Discrete Simulation of a Class of Distributed Systems Using Functional Analytic Methods

  • Vitali DymkouEmail author
  • Rudolf Rabenstein
  • Peter Steffen
Original Article


This paper investigates the discrete simulation of the solution of initial-boundary-value problems that typically arise in technical areas. Since many of them lead to unbounded and non-self-adjoint differential operators, we have to use a rather general theory as a mathematical basis. For the class of sectorial operators with a compact resolvent operator, the solution of initial-boundary-value problem can be represented by means of a certain holomorphic semigroup. It is shown that the solution can be expanded with respect to the canonical system of the considered operator. Such an expansion corresponds to a multi-dimensional functional transformation in the frequency domain. This fact leads to simple structures for the realization of the resulting system. Computationally efficient numerical algorithms can be derived by proper methods well-known from the theory of digital signal processing.


Multi-dimensional systems Partial differential equations Frequency domain Multi-functional transformations Spectral theory 


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  1. Balakrishnan, A.V. 1976Applied Functional AnalysisSpringer VerlagNew York, Heidelberg, BerlinGoogle Scholar
  2. Bose, N.K. 1982Applied Multidimensional Systems TheoryVan Nostrand ReinholdNew YorkGoogle Scholar
  3. Ditkin, V.A., Prudnikov, A.P. 1962Operational calculus in two variables and its applicationsPergamon PressOxfordGoogle Scholar
  4. Dudgeon, D.E., Mersereau, R.E. 1984Multidimensional digital signal processingPrentice-HallEnglewood CliffsGoogle Scholar
  5. Dymkou V., & Rabenstein R., & Steffen P. (2004a). An Operator Approach for Physical Modelling of Distributed Parameter Systems, Proceedings of the 6th IMA Conf. on Mathematics in Signal Processing. Cirencester.Google Scholar
  6. Dymkou V., Rabenstein R., & Steffen P. (2004b). Application of operator theory to discrete simulation of continuous systems, Proceedings of 16th International Symposium on Mathematical Theory of Networks and Systems(MTNS), Leuven, Belgium.Google Scholar
  7. Fursikov, A.V. 2001Stabilizability of a quasi-linear parabolic equation by means of a boundary control with feedbackSbornik Mathematics192593639CrossRefzbMATHMathSciNetGoogle Scholar
  8. Gantmacher, F.R. 1959Applications of the theory of matricesInterscience PublishersNew YorkGoogle Scholar
  9. Girod, B., Rabenstein, R., Stenger, A. 2001Signals and systemsWileyChichesterGoogle Scholar
  10. Gohberg I.C., & Krein M.G. (1969). Introduction to the theory of linear non-selfadjoint operators. Trans. of Math. Monogr. 18, Providence, RI: AMS.Google Scholar
  11. Goldberg, S. 1966Unbounded linear operatorsMcGraw-HillNew YorkGoogle Scholar
  12. Henry, D. 1981Geometric Theory of Semilinear Parabolic EquationsSpringer VerlagNew York, Heidelberg, BerlinGoogle Scholar
  13. Hille, E., Phillips, R.S. 1957Functional analysis and semi-groupsAmerican Mathematical Society ProvidenceRIGoogle Scholar
  14. Il’in, V.A. (1976). Necessary and sufficient conditions for the subsystem of eigenfunctions and associated functions of Keldysh’s pencil of ordinary differential operators to form the basis. DAN USSR 227 (4), 796–799 (in Russian).Google Scholar
  15. Kato, T. 1966Perturbation theory for linear operatorsSpringer VerlagNew York, Heidelberg, BerlinGoogle Scholar
  16. Keldysh, M.V. 1951About eigenvalues and eigenfunction for some classes of non-self-adjoint equationsDAN USSR771114(in Russian)zbMATHGoogle Scholar
  17. Keldysh M.V. (1971). On the completeness of eigenfunctions for certain classes of not self-adjoint linear operators, Uspekhi Mathematics Nauk, 26(4), pp.15–41 (in Russian); English translation: Russian Mathematics Surveys, 26 (4) 15–44.Google Scholar
  18. Kowalczuk, Z. 1993August). Discrete approximation of continuous-time-systems: a surveyIEE Proceedings-G,140264278Google Scholar
  19. Krein M.G. (1963). Criteria for completeness of the system of root vectors of a dissipative operator, Uspekhi Mathematics Nauk, 14, 145–152 (in Russian); English translation: American Mathematics Society Translation.Google Scholar
  20. Lidskii V.B. (1959). Conditions for the completeness of the system of root subspaces for nonselfadjoint operators with discrete spectrum, Trud. Moskov. Mat. Obshch. 8, 83–120 (in Russian); English translation: American Mathematics Soceity Translation 34(2).Google Scholar
  21. Lorenzi L., Lunardi A., Metafune G., & Pallara D. Analytic Semigroups and Reaction-Diffusion Problems, 8th International Internet Seminar 2004–2005, english.cgi/d1404894/The%20whole%20manuscriptGoogle Scholar
  22. Markus A.S. (1988). Introduction to the theory of polynomial operator pencils, AMS Translation Mathematics Monographs 71.Google Scholar
  23. Naimark, M.A. 1962Spectral analysis of non-self-adjoint operatorsAmerican Mathematics Society Translation205575MathSciNetGoogle Scholar
  24. Petrausch S., & Rabenstein R. (2004, September). A simplified design of multidimensional transfer function models, International Workshop on Spectral Methods and Multirate Signal Processing (SMMSP2004). Vienna, Austria, pp. 35–40.Google Scholar
  25. Rabenstein R., Trautmann L. (2002, June). Multidimensional transfer function models, IEEE Tr. on Circuits and Systems, I, 49(6), 852–861.Google Scholar
  26. Schüßler, H.W. 1981A signal processing approach to simulationFREQUENZ35174184Google Scholar
  27. Trautmann L., Rabenstein R. (2003). Digital sound synthesis by physical modeling using the functional transformation method. New York, Kluwer Academic/Plenum Publishers.Google Scholar
  28. Yosida, K. 1980Functional analysisSpringer VerlagGrundlehren der Math. WissGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Vitali Dymkou
    • 1
    Email author
  • Rudolf Rabenstein
    • 1
  • Peter Steffen
    • 1
  1. 1.Multimedia Communications and Signal ProcessingUniversity of Erlangen-NurembergErlangenGermany

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