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The Cauchy Problem for the Iterated Klein–Gordon Equation with the Bessel Operator

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Abstract

An analogue of the Cauchy problem for an iterated multidimensional Klein–Gordon equation with a time-dependent Bessel operator is investigated. Applying the generalized Erdélyi–Kober operator of fractional order, we reduce the formulated problem to the Cauchy problem for the polywave equation. Applying a spherical mean method, we construct an explicit formula to solve this problem for the polywave equation; then, basing on this solution, we find an integral representation of the solution of the formulated problem. The obtained formula allows one to immediately discern the character of the dependence of the solution on the initial functions and, in particular, to establish conditions for the smoothness of the classical solution. The paper will be useful for specialists engaged in the resolving of problems of higher spin theory.

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REFERENCES

  1. V. V. Katrakhov and S. M. Sitnik, ‘‘The transmutation method and boundary-value problems for singular elliptic equations,’’ Sovrem. Mat. Fundam. Napravl. 64, 211–426 (2018).

    MathSciNet  Google Scholar 

  2. S. M. Sitnik and E. L. Shishkina, The Transmutation Method for Differential Equations with Bessel Operators (Fizmatlit, Moscow, 2019) [in Russian].

    Google Scholar 

  3. E. L. Shishkina, ‘‘General Euler–Poisson–Darboux equation and hyperbolic B-Potentials,’’ Sovrem. Mat. Fundam. Napravl. 65, 157–338 (2019).

    MathSciNet  Google Scholar 

  4. J. J. Bowman and J. D. Harris, ‘‘Green’s distributions and the Cauchy problem for the iterated Klein–Gordon operator,’’ J. Math. Phys. 3, 396–402 (1962).

    Article  MathSciNet  Google Scholar 

  5. J. Lukierski, ‘‘Lagrangian Formalism for (\(2j+1\)) component higher spin theory,’’ Bull. AC, Pol. SC, Ser. Mat. Phys. XIV 2, 697–700 (1966).

    MATH  Google Scholar 

  6. D. Sorokin, ‘‘Introduction to the classical theory of higher spins,’’ AIP Conf. Proc. 767, 172 (2005).

    Article  MathSciNet  Google Scholar 

  7. D. W. Bresters, ‘‘On the Cauchy problem for iterated Klein–Gordon operators and wave operators,’’ Indag. Math. (Proc.) 72, 123–139 (1969).

    Article  MathSciNet  Google Scholar 

  8. S. E. Trione, ‘‘On the elementary retarded ultra-hyperbolic solution of the Klien–Gordon operator iterated k-times,’’ Studies Appl. Math. 79, 121–141 (1988).

    MathSciNet  Google Scholar 

  9. H. Yildirim, M. Sarikaya, and S. Oztürk, ‘‘The solution of the \(n\)-dimensional Bessel diamond operator and the Fourier–Bessel transform of their convolution,’’ Proc. Indian Acad. Sci. (Math. Sci.) 114, 375–387 (2004).

  10. E. Nane, ‘‘Higher order PDE’s and iterated processes,’’ Trans. Am. Math. Soc. 360, 2681–2692 (2008).

    Article  MathSciNet  Google Scholar 

  11. C. Bunpog, ‘‘The compound equation related to the Bessel–Helmholtz equation and the Bessel–Klein–Gordon equation,’’ Appl. Math. Sci. 7 (91), 4521–4530 (2013).

    MathSciNet  Google Scholar 

  12. A. K. Urinov and S. T. Karimov, ‘‘Solution of the Cauchy problem for Generalized Euler–Poisson–Darboux equation by the method of fractional integrals,’’ in Progress in Partial Differential Equations, Ed. by M. Reissig and M. Ruzhansky, Vol. 44 of Springer Proceedings in Mathematics and Statistics (Springer, Heidelberg, 2013), pp. 321–337.

