Nonlinear fractional-periodic boundary value problems with Hilfer fractional derivative
Existence and numerical approximations of solutions
Niels Goedegebure (Nanyang Technological University)
Kateryna Marynets (TU Delft - Electrical Engineering, Mathematics and Computer Science)
More Info
expand_more
Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.
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 ↓0. To overcome this singularity, we construct a sequence of converging solutions in a weighted continuous function space. We present a Bernstein spline-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.
help_outline