On the Convergence of a Spline Collocation Method for Nonlinear Fractional Boundary Value Problems with the Riesz-Caputo Operator

TL;DR

This paper introduces a B-spline collocation method for solving nonlinear fractional boundary value problems with the Riesz-Caputo operator, providing convergence analysis and numerical validation.

Contribution

It presents a novel collocation approach using B-splines for Riesz-Caputo fractional problems, including theoretical convergence proofs and numerical experiments.

Findings

Proved existence and uniqueness of solutions.

Established convergence of the collocation method.

Validated the approach with numerical experiments.

Abstract

Fractional boundary value problems are often used to model complex systems and processes characterized by memory effects and anomalous diffusion. In this paper, we consider fractional boundary value problems involving the Riesz-Caputo operator, which is particularly suited for modeling physical phenomena exhibiting symmetric diffusive effects. We provide an integral representation of the solution to prove existence and uniqueness of the fractional differential problem. We introduce a B-spline collocation method to approximate the solution of the problem and provide a convergence analysis, with both theoretical insights and numerical experiments.