Nonlinear fractional-periodic boundary value problems with Hilfer fractional derivative: existence and numerical approximations of solutions
Niels Goedegebure, Kateryna Marynets
TL;DR
This work develops existence theory and convergent Bernstein-splines numerics for nonlinear fractional boundary value problems driven by the Hilfer derivative ${_0}D_t^{\alpha,\beta}$ with fractional-periodic boundary conditions ${_0}I_{0}^{1-\gamma}x(0) = {_0}I_{T}^{1-\gamma}x(T)$, where $0<\alpha<1$, $0\leq\beta\leq1$ and $\gamma = \alpha+\beta-\alpha\beta$. By perturbing the IVP with a constant vector $\nu$ and enforcing the boundary via $\Delta_T(\tilde{\bf x}_0)=0$, the paper proves existence and uniqueness of solutions in the weighted space $C_{1-\gamma}[0,T]$, derives an explicit convergence bound for the iterative scheme, and shows the perturbed solution converges to the original BVP solution. Numerically, it introduces a Bernstein-splines approach on an $\varepsilon$-shifted domain with rigorous convergence guarantees and an asymptotic rate of $\mathcal{O}\left((h/\sqrt{q})^{\alpha}\right)$, enabling accurate handling of the singular behavior at $t\to0^+$. The results are demonstrated on polynomial and nonlinear examples, including a grid-search procedure to recover the non-perturbed problem, and show effective convergence across the Hilfer spectrum ($\beta\in[0,1]$). Overall, the methods generalize Caputo/Riemann-Liouville FBVPs to Hilfer-type operators and provide a practical, singularity-aware framework for numerical solutions.
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.