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On simulations of the classical harmonic oscillator equation by difference equations

Abstract

We discuss the discretizations of the second-order linear ordinary diffrential equations with constant coefficients. Special attention is given to the exact discretization because there exists a difference equation whose solutions exactly coincide with solutions of the corresponding differential equation evaluated at a discrete sequence of points. Such exact discretization can be found for an arbitrary lattice spacing.

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References

  1. Agarwal RP: Difference Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. Marcel Dekker, New York; 2000:xvi+971.

    Google Scholar 

  2. Bobenko AI, Matthes D, Suris YuB: Discrete and smooth orthogonal systems: C-approximation. International Mathematics Research Notices 2003,2003(45):2415–2459. 10.1155/S1073792803130991

    MathSciNet  Article  MATH  Google Scholar 

  3. de Souza MM: Discrete-to-continuum transitions and mathematical generalizations in the classical harmonic oscillator. preprint, 2003, hep-th/0305114v5

  4. Herbst BM, Ablowitz MJ: Numerically induced chaos in the nonlinear Schrödinger equation. Physical Review Letters 1989,62(18):2065–2068. 10.1103/PhysRevLett.62.2065

    MathSciNet  Article  Google Scholar 

  5. Hildebrand FB: Finite-Difference Equations and Simulations. Prentice-Hall, New Jersey; 1968:ix+338.

    MATH  Google Scholar 

  6. Iserles A, Zanna A: Qualitative numerical analysis of ordinary differential equations. In The Mathematics of Numerical Analysis (Park City, Utah, 1995), Lectures in Applied Mathematics. Volume 32. Edited by: Renegar J, Shub M, Smale S. American Mathematical Society, Rhode Island; 1996:421–442.

    Google Scholar 

  7. Lambert JD: Numerical Methods for Ordinary Differential Systems. John Wiley & Sons, Chichester; 1991:x+293.

    MATH  Google Scholar 

  8. Lang S: Algebra. Addison-Wesley, Massachusetts; 1965:xvii+508.

    MATH  Google Scholar 

  9. Oevel W: Symplectic Runge-Kutta schemes. In Symmetries and Integrability of Difference Equations (Canterbury, 1996), London Math. Soc. Lecture Note Ser.. Volume 255. Edited by: Clarkson PA, Nijhoff FW. Cambridge University Press, Cambridge; 1999:299–310.

    Chapter  Google Scholar 

  10. Potter D: Computational Physics. John Wiley & Sons, New York; 1973:xi+304.

    MATH  Google Scholar 

  11. Potts RB: Differential and difference equations. The American Mathematical Monthly 1982,89(6):402–407. 10.2307/2321656

    MathSciNet  Article  MATH  Google Scholar 

  12. Reid JG: Linear System Fundamentals, Continuous and Discrete, Classic and Modern. McGraw-Hill, New York; 1983.

    Google Scholar 

  13. Stuart AS: Numerical analysis of dynamical systems. Acta Numerica 1994, 3: 467–572.

    MathSciNet  Article  Google Scholar 

  14. Suris YuB: The Problem of Integrable Discretization: Hamiltonian Approach, Progress in Mathematics. Volume 219. Birkhäuser, Basel; 2003:xxii+1070.

    Book  Google Scholar 

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Correspondence to Jan L Cieśliński.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Cieśliński, J.L., Ratkiewicz, B. On simulations of the classical harmonic oscillator equation by difference equations. Adv Differ Equ 2006, 040171 (2006). https://doi.org/10.1155/ADE/2006/40171

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  • DOI: https://doi.org/10.1155/ADE/2006/40171

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation