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Mixed Problems with Integral Conditions for Hyperbolic Equations with the Bessel Operator

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Abstract

The paper considers nonlocal problems with integral conditions for hyperbolic equations with the Bessel differential operator whose statement substantially depends on the intervals where the parameter occurring in this operator varies. The well-posedness of these problems is studied according to a unified scheme based on the classical method of separation of variables, which is also used to study nonclassical problems with integral conditions for equations of elliptic–hyperbolic type containing the Bessel operator in one or two variables as well.

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REFERENCES

  1. Bateman, H. and Erdélyi, A., Higher Transcendental Functions. Vol. 2, New York–Toronto–London: McGraw-Hill, 1955. Translated under the title: Vysshie transtsendentnye funktsii. T. 2, Moscow: Nauka, 1974.

  2. Benuar, N.E. and Yurchuk, N.I., Mixed problem with integral condition for parabolic equations with the Bessel operator, Differ. Equations, 1991, vol. 27, no. 12, pp. 1482–1487.

    MathSciNet  Google Scholar 

  3. Bitsadze, A.V., Uravneniya smeshannogo tipa (Mixed Type Equations), Moscow: Izd. Akad. Nauk SSSR, 1959.

    Google Scholar 

  4. Bitsadze, A.V., Nekotorye klassy uravnenii v chastnykh proizvodnykh (Some Classes of Partial Differential Equations), Moscow: Nauka, 1981.

    Google Scholar 

  5. Bitsadze, A.V. and Pashkovskii, V.I., On the theory of Maxwell–Einstein equations, Dokl. Akad. Nauk SSSR, 1974, vol. 216, no. 2, pp. 9–10.

    MathSciNet  Google Scholar 

  6. Bitsadze, A.V. and Pashkovskii, V.I., On some classes of solutions of the Maxwell–Einstein equation, Proc. Steklov Inst. Math., 1977, vol. 134, pp. 31–35.

    Google Scholar 

  7. Watson, G.N., A Treatise on the Theory of Bessel Functions, Cambridge: Cambridge Univ. Press, 1945. Translated under the title: Teoriya Besselevykh funktsii. Chast’ pervaya., Moscow–Leningrad: Izd. Inostr. Lit., 1949.

    Google Scholar 

  8. Vekua, I.N., Novye metody resheniya ellipticheskikh uravnenii (New Methods for Solving Elliptic Equations), Moscow–Leningrad: Gostekhizdat, 1948.

    Google Scholar 

  9. Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1981.

    Google Scholar 

  10. Volkodavov, V.F. and Nikolaev, N.Ya., Kraevye zadachi dlya uravneniya Eilera–Puassona–Darbu (Boundary Value Problems for the Euler–Poisson–Darboux Equation), Kuibyshev: Izd. Kuibyshev. Gos. Ped. Inst., 1984.

  11. Vragov, V.N., Kraevye zadachi dlya neklassicheskikh uravnenii matematicheskoi fiziki (Boundary Value Problems for Nonclassical Equations of Mathematical Physics), Novosibirsk: Novosibirsk. Gos. Univ., 1983.

  12. Garipov, I.B., Solution of the Cauchy problem and some boundary value problems for a parabolic equation with the Bessel operator, Cand. Sci. (Phys.-Math.) Dissertation, Kazan: Kazan State Univ., 2002.

  13. Gafurova, S.M., Solution of the Cauchy problem and some boundary value problems for a hyperbolic equation with the Bessel operator, Cand. Sci. (Phys.-Math.) Dissertation, Kazan: Kazan State Univ., 2002.

  14. Glushko, V.P., Lineinye vyrozhdayushchiesya differentsial’nye uravneniya (Linear Degenerate Differential Equations), Voronezh: Izd. Voronezh. Gos. Univ., 1972.

    Google Scholar 

  15. Gordeziani, D.G. and Avalishvili, G.A., Solution of nonlocal problems for one-dimensional vibrations of a medium, Mat. Model., 2000, vol. 12, no. 1, pp. 94–103.

    MathSciNet  Google Scholar 

  16. Gurevich, M.I., Teoriya strui ideal’noi zhidkosti (Theory of Ideal Fluid Jets), Moscow: Nauka, 1979.

    Google Scholar 

  17. Gushchin, A.K., A condition for the compactness of operators in a certain class and its application to the analysis of the solubility of non-local problems for elliptic equations, Sb. Math., 2002, vol. 193, no. 5, pp. 649–668.

    Article  MathSciNet  Google Scholar 

  18. Gushchin, A.K. and Mikhailov, V.P., On solvability of nonlocal problems for a second-order elliptic equation, Russ. Acad. Sci. Sb. Math., 1995, vol. 81, no. 1, pp. 101–136.

