Fractional-order boundary value problems solutions using advanced numerical technique

Abstract

The main motivation of this study is to extend the use of the operational matrices approach to solve fractional-order two-point boundary value problems (TPBVPs), a method often employed in the literature for solving fractional-order initial value problems. Our proposed approach employs innovative operational matrices, specifically the integral operational matrices based on Chelyshkov polynomials (CPs), a type of orthogonal polynomials. These operational matrices enable us to integrate monomial terms into the algorithm, effectively converting the problem into easily solvable Sylvester-type equations. We provide a comprehensive comparison to demonstrate the accuracy and computational advantages of our proposed approach against existing methods, including the exact solution, the Haar wavelet method (HWM), the Bessel collocation method (BCM), the Pseudo Spectral Method (PSM), the Generalized Adams–Bashforth–Moulton Method (GABMM) and the fractional central difference scheme (FCDS) through numerical examples. Additionally, our proposed approach is well-suited for solving problems with both polynomial and non-polynomial solutions.