Shifted Jacobi Operational Matrices for Solving Sequential Fractional Boundary Value Problems

Abstract

A shifted Jacobi spectral collocation method based on operational matrices is developed for a nonlinear sequential $\Psi$-Prabhakar–ABC fractional boundary value problem with nonlocal multipoint conditions. Explicit closed‑form expressions for the operational matrices of the $\Psi$-Prabhakar–ABC derivative, the $\Psi$-Riemann–Liouville integral, and the $\Psi$-Hilfer derivative are derived, and the problem is reduced to a small nonlinear algebraic system solved via Newton–Raphson iterations. Spectral convergence is proved in Sobolev spaces and confirmed numerically, showing high accuracy with few basis polynomials.