    MathSciNet  Google Scholar 

  19. Denisova, M.Yu., Study of the main boundary value problems for some \(B \)-polyharmonic equations by the potential method, Cand. Sci. (Phys.-Math.) Dissertation, Kazan: Kazan State Univ., 2002.

  20. Dzhayani, G.V., Reshenie nekotorykh zadach dlya odnogo vyrozhdayushchegosya ellipticheskogo uravneniya i ikh prilozheniya k prizmaticheskim obolochkam (Solution of Some Problems for One Degenerate Elliptic Equation and Applications to Prismatic Shells), Tbilisi: Izd. Tbilis. Univ., 1982.

    Google Scholar 

  21. Dzhayani, G.V., Uravnenie Eilera–Paussona–Darbu (Euler–Poisson–Darboux uquation), Tbilisi: Izd. Tbilis. Univ., 1984.

    Google Scholar 

  22. Dmitriev, V.B., Nonlocal problem with integral conditions for the wave equation, Vestn. SamGU, 2006, no. 2(42), pp. 15–27.

  23. Egorov, I.E. and Fedorov, V.Evs., Neklassicheskie uravneniya matematicheskoi fiziki vysokogo poryadka (High-Order Nonclassical Equations of Mathematical Physics), Novosibirsk: Izd. Vychisl. Tsentra Sib. Otd. Ross. Akad. Nauk, 1995.

    Google Scholar 

  24. Zhegalov, V.I., Boundary value problem for a mixed type equation with a boundary condition on both characteristics and with discontinuities on the transition line, Uch. Zap. Kazan. Univ., 1962, vol. 122, no. 3, pp. 3–16.

    Google Scholar 

  25. Zhegalov, V.I., Frankl problem with displacement, Izv. VUZov. Mat., 1979, no. 9, pp. 11–20.

  26. Zhegalov, V.I., Mironov, A.N., and Utkina, E.A., Uravneniya s dominiruyushchei chastnoi proizvodnoi (Equations with Dominant Partial Derivative), Kazan: Kazan. Univ., 2014.

    Google Scholar 

  27. Zaitseva, N.V., A mixed problem for one \(B \)-hyperbolic equation with an integral condition of the first kind, Izv. Tul’sk. Gos. Univ. Estestv. Nauki, 2012, no. 2, pp. 39–50.

  28. Zaitseva, N.V., A mixed problem for one \(B \)-hyperbolic equation with an integral condition of the second kind, Izv. Smolensk. Gos. Univ., 2013, no. 4(24), pp. 397–403.

  29. Zaitseva, N.V., Initial–boundary value problem for a \(B \)-hyperbolic equation with an integral condition of the first kind in a rectangular domain, Vestn. SamGU, 2016, no. 3–4, pp. 51–62.

  30. Zaitseva, N.V., Nonlocal boundary value problem for a \(B \)-hyperbolic equation in a rectangular domain, Vestn. Samarsk. Gos. Tekh. Univ. Ser. Fiz.-Mat. Nauki, 2016, vol. 20, no. 4, pp. 589–602.

    Google Scholar 

  31. Zaitseva, N.V., Nonlocal boundary value problem with an integral condition for a mixed type equation with a singular coefficient, Differ. Equations, 2021, vol. 57, no. 2, pp. 224–234.

    Article  MathSciNet  Google Scholar 

  32. Zaitseva, N.V., Uniqueness of the solution of a nonlocal problem for an elliptic-hyperbolic equation with singular coefficients, Math. Notes, 2021, vol. 109, no. 4, pp. 563–569.

    Article  MathSciNet  Google Scholar 

  33. Il’in, V.A. and Moiseev, E.I., Nonlocal boundary value problem for the Sturm–Liouville operator in differential and difference interpretations, Dokl. Akad. Nauk SSSR, 1986, vol. 291, no. 3, pp. 534–539.

    MathSciNet  Google Scholar 

  34. Il’in, V.A. and Moiseev, E.I., Two-dimensional nonlocal boundary value problem for the Poisson operator in differential and difference interpretations, Mat. Model., 1990, vol. 2, no. 8, pp. 139–156.

    MathSciNet  Google Scholar 

  35. Il’in, V.A. and Moiseev, E.I., Uniqueness of the solution of a mixed problem for the wave equation with nonlocal boundary conditions, Differ. Equations, 2000, vol. 36, no. 5, pp. 728–733.

    Article  MathSciNet  Google Scholar 

  36. Ionkin, N.I., Solution of a boundary value problem of heat conduction theory with a nonclassical boundary condition, Differ. Uravn., 1977, vol. 13, no. 2, pp. 294–304.

    MathSciNet  Google Scholar 

  37. Ionkin, N.I. and Moiseev, E.I., On a problem for the heat equation with two-point boundary conditions, Differ. Uravn., 1979, vol. 15, no. 7, pp. 1284–1295.

