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On an operational method of solving initial-value problems for partial differential equations induced by generalized separation of variables

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Abstract

We propose an operational method of solving the Cauchy problem for partial differential equations and systems of partial differential equations. We demonstrate its superiority to the known methods. We give a number of illustrative examples of applications of the method.

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Literature Cited

  1. A. V. Bitsadze,Some Classes of Partial Differential Equations, [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  2. B. A. Bondarenko and A. Suleimanov, “Solution, of the Cauchy problem for a system of differential equations of the dynamic theory of elasticity,”Izv. Akad. Nauk Uzbek. SSR, No. 1, 5–9 (1975).

    Google Scholar 

  3. B. A. Bondarenko and A. N. Filatov,Quasipolynomial Functions and their Applications to Problems of the Theory of Elasticity [in Russian], FAN, Tashkent, (1978).

    Google Scholar 

  4. M. E. Vashchenko-Zakharchenko,The Symbolic Calculus and its Application to the Integration of Linear Differential Equations [in Russian], Kiev University Press (1862).

  5. V. S. Vladimirov,The Equations of Mathematical Physics [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  6. V. V. Vlasov,The Method of Inifial Functions in Problems of the Theory of Elasticity and Structural Mechanics [in Russian], Stroiizdat, Moscow (1975).

    Google Scholar 

  7. Yu. A. Dubinskii, “The algebra of pseudodifferential operators with analytic symbols and its applications to mathematical physics,”Usp. Mat. Nauk,37, No. 5, 97–159 (1982).

    MATH  MathSciNet  Google Scholar 

  8. Yu. A. Dubinskii “On the theory of the Cauchy problem for partial differential equations,”Dokl. Akad. Nauk SSR,259, No. 4, 781–785 (1981).

    MATH  MathSciNet  Google Scholar 

  9. P. I. Kalenyuk, Ya. E. Baranetskii and Z. N. Nitrebich,The Generalized Method of Separation of Variables [in Russian], Naukova Dumka, Kiev (1993).

    Google Scholar 

  10. P. I. Kalenyuk and Z. M. Nytrebych, “The Cauchy problem for a system of two partial differential equations of infinite order,”Mat. Met. Fiz.-Mekh. Polya, No. 35, 204–210 (1992).

    Google Scholar 

  11. P. I. Kalenyuk and Z. M. Nytrebych, “On the Cauchy problem for a class of systems of differential equations that admit separation of variables,”Visn. Derzh. Univ. “L'viv Politekhn.”, No. 277, 45–55 (1994).

    Google Scholar 

  12. P. I. Kalenyuk and Z. M. Nytrebych, “A scheme for separation of variables for a matrix bilinear functional equation and its application,”Ukr. Mat. Zh., No. 9, 1201–1209 (1992).

    Google Scholar 

  13. P. I. Kalenyuk, Z. M. Nytrebych, and P. L. Sokhan, “An operational method of constructing the solution of the Cauchy problem for a homogeneous system of partial differential equations of infinite order,” Preprint, Ukrainian Academy of Sciences Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, No. 1-95 (1995).

  14. P. I. Kalenyuk and V. Ya. Skorobogat'ko,Qualitative Methods of the Theory of Differential Equations [in Ukrainian], Naukova Dumka, Kiev (1977).

    Google Scholar 

  15. I. Lur'e,Theory of Elasticity, [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  16. V. P. Maslov,Operator Methods [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  17. Ya. Mikusinskii,The Operational Calculus, [Russian translation], Foreign Literature Publishing House, Moscow (1956).

    Google Scholar 

  18. Z. M. Nytrebych, “An operational method of constructing, the solution of the Cauchy problem for inhomogeneous Lamé equations,”Dop. Nats. Akad. Nauk Ukr., No. 7, 32–34 (1995).

    Google Scholar 

  19. Z. M. Nytrebych, “An operational method of constructing the solution of the Cauchy problem for a homogeneous system of partial differential equations,”Mat. Met. Fiz.-Mekh. Polya, No. 38, 40–45 (1995).

    Google Scholar 

  20. Ya. S. Pidstryhach, “The temperature field in thin shells,”Dop. Akad. Nauk. Ukr. RSR, No. 5, 505–507 (1958).

    Google Scholar 

  21. L. I. Slepyan,Nonstationary Elastic Waves [in Russian], Sudostroenie, Leningrad (1972).

    Google Scholar 

  22. SneddonFourier Transforms, McGraw-Hill, New York (1950).

    Google Scholar 

  23. N. Tikhonov, “Uniqueness theorems for the heat equation,”Mat. Sb.,42, No. 2, 199–215 (1935).

    MATH  Google Scholar 

  24. R. A. Eubanks and E. I. Sternberg, “On stress functions for elastokinetics and the integration of the repeated wave equation,”Quar. Appl. Math.,15, No. 2, 28–35 (1957).

    MathSciNet  Google Scholar 

  25. O. Heaviside,Electromagnetic Theory. Vols. 1–3, London (1893–1899).

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Authors
  1. P. I. Kalenyuk
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  2. Z. M. Nytrebych
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Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1 1998, pp. 136–145.

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Kalenyuk, P.I., Nytrebych, Z.M. On an operational method of solving initial-value problems for partial differential equations induced by generalized separation of variables. J Math Sci 97, 3879–3887 (1999). https://doi.org/10.1007/BF02364928

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  • DOI: https://doi.org/10.1007/BF02364928

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