ADVANCING FRACTIONAL BOUNDARY VALUE PROBLEMS SOLUTIONS USING SHIFTED VIETA–LUCAS POLYNOMIALS AND SPECTRAL APPROACHES

This study aims to extend spectral methods using operational matrices of Vieta–Lucas polynomials to address fractional-order boundary value problems (FBVPs). Although spectral methods have been used previously for solving fractional-order problems with initial conditions, the proposed approach broadens their application to FBVPs. This is accomplished by introducing newly developed fractional-order integral operational matrices within the Riemann–Liouville framework by using a basis set of shifted Vieta–Lucas polynomials. These operational matrices enable systematic incorporation of monomial expressions into the algorithm’s framework, effectively transforming FBVPs into a system of algebraic equations. The method accommodates both polynomial and non-polynomial solution forms, broadening its applicability. Comprehensive numerical experiments validate the approach’s precision and computational efficacy, with results showing excellent agreement with the exact solutions of the problems. Additionally, by comparing the outcomes of the proposed approach with those presented in the existing literature, we highlight its effectiveness as a powerful alternative for solving FBVPs.