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Relation between two Sinc-collocation methods for Volterra integral equations of the second kind and further improvement

Tomoaki Okayama

TL;DR

This paper clarifies the relationships among Sinc-based methods for Volterra equations of the second kind, showing that Stenger's Sinc-collocation and Rashidinia–Zarebnia's collocation share collocation values but are not generally equivalent, and that both achieve root-exponential convergence under Hölder conditions. It proves a matching convergence rate for both collocation approaches and introduces a DE-transformation–based Sinc-collocation method that attains significantly faster almost-exponential convergence, with numerical experiments indicating practical efficiency gains over Nyström variants. The study also develops a generalized SE/DE framework to handle endpoint values without zero-boundary requirements and provides rigorous proofs for existence, uniqueness, and error bounds of the collocation schemes. Overall, the DE-Sinc-collocation method emerges as the best-performing collocation approach in the tested scenarios, offering substantial accuracy with reduced computational cost.

Abstract

Two different Sinc-collocation methods for Volterra integral equations of the second kind have been independently proposed by Stenger and Rashidinia--Zarebnia. However, their relation remains unexplored. This study theoretically examines the solutions of these two methods, and reveals that they are not generally equivalent, despite coinciding at the collocation points. Strictly speaking, Stenger's method assumes that the kernel of the integral is a function of a single variable, but this study theoretically justifies the use of his method in general cases, i.e., the kernel is a function of two variables. Then, this study rigorously proves that both methods can attain the same, root-exponential convergence. In addition to the contribution, this study improves Stenger's method to attain significantly higher, almost exponential convergence. Numerical examples supporting the theoretical results are also provided.

Relation between two Sinc-collocation methods for Volterra integral equations of the second kind and further improvement

TL;DR

This paper clarifies the relationships among Sinc-based methods for Volterra equations of the second kind, showing that Stenger's Sinc-collocation and Rashidinia–Zarebnia's collocation share collocation values but are not generally equivalent, and that both achieve root-exponential convergence under Hölder conditions. It proves a matching convergence rate for both collocation approaches and introduces a DE-transformation–based Sinc-collocation method that attains significantly faster almost-exponential convergence, with numerical experiments indicating practical efficiency gains over Nyström variants. The study also develops a generalized SE/DE framework to handle endpoint values without zero-boundary requirements and provides rigorous proofs for existence, uniqueness, and error bounds of the collocation schemes. Overall, the DE-Sinc-collocation method emerges as the best-performing collocation approach in the tested scenarios, offering substantial accuracy with reduced computational cost.

Abstract

Two different Sinc-collocation methods for Volterra integral equations of the second kind have been independently proposed by Stenger and Rashidinia--Zarebnia. However, their relation remains unexplored. This study theoretically examines the solutions of these two methods, and reveals that they are not generally equivalent, despite coinciding at the collocation points. Strictly speaking, Stenger's method assumes that the kernel of the integral is a function of a single variable, but this study theoretically justifies the use of his method in general cases, i.e., the kernel is a function of two variables. Then, this study rigorously proves that both methods can attain the same, root-exponential convergence. In addition to the contribution, this study improves Stenger's method to attain significantly higher, almost exponential convergence. Numerical examples supporting the theoretical results are also provided.

Paper Structure

This paper contains 24 sections, 24 theorems, 92 equations, 4 figures.

Table of Contents

  1. Introduction and summary
  2. Preliminaries
  3. Sinc approximation and Sinc indefinite integration
  4. SE-Sinc approximation and SE-Sinc indefinite integration
  5. DE-Sinc approximation and DE-Sinc indefinite integration
  6. Generalized SE/DE-Sinc approximation
  7. Sinc-Nyström methods
  8. SE-Sinc-Nyström method
  9. DE-Sinc-Nyström method
  10. Existing Sinc-collocation methods
  11. Sinc-collocation method by Stenger
  12. Sinc-collocation method by Rashidinia and Zarebnia
  13. Main result 1: Relation between the two methods and their convergence
  14. Sinc-collocation method combined with the DE transformation
  15. Derivation of the DE-Sinc-collocation method
  16. ...and 9 more sections

Key Result

Theorem 2.1

Assume that $f$ is analytic on $\psi^{\text{\tiny{\rm{SE}}}}(\mathscr{D}_d)$ for $d$ with $0<d<\uppi\>\!$, and there exists constants $K$ and $\alpha$ such that holds for all $z\in\psi^{\text{\tiny{\rm{SE}}}}(\mathscr{D}_d)$. Let $N$ be a positive integer, and let $h$ be selected by the formula Then, there exists a constant $C$ independent of $N$ such that

Figures (4)

  • Figure 1: Errors with respect to $N$ for \ref{['eq:example1']}.
  • Figure 2: Errors with respect to the computation time for \ref{['eq:example1']}.
  • Figure 3: Errors with respect to $N$ for \ref{['eq:example2']}.
  • Figure 4: Errors with respect to the computation time for \ref{['eq:example2']}.

Theorems & Definitions (31)

  • Theorem 2.1: Stenger stenger93:_numer
  • Theorem 2.2: Okayama et al. okayama09:_error
  • Theorem 2.3: Tanaka et al. tanaka09:_desinc
  • Theorem 2.4: Okayama et al. okayama09:_error
  • Definition 1
  • Definition 2
  • Theorem 2.5: Okayama okayama13:_note
  • Theorem 2.6: Okayama okayama13:_note
  • Theorem 3.1: Okayama et al. okayama13:_theo
  • Theorem 3.2: Okayama et al. okayama13:_theo
  • ...and 21 more