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The Cauchy Problem for One Second Order Partial Differential Equation with Carleman Shift

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Abstract

We considered the second order partial differential equation with Carleman shift and singular characteristics. The well-posed Cauchy problem in the sense of Hadamard is given. The solution of the problem is constructed in the explicit form on the basis of Riemann method.

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REFERENCES

  1. Ch. Babbage, ‘‘An essay towards the calculus of functions, Part II,’’ Philos. Trans. R. Soc. London 106, 179–256 (1816).

    Article  Google Scholar 

  2. W. B. Fite, ‘‘Properties of the solution of certain functional differential equations,’’ Trans. Am. Math. Soc., No. 3, 311–319 (1921).

  3. T. Carleman, ‘‘Sur la theorie des equations integrales et ses applications,’’ in Proceedings of the International Mathematical Congress (Zurich, 1932), Vol. 1, pp. 138–151.

  4. S. F. Lacroix, Traite du calcul differentiel et du calcul integral (Paris, 1819), Vol. 3, Chap. 8.

    Google Scholar 

  5. N. K. Karapetyants and S. G. Samko, Equations with Involutive Operators (Birkhäuser, Boston, MA, 2001).

    Book  Google Scholar 

  6. G. S. Litvinchuk, Boundary Value Problems and Singular Integral Equations with Shift (Nauka, Moscow, 1973) [in Russian].

    Google Scholar 

  7. I. G. Petrovsky, Lectures on the Theory of Ordinary Differential Equations (Nauka, Moscow, 1970) [in Russian].

    Google Scholar 

  8. I. M. Gul, ‘‘The Cauchy problem for some partial differential equations with functional arguments,’’ Russ. Math. Surv. 10, 153–156 (1955).

    MathSciNet  Google Scholar 

  9. A. B. Nersesyan, ‘‘On the Cauchy problem for a partial differential equation with a deviating argument,’’ in Proceedings of the 2nd Mathematical All-Union Interuniversity Conference on the Theory and Applications of Differential Equations with Deviating Argument (Chernovtsy, 1968), pp. 116–117.

  10. A. A. Andreev, ‘‘On the correctness of boundary value problems for some partial differential equations with a Carleman shift,’’ in Differential Equations and Their Applications, Proceedings of the 2nd International Seminar, Samara (1998), pp. 5–18.

  11. A. A. Andreev, ‘‘On analogues of classical boundary value problems for a differential equation with deviating argument,’’ Differ. Equat. 40, 1192–1195 (2004).

    Article  MathSciNet  Google Scholar 

  12. A. A. Andreev and Yu. A. Senitsky, ‘‘On the Cauchy problem for the Euler–Poisson–Darboux equation of particular form with deviating argument,’’ in Nonclassical Equations of Mathematical Physics (Novosibirsk, IM SOAN USSR, 1987), pp. 51–53 [in Russian].

    Google Scholar 

  13. A. A. Andreev and A. V. Linkov, ‘‘On well-posed problems for one model partial differential equation with deviating argument,’’ in Equations of Non-Classical Type (IM SOAN USSR, Novosibirsk, 1997), pp. 3–11 [in Russian].

    Google Scholar 

  14. A. A. Andreev and E. N. Ogorodnikov, ‘‘On the well-posedness of initial boundary value problems for a hyperbolic equation with order degeneration and involutive deviation,’’ J. Samara State Tech. Univ., Ser. Phys. Math. Sci., No. 9, 32–36 (2000).

  15. A. A. Andreev and I. P. Shindin, ‘‘On the well-posedness of boundary value problems for a partial differential equation with deviating argument,’’ in Analytical Methods in the Theory of Differential and Integral Equations (Kuibysh. Gos. Univ., Kuibyshev, 1987), pp. 3–6.

    Google Scholar 

  16. A. N. Zarubin, ‘‘An analogue of the Tricomi problem for a mixed-type equation with a retarded argument,’’ Differ. Equat. 32, 350–356 (1996).

    Google Scholar 

  17. A. A. Andreev and I. N. Saushkin, ‘‘On an analog of Tricomi problem for a certain model equation with involutive deviation in infinite domain,’’ J. Samara State Tech. Univ., Ser. Phys. Math. Sci. 34, 10–16 (2005).

    Google Scholar 

  18. A. P. Khromov, ‘‘The mixed problem for the differential equation with involution and potential of the special kind,’’ Izv. Sarat. Univ., New Ser., Ser.: Mat. Mekh. Inform. 10 (4), 17–22 (2010).

    Google Scholar 

  19. M. Sh. Burlutskaya, ‘‘Some properties of functional-differential operators with involution \(\nu(x)=1-x\) and their applications,’’ Russ. Math. (Iz. VUZ) 65 (5), 69–76 (2021).

  20. F. R. Gantmakher, Theory of Matrices (Nauka, Moscow, 1988) [in Russian].

    MATH  Google Scholar 

  21. A. V. Bitsadze, ‘‘On the theory of one class of mixed type equations,’’ in Some Problems of Mathematics and Mechanics (1970), pp. 112–119.

  22. S. A. Tersenov, Introduction to the Theory of Equations Degenerate on the Boundary, Textbook for Universities (Novosibirsk Univ., Novosibirsk, 1973) [in Russian].

    Google Scholar 

  23. A. A. Andreev, ‘‘On a class of systems of differential equations of hyperbolic type,’’ in Partial Differential Equations (Ryazan. Gos. Ped. Inst., Ryazan, 1980), pp. 3–9 [in Russian].

    Google Scholar 

  24. E. A. Maksimova, ‘‘On Cauchy problem for system of n Euler–Poisson–Darboux equations in the plane,’’ J. Samara State Tech. Univ., Ser. Phys. Math. Sci. 1 (26), 21–30 (2012).

    Google Scholar 

  25. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Ed. by H. Bateman (McGraw-Hill, New York, 1953), Vol. 1.

    MATH  Google Scholar 

  26. R. S. Khairullin, The Cauchy Problem for the Euler–Poisson–Darboux Equation (Kazan Univ., Kazan, 2014) [in Russian].

    Google Scholar 

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Correspondence to E. A. Maksimova.

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(Submitted by A. B. Muravnik)

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Maksimova, E.A. The Cauchy Problem for One Second Order Partial Differential Equation with Carleman Shift. Lobachevskii J Math 43, 1366–1372 (2022). https://doi.org/10.1134/S1995080222090189

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  • DOI: https://doi.org/10.1134/S1995080222090189

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