    MathSciNet  Google Scholar 

  38. Kamynin, L.I., On a boundary value problem in the theory of heat conduction with nonclassical boundary conditions, Zh. Vychisl. Mat. Mat. Fiz., 1964, vol. 4, no. 6, pp. 1006–1024.

    Google Scholar 

  39. Katrakhov, V.V., General boundary value problems for one class of singular and degenerate equations, Dokl. Akad. Nauk SSSR, 1980, vol. 251, no. 6, pp. 1296–1300.

    MathSciNet  Google Scholar 

  40. Katrakhov, V.V., General boundary value problems for one class of singular and degenerate elliptic equations, Math. USSR-Sb., 1981, vol. 40, no. 3, pp. 325–347.

    Article  Google Scholar 

  41. Katrakhov, V.V. and Sitnik, S.M., Composition method for constructing \(B \)-elliptic, \(B \)-parabolic, and \(B \)-hyperbolic transformation operators, Dokl. Math., 1995, vol. 50, no. 1, pp. 70–77.

    MathSciNet  Google Scholar 

  42. Katrakhov, V.V. and Sitnik, S.M., Transformation operator method and boundary value problems for singular elliptic equations, Sovrem. Mat. Fundam. Napravl., 2018, vol. 64, no. 2, pp. 211–426.

    MathSciNet  Google Scholar 

  43. Keldysh, M.V., On some cases of degeneration of elliptic type equations on the boundary of a domain, Dokl. Akad. Nauk SSSR, 1951, vol. 77, no. 2, pp. 181–183.

    Google Scholar 

  44. Kipriyanov, I.A., On boundary value problems for partial differential equations with the Bessel operator, Dokl. Akad. Nauk SSSR, 1964, vol. 158, no. 2, pp. 275–278.

    MathSciNet  Google Scholar 

  45. Kipriyanov, I.A. and Kononenko, V.I., Fundamental solutions of \(B \)-elliptic equations, Differ. Uravn., 1967, vol. 3, no. 1, pp. 114–129.

    MathSciNet  Google Scholar 

  46. Kipriyanov, I.A. and Ivanov, L.A., Fundamental solutions of homogeneous \(B \)-hyperbolic equations, Sib. Math. J., 1980, vol. 21, no. 4, pp. 552–557.

    Article  MathSciNet  Google Scholar 

  47. Kipriyanov, I.A., Singulyarnye ellipticheskie kraevye zadachi (Singular Elliptic Boundary Value Problems), Moscow: Nauka, 1997.

    Google Scholar 

  48. Kozhanov, A.I., On a nonlocal boundary value problem with variable coefficients for the heat equation and the Aller equation, Differ. Equations, 2004, vol. 40, no. 6, pp. 815–826.

    Article  MathSciNet  Google Scholar 

  49. Kozhanov, A.I., Time-nonlocal boundary value problem for linear parabolic equations, Sib. Zh. Ind. Mat., 2004, vol. 7, no. 1(17), pp. 51–60.

    MathSciNet  Google Scholar 

  50. Kozhanov, A.I. and Pul’kina, L.S., Boundary value problems with an integral boundary condition for multidimensional hyperbolic equations, Dokl. Akad. Nauk, 2005, vol. 404, no. 5, pp. 589–592.

    MathSciNet  Google Scholar 

  51. Kozhanov, A.I., On the solvability of some spatially nonlocal problems for linear parabolic equations, Vestn. SamGU, 2008, no. 3(62), pp. 165–174.

  52. Kozhanov, A.I., Nonlocal problems for linear hyperbolic equations with boundary conditions containing the time derivative, Dokl. AMAN, 2010, vol. 12, no. 1, pp. 40–52.

    Google Scholar 

  53. Kudryavtsev, L.D., Pryamye i obratnye teoremy vlozheniya. Prilozheniya k resheniyu variatsionnym metodom ellipticheskikh uravnenii (Direct and Converse Embedding Theorems. Applications to the Solution of Elliptic Equations by the Variation Method), Moscow: Izd. Akad. Nauk SSSR, 1959.

    Google Scholar 

  54. Ladyzhenskaya, O.A. and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa (Linear and Quasilinear Equations of Elliptic Type), Moscow: Nauka, 1964.

    Google Scholar 

  55. Lerner, M.E. and Repin, O.A., A problem with two nonlocal boundary conditions for a mixed-type equation, Sib. Math. J., 1999, vol. 40, no. 6, pp. 1064–1078.

    Article  Google Scholar 

  56. Lerner, M.E. and Repin, O.A., On problems of the type of the Frankl problem for some elliptic equations with degeneracy of various kinds, Differ. Uravn., 1999, vol. 35, no. 8, pp. 1098–1104.