  13. A. Erdelyi, ‘‘On fractional integration and its application to the theory of Hankel transforms,’’ Quart. J. Math. 11 (44), 293–303 (1940).

    Article  MathSciNet  Google Scholar 

  14. A. Erdelyi and H. Kober, ‘‘Some remarks on Hankel transforms,’’ Quart. J. Math. 11 (43), 212–221 (1940).

    Article  MathSciNet  Google Scholar 

  15. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Sci., New York etc., 1993).

  16. A. Erdelyi, ‘‘Some applications of fractional integration,’’ Boeing Sci. Res. Labor. Docum. Math. Not., No. 316, Dl-82-0286 (1963).

  17. A. Erdelyi, ‘‘An application of fractional integrals,’’ J. Anal. Math. 14, 113–126 (1965).

    Article  MathSciNet  Google Scholar 

  18. I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory (North-Holland, Amsterdam, 1966).

    MATH  Google Scholar 

  19. I. N. Sneddon, ‘‘The use in mathematical analysis of Erdélyi–Kober’ operators and of some of their applications,’’ Lect. Notes Math. 457, 37–79 (1975).

    Article  Google Scholar 

  20. V. Kiryakova, Generalized Fractional Calculus and Applications (Longman Sci. Tech., Wiley, Harlow, New York, 1994).

  21. J. S. Lowndes, ‘‘A generalization of the Erdélyi–Kober operators,’’ Proc. Edinburgh Math. Soc. 17, 139–148 (1970).

    Article  MathSciNet  Google Scholar 

  22. P. Heywood, ‘‘Improved boundedness conditions for Lowndes’ operators,’’ Proc. R. Soc. Edinburgh, A 73 (9), 291–199 (1975).

  23. P. Heywood and P. G. Rooney, ‘‘On the boundedness of Lowndes’ operators,’’ J. London Math. Soc. 10, 241–248 (1975).

    Article  MathSciNet  Google Scholar 

  24. Sh. T. Karimov, ‘‘On some generalizations of properties of the Lowndes operator and their applications to partial differential equations of high order,’’ Filomat 32, 873–883 (2018).

    Article  MathSciNet  Google Scholar 

  25. Sh. T. Karimov, ‘‘About some generalizations of the properties of the Erdelyi–Kober operator and their application,’’ Vestn. KRAUNC, Fiz.-Mat. Nauki 18 (2), 20–40 (2017). https://doi.org/10.18454/2079-6641-2017-18-2-20-40

    Article  MATH  Google Scholar 

  26. A. Weinstein, ‘‘On a class of partial differential equation of even order,’’ Ann. Math. Pura. Appl. 39, 245–254 (1995).

    Article  MathSciNet  Google Scholar 

  27. D. Krahn, ‘‘On the iterated wave equation,’’ Proc. Ser. A, Math. Sci. 60, 492–505 (1957).

    MathSciNet  MATH  Google Scholar 

  28. R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, New York, 1962), Vol. 2.

    MATH  Google Scholar 

  29. L. C. Evans, Partial Differential Equation (AMS, Berkeley, 1997).

    Google Scholar 

  30. Sh. T. Karimov, ‘‘Method of solving the Cauchy problem for one-dimensional polywave equation With singular Bessel operator,’’ Russ. Math. 61 (8), 22–35 (2017). https://doi.org/10.3103/S1066369X17080035

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, New York, 1972).

  32. L. A. Ivanov, ‘‘The Cauchy problem for some operators with singularities,’’ Differ. Uravn. 18, 1020–1028 (1982).

    MathSciNet  MATH  Google Scholar 

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  1. Fergana State University, 150100, Fergana, Uzbekistan

    Sh. T. Karimov

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Karimov, S.T. The Cauchy Problem for the Iterated Klein–Gordon Equation with the Bessel Operator. Lobachevskii J Math 41, 772–784 (2020). https://doi.org/10.1134/S1995080220050042

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  • DOI: https://doi.org/10.1134/S1995080220050042

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