Nonlinear fractional-periodic boundary value problems with Hilfer fractional derivative: existence and numerical approximations of solutions

TL;DR

This paper establishes existence conditions for solutions to nonlinear boundary value problems involving the Hilfer fractional derivative, introduces a Bernstein splines-based numerical method, and demonstrates its convergence and effectiveness through numerical examples.

Contribution

It generalizes boundary conditions for Hilfer derivatives, develops a convergent numerical approximation method, and applies it to nonlinear problems with singular solutions.

Findings

Proven existence of analytical solutions under new boundary conditions.

Developed a Bernstein splines-based numerical method with proven convergence.

Numerical examples confirm empirical convergence and ability to handle singularities.

Abstract

We prove conditions for existence of analytical solutions for boundary value problems with the Hilfer fractional derivative, generalizing the commonly used Riemann-Liouville and Caputo operators. The boundary values, referred to in this paper as fractional-periodic, are fractional integral conditions generalizing recurrent solution values for the non-Caputo case of the Hilfer fractional derivative. Analytical solutions to the studied problem are obtained using a perturbation of the corresponding initial value problem with enforced boundary conditions. In general, solutions to the boundary value problem are singular for $t\downarrow 0$. To overcome this singularity, we construct a sequence of converging solutions in a weighted continuous function space. We present a Bernstein splines-based implementation to numerically approximate solutions. We prove convergence of the numerical method, providing convergence criteria and asymptotic convergence rates. Numerical examples show empirical convergence results corresponding with the theoretical bounds. Moreover, the method is able to approximate the singular behavior of solutions and is demonstrated to converge for nonlinear problems. Finally, we apply a grid search to obtain correspondence to the original, non-perturbed system.