    MathSciNet  Google Scholar 

  57. Lerner, M.E. and Repin, O.A., Nonlocal boundary value problems in a vertical half-strip for a generalized axisymmetric Helmholtz equation, Differ. Equations, 2001, vol. 37, no. 11, pp. 1640–1642.

    Article  MathSciNet  Google Scholar 

  58. Lomov, I.S., A theorem on the unconditional basis property of root vectors of loaded second-order differential operators, Differ. Uravn., 1991, vol. 27, no. 9, pp. 1550–1563.

    Google Scholar 

  59. Lomov, I.S., On the basis property of systems of irregular root vectors of high-order differential operators, Differ. Uravn., 1993, vol. 29, no. 1, pp. 74–86.

    Google Scholar 

  60. Lomov, S.A. and Lomov, I.S., Osnovy matematicheskoi teorii pogranichnogo sloya (Fundamentals of the Mathematical Theory of the Boundary Layer), Moscow: Izd. Mosk. Gos. Univ., 2011.

    Google Scholar 

  61. Lomov, I.S., Nonclassical statements of problems for degenerate partial differential equations, Differ. Equations, 2012, vol. 48, no. 5, pp. 730–736.

    Article  MathSciNet  Google Scholar 

  62. Lomov, I.S., Spektral’nyi metod V.A. Il’ina. Nesamosopryazhennye operatory (V.A. Ilyin’s Spectral Method. Nonself-Adjoint Operators), Moscow: Izd. Mosk. Gos. Univ., 2019.

    Google Scholar 

  63. Lomov, I.S., Uniform convergence of expansions in root functions of a differential operator with integral boundary conditions, Differ. Equations, 2019, vol. 55, no. 4, pp. 471–482.

    Article  MathSciNet  Google Scholar 

  64. Lyakhov, L.N., Fourier–Bessel transform of the kernel of a singular integral operator generated by a generalized displacement, in Neklassicheskie uravneniya matematicheskoi fiziki (Nonclassical Equations of Mathematical Physics), Novosibirsk, 1986, pp. 75–83.

  65. Lyakhov, L.N., Inversion of Riesz \(B \)-potentials, Dokl. Math., 1992, vol. 44, no. 3, pp. 717–720.

    MathSciNet  Google Scholar 

  66. Lyakhov, L.N., \(B \)-gipersingulyarnye integraly i ikh prilozheniya k opisaniyu funktsional’nykh klassov Kipriyanova i k integral’nym uravneniyam s \(B\)-potentsial’nymi yadrami (\(B \)-Hypersingular Integrals and Applications to the Description of Kipriyanov Function Classes and to Integral Equations with \(B \)-Potential Kernels), Lipetsk: Izd. LGPU, 2007.

  67. Lyakhov, L.N., Polovinkin, I.P., and Shishkina, E.L., On a Kipriyanov problem for a singular ultrahyperbolic equation, Differ. Equations, 2014, vol. 50, no. 4, pp. 513–525.

    Article  MathSciNet  Google Scholar 

  68. Lyakhov, L.N., Polovinkin, I.P., and Shishkina, E.L., Formulas for the solution of the Cauchy problem for a singular wave equation with Bessel time operator, Dokl. Math., 2014, vol. 90, no. 3, pp. 737–742.

    Article  MathSciNet  Google Scholar 

  69. Marichev, O.I., Kilbas, A.A., and Repin, O.A., Kraevye zadachi dlya uravnenii v chastnykh proizvodnykh s razryvnymi koeffitsientami (Boundary Value Problems for Partial Differential Equations with Discontinuous Coefficients), Samara: Izd. Samar. Gos. Ekon. Univ., 2008.

    Google Scholar 

  70. Matiichuk, M.I., Parabolichni singulyarni kraiovi zadachi (Parabolic Singular Boundary Value Problems), Kyiv: Inst. Mat. NAN Ukraini, 1999.

    Google Scholar 

  71. Moiseev, E.I., Uravneniya smeshannogo tipa so spektral’nym parametrom (Mixed Type Equations with a Spectral Parameter), Moscow: Izd. Mosk. Gos. Univ., 1988.

    Google Scholar 

  72. Moiseev, E.I., On the solution of a nonlocal boundary value problem by the spectral method, Differ. Equations, 1999, vol. 35, no. 8, pp. 1105–1112.

    MathSciNet  Google Scholar 

  73. Muravnik, A.B., Stabilization of solutions of certain singular quasilinear parabolic equations, Math. Notes, 2003, vol. 74, no. 6, pp. 812–818.

    Article  MathSciNet  Google Scholar 

  74. Muravnik, A.B., On stabilization of solutions of singular elliptic equations, J. Math. Sci., 2008, vol. 150, no. 5, pp. 2408–2421.

