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Hamiltonian Boundary Value Methods (HBVMs) for functional differential equations with piecewise continuous arguments

Gianmarco Gurioli, Weijie Wang, Xiaoqiang Yan

Abstract

In this paper, a class of high-order methods to numerically solve Functional Differential Equations with Piecewise Continuous Arguments (FDEPCAs) is discussed. The framework stems from the expansion of the vector field associated with the reference differential equation along the shifted and scaled Legendre polynomial orthonormal basis, working on a suitable extension of Hamiltonian Boundary Value Methods. Within the design of the methods, a proper generalization of the perturbation results coming from the field of ordinary differential equations is considered, with the aim of handling the case of FDEPCAs. The error analysis of the devised family of methods is performed, while a few numerical tests on Hamiltonian FDEPCAs are provided, to give evidence to the theoretical findings and show the effectiveness of the obtained resolution strategy.

Hamiltonian Boundary Value Methods (HBVMs) for functional differential equations with piecewise continuous arguments

Abstract

In this paper, a class of high-order methods to numerically solve Functional Differential Equations with Piecewise Continuous Arguments (FDEPCAs) is discussed. The framework stems from the expansion of the vector field associated with the reference differential equation along the shifted and scaled Legendre polynomial orthonormal basis, working on a suitable extension of Hamiltonian Boundary Value Methods. Within the design of the methods, a proper generalization of the perturbation results coming from the field of ordinary differential equations is considered, with the aim of handling the case of FDEPCAs. The error analysis of the devised family of methods is performed, while a few numerical tests on Hamiltonian FDEPCAs are provided, to give evidence to the theoretical findings and show the effectiveness of the obtained resolution strategy.
Paper Structure (10 sections, 12 theorems, 106 equations, 4 figures, 4 tables)

This paper contains 10 sections, 12 theorems, 106 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Perturbation results
  4. Accuracy analysis
  5. Formulation of the method and error analysis
  6. Numerical experiments
  7. Problem 1
  8. Problem 2
  9. Problem 3
  10. Conclusions and perspectives

Key Result

lemma thmcounterlemma

(Brugnano20-1) Let $G:[0,h]\rightarrow V$, with $V$ a vector space, admit a Taylor expansion at 0. Then, for all $j\ge0$:

Figures (4)

  • Figure 1: Numerical results for problem \ref{['5.1']}--\ref{['ICtest']} with and \ref{['5.4']} by using HBVM($2$,$2$) (left plots) and HBVM($10$,$2$) (right plots) in the time interval $[0,10^5]$, with step size $h=1/50$ (we refer to the text for more details).
  • Figure 2: Numerical results for problem \ref{['5.1']}--\ref{['ICtest']} with \ref{['5.5']} and Brugnano22-1 by using HBVM($2$,$2$) (left plots) and HBVM($10$,$2$) (right plots) in the time interval $[0,500]$, with step size $h=1/2$ (we refer to the text for more details).
  • Figure 3: Relative errors among time of the numerical Hamiltonian obtained by HBVM($2$,$2$) and HBVM($10$,$2$), when solving \ref{['5.1']}--\ref{['ICtest']} with \ref{['5.5']}, with respect to the corresponding Hamiltonian values computed via HBVM($22$,$20$) (time interval $[0,500]$, step size $h=1/2$).
  • Figure 4: Numerical results for problem \ref{['5.1']}--\ref{['ICtest']} with \ref{['Problem3']} by using HBVM($k$,$2$), for $k=2$ (left plots), $k=4$ (middle plots) and $k=10$ (right plots), in the time interval $[0,2\cdot 10^5h]$ (we refer to the text for more details).

Theorems & Definitions (14)

  • remark thmcounterremark
  • lemma thmcounterlemma
  • corollary thmcountercorollary
  • lemma thmcounterlemma
  • theorem 1
  • corollary thmcountercorollary
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem 2
  • theorem 3
  • ...and 4 more