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aequationes mathematicae

, Volume 30, Issue 1, pp 223–238 | Cite as

Initial-value problems for linear partial difference equations

  • A. Gameiro Pais
Research Papers
  • 36 Downloads

Abstract

The method presented in [4] for the solution of linear difference equations in a single variable is extended to some equations in two variables. Every linear combination of a given functionf and of its partial differences can be obtained by the discrete convolution product off by a suitable functionl (which depends on the considered linear combination), and we want to solve in a convolutional form difference equations in the whole plane. However, the convolution of two functions may not be possible if their supports contain half straight lines with opposite directions. To avoid this, we take four sets of functions corresponding to the quadrants such thatl belong to every set, every set endowed with the convolution and with the usual addition is a ring, and there is an inverse ofl in each of the four rings. This is attained by taking, for each ring, a set of functions whose supports belong to suitable cones. After choosing such rings, a very natural initial-value first-order Cauchy Problem (in partial differences) is reduced to a convolutional form. This is done either by a direct method or by introducing the forward difference functionsδ i f(i=1,2) in a general way depending on the shape of the support off so that Laplace-like formulas with initial and final values) hold. Applications to difference equations in the whole plane and to partial differential problems are made.

AMS (1980) subject classification

Primary 39A10 

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References

  1. [1]
    Fenyö, I.,Eine neue Method zur Lösung von Differenzengleichungen nebst Anwendungen. Periodica Polytechnica3 (1959), 135–151.Google Scholar
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    Godounov, S. andRiabenki, V.,Schémas aux differences. Introduction à la théorie. Traduit du russe par Irina Pétrova Éditions Mir, Moscow, 1977.Google Scholar
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    Mikusinski, J.,Operational calculus. Pergamon Press, London-New York-Toronto-Sydney, 1967.Google Scholar
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    Pais, A. G.,A Laplace-like method for solving linear difference equations. Aequationes Math.27 (1984), 49–56.Google Scholar
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    Richtmyer, R. andMorton, K.,Difference methods for initial-value problems.Second edition. Interscience Publishers, John Wiley & Sons, New York-London-Sydney, 1967.Google Scholar

Copyright information

© Birkhäuser Verlag 1986

Authors and Affiliations

  • A. Gameiro Pais
    • 1
  1. 1.Centro de Matemática e Aplicações FundamentaisLisboa CodexPortugal

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