    Article  MathSciNet  Google Scholar 

  75. Muravnik, A.B., Functional differential parabolic equations: Integral transformations and qualitative properties of solutions of the Cauchy problem, J. Math. Sci., 2016, vol. 216, no. 3, pp. 345–496.

    Article  MathSciNet  Google Scholar 

  76. Mukhlisov, F.G., On the existence and uniqueness of solutions of certain partial differential equations with the Bessel differential operator, Izv. VUZov. Mat., 1982, no. 10, pp. 58–71.

  77. Mukhlisov, F.G., Generalized solution of a Dirichlet-type problem for some singular elliptic equations, Sib. Math. J., 1990, vol. 31, no. 5, pp. 770–780.

    Article  MathSciNet  Google Scholar 

  78. Natanson, I.P., Teoriya funktsii veshchestvennoi peremennoi (Theory of Functions of a Real Variable), Moscow: Nauka, 1974.

    Google Scholar 

  79. Nakhushev, A.M., On some boundary value problems for hyperbolic equations and equations of mixed type, Differ. Uravn., 1969, vol. 5, no. 1, pp. 44–59.

    Google Scholar 

  80. Nakhushev, A.M., On the theory of linear boundary value problems for a second-order equation of mixed hyperbolic-parabolic type, Differ. Uravn., 1978, vol. 15, no. 1, pp. 66–73.

    MathSciNet  Google Scholar 

  81. Nakhushev, A.M., Zadachi so smeshcheniem dlya uravnenii v chastnykh proizvodnykh (Problems with Shift for Partial Differential Equations), Moscow: Nauka, 2006.

    Google Scholar 

  82. Nakhushev, A.M., Nagruzhennye uravneniya i ikh primeneniya (Loaded Equations and Applications), Moscow: Nauka, 2012.

    Google Scholar 

  83. Nakhusheva, Z.A., Nelokal’nye kraevye zadachi dlya osnovnykh i smeshannogo tipov differentsial’nykh uravnenii (Nonlocal Boundary Value Problems for Basic and Mixed Types of Differential Equations), Nal’chik: Izd. KBNTs RAN, 2011.

    Google Scholar 

  84. Nikol’skii, S.M., Variational problem for an elliptic type equation with degeneracy on the boundary, Proc. Steklov Inst. Math., 1981, vol. 150, pp. 227–254.

    Google Scholar 

  85. Olver, F.W.J., Introduction to Asymptotics and Special Functions, New York–London: Academic Press, 1974. Translated under the title: Vvedenie v asimptoticheskie metody i spetsial’nye funktsii, Moscow: Mir, 1986.

    Google Scholar 

  86. Oleinik, O.A. and Radkevich, E.V., Uravneniya s neotritsatel’noi kharakteristicheskoi formoi (Equations with Nonnegative Characteristic Form), Moscow: Izd. Mosk. Gos. Univ., 2010.

    Google Scholar 

  87. Pul’kin, S.P., Izbrannye trudy (Selected Works), Samara: Univers Grupp, 2007.

    Google Scholar 

  88. Pul’kina, L.S., On one nonclassical problem for a degenerate hyperbolic equation, Sov. Math. (Iz. VUZ), 1991, vol. 35, no. 11, pp. 49–51.

    Google Scholar 

  89. Pul’kina, L.S., A mixed problem with integral condition for the hyperbolic equation, Math. Notes, 2003, vol. 74, no. 3, pp. 411–421.

    Article  MathSciNet  Google Scholar 

  90. Pul’kina, L.S., A nonlocal problem with integral conditions for a hyperbolic equation, Differ. Equations, 2004, vol. 40, no. 7, pp. 947–953.

    Article  MathSciNet  Google Scholar 

  91. Pul’kina, L.S., Zadachi s neklassicheskimi usloviyami dlya giperbolicheskikh uravnenii (Problems with Nonclassical Conditions for Hyperbolic Equations), Samara: Izd. Samarsk. Univ., 2012.

    Google Scholar 

  92. Pul’kina, L.S., Boundary-value problems for a hyperbolic equation with nonlocal conditions of the I and II kind, Russ. Math., 2012, vol. 56, no. 4, pp. 62–69.

    Article  Google Scholar 

  93. Rajabov, N., Integral’nye predstavleniya i granichnye zadachi dlya nekotorykh differentsial’nykh uravnenii s singulyarnoi liniei ili singulyarnymi poverkhnostyami (Integral Representations and Boundary Value Problems for Some Differential Equations with a Singular Line or Singular Surfaces), Dushanbe: Izd. TGU, 1980.

