Analytical and numerical study of nonlinear Riesz-Caputo boundary value problems

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 nonlinear fractional boundary value problems of arbitrary order α involving the Riesz-Caputo operator, which is particularly suited for modelling physical phenomena exhibiting symmetric diffusive effects. By reducing the problem to a dominant singular integral equation of Cauchy type and inverting it in closed form, we derive an integral representation of the solution, which we use to prove existence and uniqueness. The closed-form inversion also describes the typical behaviour of the solution near the endpoints of the interval and provides an exact singular solution, which we use as a test case. We then analyse a B-spline collocation method for the approximation of the solution, proving its convergence and illustrating it by numerical experiments.