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Correctness of multi-dimensional Darboux problems for the wave equation

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Abstract

It is proved that multi-dimensional Darboux problems for the wave equation are correct in the domain\(D_3 \subset E_{m + 1} \) bounded by the surfaces ¦x ¦=t + ɛ and ¦x ¦=1- t and the planet = 0, 0 ≤ ɛ < 1. The behavior of the solutions as ɛ→0 is studied.

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References

  1. M. H. Protter, “New boundary-value problems for the wave equation and equations of mixed type,”J. Rat. Mech. Anal.,3, No. 4, 435–446 (1954).

    Google Scholar 

  2. Wand Guand Wind, “The Goursat problems in space,”Sci. Rec. New Ser.,1, No. 5, 7–10 (1957).

    Google Scholar 

  3. S. G. Mikhlin,Multi-Dimensional Singular Integral Equations [in Russian], Fizmatgiz, Moscow (1962).

    Google Scholar 

  4. S. A. Aldashev, “On some boundary-value problems for the multi-dimensional wave equation,”Dokl. Akad. Nauk SSSR,265, No. 6, 1289–1292 (1982).

    Google Scholar 

  5. Tong Kwand Chang, “On a boundary value problem for the wave equation,”Sci. Rec. New Ser.,1, No. 5, 1–2 (1957).

    Google Scholar 

  6. S. A. Aldashev,Boundary-Value Problems for Multi-Dimensional Hyperbolic Equations and Mixed Equations [in Russian], Doctor of Science Thesis (Physics and Mathematics), Kiev (1990).

  7. He Kan Cher, “On the nonuniqueness of solutions to some boundary-value problems for degenerating equations,”Dokl. Akad. Nauk SSSR,272, No. 5, 1066–1068 (1983).

    Google Scholar 

  8. S. A. Aldashev, “On some local and nonlocal boundary-value problems for the wave equation,”Differents. Uravn.,19, No. 1, 3–8 (1983).

    Google Scholar 

  9. N. I. Popivanov and M. Schneider, “The Darboux problem in R3 for a class of degenerated hyperbolic equation,”Rep. Bulg. Acad. Sci.,41, No. 11, 7–9 (1988).

    Google Scholar 

  10. A. V. Bitsadze,Mixed Equations [in Russian], Izd. Akad. Nauk SSSR, Moscow (1959).

    Google Scholar 

  11. E. T. Copson, “On the Riemann-Green function,”J. Rath. Mech. Anal.,1, 324–348 (1958).

    Google Scholar 

  12. S. G. Samko, A. A. Kilbas, and O. I. Marichev,Integrals and Derivatives of Fractional Order and Some Applications [in Russian], Nauka i Tekhnika, Minsk (1987).

    Google Scholar 

  13. H. Bateman and A. Erdélyi,Higher Transcendental Functions, Vol.1, McGraw-Hill, New York (1954).

    Google Scholar 

  14. V. Li,On Spatial Goursat-Type Problems, [in Russian], Candidate of Science Thesis (Physics and Mathematics), Alma-Ata (1969).

  15. He Kan Cher,Darboux-Protter Problems for the Two-Dimensional Wave Equation [in Russian], Preprint, IPM, Far-Eastern Center of the Academy of Sciences of the USSR, Vladivostok (1990).

    Google Scholar 

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Authors and Affiliations

  1. Alma-Ata Institute of Railroad Engineers, Alma-Ata

    S. A. Aldashev

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  1. S. A. Aldashev
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1299–1306, September, 1993.

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Aldashev, S.A. Correctness of multi-dimensional Darboux problems for the wave equation. Ukr Math J 45, 1456–1464 (1993). https://doi.org/10.1007/BF01058644

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  • DOI: https://doi.org/10.1007/BF01058644

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