arXiv subjects

Kateryna Marynets

Publications and source records attributed to Kateryna Marynets.

4 recordsLinked to original sources

arXiv2026

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

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.

math.NA↗

arXiv2025

A numerical Bernstein splines approach for nonlinear initial value problems with Hilfer fractional derivative

The Hilfer fractional derivative generalizes and interpolates between the commonly used Riemann-Liouville and Caputo fractional derivative. In general, solutions to Hilfer fractional derivative initial value problems are singular for $t \downarrow t_0$, providing a challenge in numerical approximation. In this work, we present a Bernstein splines method to obtain accurate approximations to solutions of nonlinear Hilfer fractional derivative initial value problems, including solutions with singular behavior. We provide explicit convergence requirements and asymptotic convergence rates to analytical solutions. Moreover, the method is efficiently implemented, using a vectorized approach and parallelization in polynomial order. Numerical experiments show empirical convergence corresponding to analytical results and a significantly higher accuracy compared to the commonly used fractional Adams-Bashforth-Moulton predictor-corrector method. Furthermore, our method is able to simulate nonlinear systems with Hilfer fractional derivatives, as demonstrated by the application to the fractional Van der Pol oscillator.

math.NA↗

arXiv2022

Approximation approach to the fractional BVP with the Dirichlet type boundary conditions

We use a numerical-analytic technique to construct a sequence of successive approximations to the solution of a system of fractional differential equations, subject to Dirichlet boundary conditions. We prove the uniform convergence of the sequence of approximations to a limit function, which is the unique solution to the boundary value problem under consideration, and give necessary and sufficient conditions for the existence of solutions. The obtained theoretical results are confirmed by a model example.

math.NA↗

arXiv2021

Study of a nonlinear fractional boundary value problem via the dichotomy-type technique

We present a new view onto the successive approximations' approach in study of the two-point nonlinear fractional boundary value problems. In order to reduce the original problem and further construct its approximate solution we use the co-called 'freezing' technique and the dichotomy-type approach. These lead to improvement of the sufficient conditions for application of the aforementioned method and sharpen the obtained estimates.

math.CA↗