General Euler--Poisson--Darboux equation and hyperbolic $B$-potentials

E. L. Shishkina

Contemporary Mathematics. Fundamental Directions, Partial differential equations, Tome 65 (2019) no. 2, pp. 157-338.

Résumé

In this work, we develop the theory of hyperbolic equations with Bessel operators. We construct and invert hyperbolic potentials generated by multidimensional generalized translation. Chapter 1 contains necessary notation, definitions, auxiliary facts and results. In Chapter 2, we study some generalized weight functions related to a quadratic form. These functions are used below to construct fractional powers of hyperbolic operators and solutions of hyperbolic equations with Bessel operators. Chapter 3 is devoted to hyperbolic potentials generated by multidimensional generalized translation. These potentials express negative real powers of the singular wave operator, i. e. the wave operator where the Bessel operator acts instead of second derivatives. The boundedness of such an operator and its properties are investigated and the inverse operator is constructed. The hyperbolic Riesz $B$-potential is studied as well in this chapter. In Chapter 4, we consider various methods of solution of the Euler–Poisson–Darboux equation. We obtain solutions of the Cauchy problems for homogeneous and nonhomogeneous equations of this type. In Conclusion, we discuss general methods of solution for problems with arbitrary singular operators.

Bibliographie

[1] M. Abramowitz, I. Stegun, Handbook Of Mathematical Functions, Nauka, M., 1979 (Russian translation)

[2] J. Hadamard, Cauchy's Problem in Linear Partial Differential Equations of Hyperbolic Type, Nauka, M., 1978 (Russian translation)

[3] O. P. Barabash, E. L. Shishkina, “Solution of the general Euler–Poisson–Darboux equation containing the Bessel operator with respect to all variables”, Bull. Tambov Univ. Ser. Nat. Tech. Sci., 2016, no. 6, 2146–2151 (in Russian)

[4] J. Bergh, J. Lefstrem, Interpolation Spaces, Mir, M., 1980 (Russian translation)

[5] Yu. M. Berezanskiy, “On operator generated by ultrahyperbolic differential expression”, Ukr. Math. J., 11:3 (1959), 315–321 (in Russian)

[6] L. Bers, F. John, M. Schechter, Partial Differential Equations, Mir, M., 1966 (in Russian)

[7] O. V. Besov, V. P. Il'in, S. M. Nikol'skiy, Integral Representations of Functions and Embedding Theorems, Nauka, M., 1975 (in Russian)

[8] A. S. Blagoveshchenskiy, “On some correct problems for ultrahyperbolic and wave equation with data on characteristic cone”, Rep. Acad. Sci. USSR, 140:5 (1961), 990–993 (in Russian)

[9] A. S. Blagoveshchenskiy, “On the characteristic problem for an ultrahyperbolic equation”, Math. Digest, 63:1 (1964), 137–168 (in Russian)

[10] G. N. Watson, A Treatise on the Theory of Bessel Functions, v. 1, IL, M., 1949 (Russian translation)

[11] V. S. Vladimirov, Equations of Mathematical Physics, Textbook, Nauka, M., 1981 (in Russian) | MR

[12] V. S. Vladimirov, V. V. Zharinov, Equations of Mathematical Physics, Fizmatlit, M., 2004 (in Russian)

[13] V. Ya. Volk, “On inversion formulas for a differential equation with singularity at $x=0$”, Progr. Math. Sci., 111:4 (1953), 141–151 (in Russian)

[14] “Abstract Euler–Poisson–Darboux equation containing powers of an unbounded operator”, Differ. Equ., 37:5 (2001), 706–709 (in Russian) | MR | Zbl

[15] I. M. Gel'fand, G. E. Shilov, Generalized Functions and Operations over Them, Textbook, Fizmatlit, M., 1958 (in Russian)

[16] I. M. Gel'fand, G. E. Shilov, Some Issues of the Theory of Differential Equations, Generalized Functions, 3, Fizmatlit, M., 1958 (in Russian)

[17] A. V. Glushak, “On one abstract Euler–Poisson–Darboux equation with lower-order term containing a singularity”, Bull. Higher Edu. Inst. Ser. Math., 1995, no. 3, 3–7 (in Russian) | MR | Zbl

[18] A. V. Glushak, “On perturbation of abstract Euler–Poisson–Darboux equation”, Math. Notes, 60:3 (1996), 363–369 (in Russian) | DOI | MR | Zbl

[19] A. V. Glushak, “Regular and singular perturbations of abstract Euler–Poisson–Darboux equation”, Math. Notes, 66:3 (1999), 364–371 (in Russian) | DOI | MR | Zbl

[20] A. V. Glushak, “Nonlocal problem for abstract Euler–Poisson–Darboux equation”, Bull. Higher Edu. Inst. Ser. Math., 2016, no. 6, 27–35 (in Russian) | Zbl

[21] A. V. Glushak, “Operator formula of translation of solution for the Cauchy problem for abstract Euler–Poisson–Darboux equation”, Math. Notes, 105:5 (2019), 656–665 (in Russian) | DOI | MR | Zbl

[22] A. V. Glushak, O. A. Pokruchin, “Solvability criterion for the Cauchy problem for abstract Euler–Poisson–Darboux equation”, Differ. Equ., 52:1 (2016), 41–59 (in Russian) | DOI | MR | Zbl

[23] A. V. Glushak, V. A. Popova, “Inverse problem for abstract Euler–Poisson–Darboux equation”, Contemp. Math. Fundam. Directions, 15, 2006, 126–141 (in Russian)