    Google Scholar 

  94. Repin, O.A., Kraevye zadachi so smeshcheniem dlya uravnenii giperbolicheskogo i smeshannogo tipov (Boundary Value Problems with Shift for Equations of Hyperbolic and Mixed Types), Samara: Izd. Saratov. Univ. Samar. Fil., 1992.

    Google Scholar 

  95. Sabitov, K.B., Boundary value problem for a parabolic-hyperbolic equation with a nonlocal integral condition, Differ. Equations, 2010, vol. 46, no. 10, pp. 1472–1481.

    Article  MathSciNet  Google Scholar 

  96. Sabitov, K.B., Nonlocal problem for a parabolic-hyperbolic equation in a rectangular domain, Math. Notes, 2011, vol. 89, no. 4, pp. 562–567.

    Article  MathSciNet  Google Scholar 

  97. Sabitov, K.B. and Martem’yanova, N.V., A nonlocal inverse problem for a mixed-type equation, Russ. Math., 2011, vol. 55, no. 2, pp. 61–74.

    Article  Google Scholar 

  98. Sabitov, K.B. and Zaitseva, N.V., Initial value problem for \(B \)-hyperbolic equation with integral condition of the second kind, Differ. Equations, 2018, vol. 54, no. 1, pp. 121–133.

    Article  MathSciNet  Google Scholar 

  99. Sabitov, K.B. and Zaitseva, N.V., Second initial–boundary value problem for the \(B \)-hyperbolic equation, Izv. VUZov. Mat., 2019, no. 10, pp. 75–86.

  100. Savchuk, A.M. and Shkalikov, A.A., Sturm–Liouville operators with singular potentials, Math. Notes, 1999, vol. 66, no. 6, pp. 741–753.

    Article  MathSciNet  Google Scholar 

  101. Sadovnichaya, I.V., Classical equiconvergence problem for the Sturm–Liouville operator with a singular potential, Differ. Equations, 2019, vol. 55, no. 4, pp. 490–499.

    Article  MathSciNet  Google Scholar 

  102. Salakhitdinov, M.S. and Mirsaburov, M., Nelokal’nye zadachi dlya uravnenii smeshannogo tipa s singulyarnymi koeffitsientami (Nonlocal Problems for Equations of Mixed Type with Singular Coefficients), Tashkent: Universitet, 2005.

    Google Scholar 

  103. Serbina, L.I., Nelokal’nye matematicheskie modeli perenosa v vodonasosnykh sistemakh (Nonlocal Mathematical Models of Transport in Water Pumping Systems), Moscow: Nauka, 2007.

    Google Scholar 

  104. Sitnik, S.M., Transformation operators for singular differential equations with the Bessel operator, in Sb. nauchn. tr.: Krayevye zadachi dlya neklassicheskikh uravnenii matematicheskoi fiziki (Collect. Sci. Works: Boundary Value Problems for Nonclassical Equations of Mathematical Physics), Novosibirsk, 1989, pp. 179–185.

  105. Sitnik, S.M. and Katrakhov, V.V., A composition method for constructing \(B \)-elliptic, \(B \)-parabolic, and \(B \)-hyperbolic operators, Dokl. Math., 1995, vol. 50, no. 1, pp. 70–77.

    MathSciNet  Google Scholar 

  106. Sitnik, S.M., Application of Bushman–Erdélyi transformation operators and their generalizations in the theory of differential equations with singularities in the coefficients, Doctoral (Phys.-Math.) Dissertation, Voronezh: Voronezh State Univ., 2016.

  107. Sitnik, S.M. and Shishkina, E.L., Metod operatorov preobrazovaniya dlya differentsial’nykh uravnenii s operatorami Besselya (Transformation Operator Method for Differential Equations with Bessel Operators), Moscow: Fizmatlit, 2019.

    Google Scholar 

  108. Skubachevskii, A.L., Nonclassical boundary-value problems. I, J. Math. Sci., 2008, vol. 155, no. 2, pp. 199–334.

    Article  MathSciNet  Google Scholar 

  109. Skubachevskii, A.L., Nonclassical boundary-value problems. II, J. Math. Sci., 2010, vol. 166, no. 4, pp. 377–561.

    Article  MathSciNet  Google Scholar 

  110. Skubachevskii, A.L. and Gurevich, P.L., Primenenie metodov nelineinogo funktsional’nogo analiza k nelokal’nym problemam protsessov raspredeleniya tepla. Uchebn. posobie (Application of Nonlinear Functional Analysis Methods to Nonlocal Problems of Heat Distribution Processes. A Textbook), Moscow: RUDN, 2008.

    Google Scholar 

  111. Smirnov, M.M., Vyrozhdayushchiesya ellipticheskie i giperbolicheskie uravneniya (Degenerate Elliptic and Hyperbolic Equations), Moscow: Nauka, 1966.

