized orthogonal polynomials, called Chebyshev polynomials TU¯, is introduced. This basis is constructed by normalizing the first- and second-kind Chebyshev polynomials, forming their structural sum, and applying a weighted Gram–Schmidt process, yielding a normalized orthogonal system with enhanced approximation flexibility. Based on the TU¯ basis, the main problem is transformed into a system of algebraic equations using the collocation method, reducing computational complexity. A rigorous error analysis ensures the convergence of the method and validates its reliability. Numerical results demonstrate that the proposed method achieves higher accuracy and faster convergence compared with classical Chebyshev-based approaches, confirming the effectiveness of the TU¯ basis in solving fractional integro-differential problems