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Existence of solution to some mixed problems for linear differential-algebraic systems of partial differential equations

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Abstract

We consider a linear differential-algebraic system of PDEs with special matrix coefficients. Two cases are investigated. In the first one, the system has a small index, and the matrix at unknown vector-function, while the system written in the canonical form, is arbitrary. In the second case, the system has an arbitrary index, while a matrix at the small term is triangular. In both the cases, using the methods of characteristics and successive approximations, we prove the existence of a unique classical solution of mixed problems for the considered differential-algebraic systems of PDEs.

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References

  1. Sobolev, S. L. “On a new problem of mathematical physics”, Izv. Akad. Nauk SSSR, Ser. Mat. 18, No. 1, 3–50 (1954). [in Russian]

    MathSciNet  MATH  Google Scholar 

  2. Demidenko, G. V., Uspenskii, S. V. Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative (Marcel Dekker, New York-Basel, 2003).

    MATH  Google Scholar 

  3. Ruschinskii, V. M. “Linear and non-linear space model of boiler generators”, Voprosi identifik. i modelir., 8–15 (1968). [in Russian]

  4. Soto, M. S., Tischendorf, C. “Numerical analysis of DAEs from coupled circuit and semiconductor simulation”, Appl. Numer. Math. 53, 471–488 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  5. Lucht, W. “Partial differential-algebraic systems of second order with symmetric convection”, Appl. Numer. Math. 53, 357–371 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  6. Gaidomak, S. V., Chistyakov, V. F. “On systems of not Cauchy-Kovalevskaya type with index (1, k)”, Vichislit. Tekhnol. 10, No. 2, 45–59 (2005). [in Russian]

    MATH  Google Scholar 

  7. Lucht, W., Strehmel, K., Eichler-Liebenow, C. “Indexes and special discretization methods for linear partial differential algebraic equations”, BIT 39, No. 3, 484–512 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  8. Campbell, S. L., Marszalek, W. “The index of an infinite dimensional impliscit system”, Math. and Comp. Model. of Syst. 5, No. 1, 18–42 (1999).

    Article  MATH  Google Scholar 

  9. Gaidomak, S. V. “Boundary value problem for a first-order linear parabolic system”, Comput. Math. Math. Phys., 54, No. 4, 620–630 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  10. Gaidomak, S. V. “Numerical solution of linear differential-algebraic systems of partial differential equations”, Comput. Math. Math. Phys., 55, No. 9, 1501–1514 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  11. Gaidomak, S. V. “On the stability of an implicit spline collocation difference scheme for linear partial differential algebraic equations”, Comput. Math. Math. Phys., 53, No. 9, 1272–1291 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  12. Bormotova, O. V., Gaidomak, S. V., Chistyakov, V. F. “On the solvability of degenerate systems of partial differential equations”, Russian Math. 49, No. 4, 15–26 (2005).

    MathSciNet  MATH  Google Scholar 

  13. Courant, R., Hilbert, D. Methods of Mathematical Physics, vol. 2, Differential Equations (Wiley-VCH. Verlag GmbH & Co. KGaA, 2004)

  14. Petrovsky, I. G. Lectures on Partial Differential Equations (Interscience, New York 1954).

    MATH  Google Scholar 

  15. Roždestvenskii, B. L., Yanenko, N. N. Systems of Quasilinear Equations and Their Applications to Gas Dynamics (AMS; Translations of Mathematical Monographs. V. 55).

  16. Gaidomak, S. V. “The canonical structure of a pencil of degenerate matrix functions”, Russian Math. 56, No. 2, 19–28 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  17. Vladimirov, V. S. Equations of Mathematical Physics (Nauka, Moscow, 1984).

    Google Scholar 

  18. Godunov, S. K. Equations of Mathematical Physics (Nauka, Moscow, 1979). [in Russian]

    Google Scholar 

  19. Gaidomak, S. V. “Spline collocation method for linear singular hyperbolic systems”, Comput. Math. Math. Phys., 48, No. 7, 1161–1180 (2008).

    Article  MathSciNet  Google Scholar 

  20. Gantmacher, F. R. The Theory of Matrices (Chelsey Publ. Comp. New York, 1959).

    MATH  Google Scholar 

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Acknowledgments

The work is fulfilled within the project No.0348-216-0009 “Qualitative theory and numerical analysis of differential-algebraic equations” of the Siberian Branch of the Russian Academy of Science.

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Authors and Affiliations

  1. Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, 134 Lermontov str., Irkutsk, 664033, Russia

    S. V. Svinina & A. K. Svinin

Authors
  1. S. V. Svinina
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  2. A. K. Svinin
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Correspondence to S. V. Svinina or A. K. Svinin.

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Russian Text © The Author(s), 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 4, pp. 73–84.

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Svinina, S.V., Svinin, A.K. Existence of solution to some mixed problems for linear differential-algebraic systems of partial differential equations. Russ Math. 63, 64–74 (2019). https://doi.org/10.3103/S1066369X19040078

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  • DOI: https://doi.org/10.3103/S1066369X19040078

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