    Google Scholar 

  112. Smirnov, M.M., Uravneniya smeshannogo tipa (Mixed Type Equations), Moscow: Nauka, 1970.

    Google Scholar 

  113. Soldatov, A.P., Solution of a boundary value problem in function theory with displacement, Differ. Uravn., 1974, vol. 10, no. 1, pp. 143–152.

    Google Scholar 

  114. Kovaleva, L.A. and Soldatov, A.P., On a nonlocal problem in function theory, Differ. Equations, 2010, vol. 46, no. 3, pp. 400–414.

    Article  MathSciNet  Google Scholar 

  115. Tersenov, S.A., Vvedenie v teoriyu uravnenii, vyrozhdayushchikhsya na granitse (Introduction to the Theory of Equations That Degenerate at the Boundary), Novosibirsk: NGU, 1973.

    Google Scholar 

  116. Tricomi, F.G., Lezioni sulle equazioni a derivate parziali, Torino: Editrice Gheroni, 1954. Translated under the title: Lektsii po uravneniyam v chastnykh proizvodnykh, Moscow: Inostr. Lit., 1957.

    Google Scholar 

  117. Utkina, E.A., On a boundary value problem with displacements in four-dimensional space, Izv. VUZov. Mat., 2009, no. 3, pp. 50–55.

  118. Utkina, E.A., Problem with displacements for the three-dimensional Bianchi equation, Differ. Equations, 2010, vol. 46, no. 4, pp. 538–542.

    Article  MathSciNet  Google Scholar 

  119. Frank, P., Differential- und Integralgleichungen der Mechanik und Physik, Braunschweig: Druck und Verlag von Friedrich Vieweg und Sohn, 1935. Translated under the title: Differentsial’nye i integral’nye uravneniya matematicheskoi fiziki. Chast’ 2 , Leningrad: Glav. Red. Obshchetekh. Lit., 1937.

    Google Scholar 

  120. Frankl, F.I., On Chaplygin’s problems for mixed sub- and supersonic flows, Izv. Akad. Nauk SSSR. Ser. Mat., 1945, vol. 9, no. 2, pp. 121–142.

    MathSciNet  Google Scholar 

  121. Frankl, F.I., Flow around profiles with a subsonic flow with a supersonic zone ending with a direct shock wave, Prikl. Mat. Mekh., 1956, vol. 20, no. 2, pp. 196–202.

    Google Scholar 

  122. Frankl, F.I., Izbrannye trudy po gazovoi dinamike (Selected Works on Gas Dynamics), Moscow: Nauka, 1973.

  123. Chebotareva, E.V., Solving boundary value problems for multidimensional degenerate \(B \)-elliptic equations by the potential method, Cand. Sci. (Phys.-Math.) Dissertation, Kazan: Kazan State Univ., 2010.

  124. Shishkina, E.L., Generalized divergence theorem and second Green’s identity for \(B \)-elliptic and \(B \)-hyperbolic operators, Prikl. Mat. Fiz., 2019, vol. 51, no. 4, pp. 506–513.

    Google Scholar 

  125. Shishkina, E.L., The general Euler–Poisson–Darboux equation and hyperbolic \(B \)-potentials, Sovrem. Mat. Fundam. Napravl., 2019, vol. 65, no. 2, pp. 157–338.

    MathSciNet  Google Scholar 

  126. Shkalikov, A.A., On the basis property of eigenfunctions of an ordinary differential operator with integral boundary conditions, Vestn. MGU. Ser. 1. Mat. Mekh., 1982, no. 6, pp. 12–21.

  127. Yurchuk, N.I., Mixed problem with integral condition for some hyperbolic equations, Differ. Uravn., 1986, vol. 22, no. 12, pp. 2117–2126.

    Google Scholar 

  128. Yanushauskas, A.I., Analiticheskaya teoriya ellipticheskikh uravnenii (Analytic Theory of Elliptic Equations), Novosibirsk: Nauka Sib. Otd., 1979.

    Google Scholar 

  129. Avalishvili, G., Avalishvili, M., and Gordeziani, D., On integral nonlocal boundary value problems for some partial differential equations, Bull. Georgian Natl. Acad. Sci., 2011, vol. 5, no. 1, pp. 31–37.

    MathSciNet  Google Scholar 

  130. Bers, L., On a class of differential equations in mechanics of continua, Q. Appl. Math., 1943, vol. 5, no. 1, pp. 168–188.

    Article  MathSciNet  Google Scholar 

  131. Bers, L., A remark on an applications of pseudo-analytic functions, Am. J. Math., 1956, vol. 78, no. 3, pp. 486–496.

    Article  MathSciNet  Google Scholar 

  132. Bers, L. and Gelbart, A., On a class of functions defined by partial differential equations, Trans. Am. Math. Soc., 1944, vol. 56, pp. 67–93.

