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

Niels Goedegebure, Kateryna Marynets

TL;DR

This work addresses nonlinear initial-value problems driven by the Hilfer fractional derivative, whose solutions are often singular near the initial time. It develops a Bernstein splines-based numerical method within a transformed, weighted function space to achieve accurate approximations and rigorous convergence rates for general Hilfer parameters. The authors prove asymptotic convergence results, implement a highly efficient vectorized algorithm, and demonstrate superior accuracy compared to the predictor-corrector approach on test problems and a nonlinear Hilfer-fractional Van der Pol oscillator. The methodology effectively handles singular behavior and memory effects, offering a practical tool for solving a broad class of Hilfer FIVPs with potential applications in control and physics.

Abstract

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.

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

TL;DR

This work addresses nonlinear initial-value problems driven by the Hilfer fractional derivative, whose solutions are often singular near the initial time. It develops a Bernstein splines-based numerical method within a transformed, weighted function space to achieve accurate approximations and rigorous convergence rates for general Hilfer parameters. The authors prove asymptotic convergence results, implement a highly efficient vectorized algorithm, and demonstrate superior accuracy compared to the predictor-corrector approach on test problems and a nonlinear Hilfer-fractional Van der Pol oscillator. The methodology effectively handles singular behavior and memory effects, offering a practical tool for solving a broad class of Hilfer FIVPs with potential applications in control and physics.

Abstract

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 , 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.

Paper Structure

This paper contains 11 sections, 12 theorems, 76 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary definitions and results
  3. Fractional calculus
  4. Polynomial splines approximation
  5. Bernstein spline approximation results
  6. Numerical implementation
  7. Numerical results
  8. Polynomial example and convergence results
  9. Comparison with Predictor-Corrector method
  10. Fractional Van der Pol oscillator
  11. Conclusions

Key Result

Proposition 2.1

For $\alpha, \beta > 0$ and $a\leq t\leq b$, if $f \in L^p[a, b]$ with $1\leq p \leq \infty$, we have: almost everywhere. Furthermore, if $\alpha+\beta>1$, equality eq:semigroupeq holds pointwise.

Figures (3)

  • Figure 1: Results of convergence in $q$, $h$ and $\varepsilon$ for system \ref{['eq:num_pol_sys']} with $\alpha, \beta = 0.5$, $k=0.9$, $\tilde{y}_0 = 1$.
  • Figure 2: Comparison of approximations of the predictor-corrector method (left) with our Bernstein splines setup (middle) and respective errors (right) for decreasing knot size values $h$ for system \ref{['eq:diethelm_sys_lin_a']}.
  • Figure 3: Results for the Hilfer-derivative fractional Van der Pol oscillator \ref{['eq:res_vdp_sys']} for ${\tilde{x}}_0 = 1$, $\alpha = 1/2$ and various $\beta$.

Theorems & Definitions (34)

  • Definition 2.1: Riemann-Liouville fractional integral, Samko1987
  • Proposition 2.1: Semigroup property of fractional integration, Samko1987
  • Proposition 2.2: $L^p$ boundedness in $f$, Samko1987
  • Definition 2.2: Beta function, Samko1987
  • Definition 2.3: Incomplete beta function, Osborn1968
  • Proposition 2.3: Fractional integral of monomials
  • proof
  • Proposition 2.4: Fractional integral of monomial with right-local support
  • proof
  • Definition 2.4: Hilfer fractional derivative, Hilfer2000
  • ...and 24 more