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

Source abstract

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 , 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 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 basis in solving fractional integro‐differential problems