    Article  MathSciNet  Google Scholar 

  133. Bouziani, A. and Mesloub, S., A strong solution of an evolution problem with integral condition, Georgian Math. J., 2002, vol. 9, no. 12, pp. 149–159.

    MathSciNet  Google Scholar 

  134. Bouziani, A., On the solvability of a nonlocal problem arising in dynamics of moisture transfer, Georgian Math. J., 2003, no. 4, pp. 607–622.

  135. Bouziani, A. and Benouar, N., On integral nonlocal boundary value problems for some partial differential equations, Bull. Belg. Math. Soc., 1996, no. 3, pp. 137–145.

  136. Byszewski, L., Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation \(u_{xt}=F(x,t,u,u_x)\), J. Appl. Math. Stochastic Anal., 1990, vol. 3, no. 3, pp. 163–168.

    Article  MathSciNet  Google Scholar 

  137. Cannon, J.R., The solution of heat equation subject to the specification of energy, Q. Appl. Math., 1963, vol. 21, no. 2, pp. 155–160.

    Article  MathSciNet  Google Scholar 

  138. Cannon, J.R., Dirichlet problem for an equation of mixed type with a discontinuous coefficient, Ann. Math. Pura Appl., 1963, vol. 62, pp. 371–377.

    Article  MathSciNet  Google Scholar 

  139. Carroll, R.W. and Showalter, R.E., Singular and Degenerate Cauchy Problems, New York: Academic Press, 1976.

    Google Scholar 

  140. Fitouhi, A., Jebabli, I., Shishkina, E.L., and Sitnik, S.M., Applications of the integral transforms composition method to wave-type singular differential equations and index shift transmutations, Electron. J. Differ. Equat., 2018, vol. 130, pp. 1–27.

    MathSciNet  Google Scholar 

  141. Gilbert, R.P., Function Theoretic Method in Partial Differential Equations, New York–London: Academic Press, 1969.

    Google Scholar 

  142. Popivanov, N. and Schneider, M., The Darboux problem in \(R^3 \) for a class of degenerating hyperbolic equations, J. Math. Anal. Appl., 1993, vol. 175, pp. 537–579.

    Article  MathSciNet  Google Scholar 

  143. Sabitov, K.B. and Zaitseva, N.V., Initial–boundary value problem for hyperbolic equation with singular coefficient and integral condition of second kind, Lobachevskii J. Math., 2018, vol. 39, no. 9, pp. 1419–1427.

    Article  MathSciNet  Google Scholar 

  144. Shishkina, E.L. and Sitnik, S.M., General form of the Euler–Poisson–Darboux equation and application of the transmutation method, Electron. J. Differ. Equat., 2017, vol. 177, pp. 1–20.

    MathSciNet  Google Scholar 

  145. Weinstein, A., Discontinuous integrals and generalized theory of potential, Trans. Am. Math. Soc., 1948, vol. 63, no. 2, pp. 342–354.

    Article  MathSciNet  Google Scholar 

  146. Weinstein, A., Generalized axially symmetric potential theory, Bull. Am. Math. Soc., 1953, vol. 59, pp. 20–38.

    Article  MathSciNet  Google Scholar 

  147. Zaitseva, N.V., Boundary value problem for B-hyperbolic equation with an integral condition of the second kind, J. Theor. Appl. Technol., 2015, vol. 78, no. 3, pp. 497–508.

    Google Scholar 

  148. Zaitseva, N.V., Keldysh type problem for \(B \)-hyperbolic equation with integral boundary value condition of the first kind, Lobachevskii J. Math., 2017, vol. 38, no. 1, pp. 162–169.

    Article  MathSciNet  Google Scholar 

  149. Zaitseva, N.V., First initial–boundary value problem for \(B \)-hyperbolic equation, Lobachevskii J. Math., 2019, vol. 40, no. 2, pp. 240–247.

    Article  MathSciNet  Google Scholar 

  150. Zaitseva, N.V., Boundary value problem with integral condition for the mixed type equation with a singular coefficient, in Transmutation Operators and Applications. Trends in Mathematics, Kravchenko, V.V. and Sitnik, S.M., Eds., Cham, Switzerland: Springer, 2020, pp. 671–686.

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This work was carried out with financial support from the Ministry of Science and Higher Education of the Russian Federation within the framework of the program of the Moscow Center for Fundamental and Applied Mathematics under agreement no. 075-15-2022-284.

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Zaitseva, N.V. Mixed Problems with Integral Conditions for Hyperbolic Equations with the Bessel Operator. Diff Equat 59 (Suppl 1), 1–72 (2023). https://doi.org/10.1134/S00122661230130013

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