Existence, Multiplicity, and Spectral Approximation for Nonlinear Caputo FBVPs: A Fixed-Point and Operational Matrix Approach

Abstract

A class of fractional boundary value problems (FBVPs) under Caputo fractional derivatives, both linear and nonlinear, is examined in this work. Using fixed-point index theory and the accompanying Green's function, we first prove the existence of solutions. To achieve accurate numerical results, an efficient operational matrix method is implemented. The reliability of the algorithm is validated through a series of examples, where the numerical results show excellent agreement with exact solutions. We also investigate the influence of the fractional order α on the system dynamics and show that there is a smooth transition from memory-dependent fractional regimes to classical integer-order dynamics. In the nonlinear cases, the results demonstrate the existence of multiple positive solutions and bifurcation phenomena that affirm the fractional parameter as a key control parameter in the qualitative behavior of the system.