[24] A. V. Glushak, T. G. Romanchenko, “Relation formulas between solutions of abstract singular differential equations”, Sci. Bull. Belgorod State Univ. Ser. Math. Phys., 42:6 (2016), 36–39 (in Russian)

[25] M. L. Gol'dman, “Generalized kernels of fractional order”, Differ. Equ., 7:12 (1971), 2199–2210 (in Russian) | Zbl

[26] M. L. Gol'dman, “Integral properties of generalized Bessel potentials”, Rep. Russ. Acad. Sci., 414:2 (2007), 159–164 (in Russian) | Zbl

[27] M. L. Gol'dman, “Permutably invariant hulls of generalized Bessel and Riesz potentials”, Rep. Russ. Acad. Sci., 423:1 (2008), 14–18 (in Russian) | DOI | Zbl

[28] M. L. Gol'dman, “Cone of permutations for generalized Bessel potentials”, Proc. Math. Inst. Russ. Acad. Sci., 260, 2008, 151–163 (in Russian) | Zbl

[29] M. L. Gol'dman, “Optimal embeddings of Bessel-type and Riesz-type potentials”, Rep. Russ. Acad. Sci., 428:3 (2009), 305–309 (in Russian) | Zbl

[30] M. L. Gol'dman, “On optimal embeddings of generalized Bessel and Riesz potentials”, Proc. Math. Inst. Russ. Acad. Sci., 269, 2010, 91–111 (in Russian) | Zbl

[31] M. L. Gol'dman, O. M. Gusel'nikova, “Optimal embeddings of Bessel-type and Riesz-type potentials. V. 1”, Bull. RUND Univ. Ser. Math. Inform. Phys., 2011, no. 3, 4–16 (in Russian)

[32] M. L. Gol'dman, A. V. Malysheva, “On estimate of uniform continuity module of generalized Bessel potential”, Proc. Math. Inst. Russ. Acad. Sci., 283, 2013, 80–91 (in Russian) | Zbl

[33] A. M. Gordeev, “Some boundary-value problems for generalized Euler–Poisson–Darboux equation”, Volga Math. Digest, 1968, no. 6, 56–61 (in Russian)

[34] D. S. Donchev, S. M. Sitnik, E. L. Shishkina, “On generalization of the binomial theorem arising in the theory of differential equations”, Sci. Bull. Belgorod State Univ. Ser. Math. Phys., 49:27 (2017), 19–25 (in Russian)

[35] D. S. Donchev, S. M. Sitnik, E. L. Shishkina, “On refinements of the neoclassical inequality and its applications in the theory of stochastic differential equations and Brownian motion”, Chelyabinsk Phys. Math. J., 2:3 (2017), 257–265 (in Russian)

[36] Ya. I. Zhitomirskiy, “The Cauchy problem for systems of linear partial differential equations with Bessel-type differential operators”, Math. Digest, 36:2 (1955), 299–310 (in Russian)

[37] G. Ya. Zagorskiy, Mixed Problems for Systems of Partial Differential Equations of Parabolic Type, Izd-vo L'vovskogo un-ta, L'vov, 1961 (in Russian)

[38] V. A. Il'in, “Kernels of fractional order”, Math. Digest, 41:4 (1957), 459–480 (in Russian) | Zbl

[39] F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, IL, M., 1958 (Russian translation)

[40] Sh. T. Karimov, “Multidimensional Erdélyi–Kober operator and its applications to solving of Cauchy problem for three–dimensional hyperbolic equation with singular coefficients”, Uzb. Math. J., 2013, no. 1, 70–80 (in Russian)

[41] Sh. T. Karimov, “On a method of solution of the Cauchy problem for a generalized Euler–Poisson–Darboux equation”, Uzb. Math. J., 2013, no. 3, 57–69 (in Russian)

[42] Sh. T. Karimov, “Solution of the Cauchy problem for a multidimensional hyperbolic equation with singular coefficients by the method of fractional integrals”, Rep. Acad. Sci. Resp. Uzbekistan, 2013, no. 1, 11–13 (in Russian)

[43] Sh. T. Karimov, “Solution of the Cauchy problem for three-dimensional hyperbolic equation with singular coefficients and spectral parameter”, Uzb. Math. J., 2014, no. 2, 55–65 (in Russian)

[44] Sh. T. Karimov, “On some generalizations of properties of the Erdélyi–Kober operator and their applications”, Bull. KRAUNTz. Phys. Math. Sci., 2017, no. 2, 20–40 (in Russian) | Zbl

[45] Sh. T. Karimov, “On a method of solution of the Cauchy problem for one-dimensional polywave equation with singular Bessel operator”, Bull. Acad. Sci. USSR. Ser. Math., 2017, no. 8, 27–41 (in Russian) | Zbl

[46] Sh. T. Karimov, “On one method of solution for an analog of the Cauchy problem for a polycaloric equation with singular Bessel operator”, Ukr. Math. J., 69:10 (2017), 1372–1384 (in Russian)

[47] D. B. Karp, S. M. Sitnik, “Fractional Hankel transforms and its applications”, Abstr. of Voronezh. Spring Math. School (17–23 Apr. 1996), VGU, Voronezh, 1996, 92 (in Russian)

[48] V. V. Katrakhov, “General boundary-value problems for a class of singular and degenerated elliptic equations”, Math. Digest, 112:3 (1980), 354–379 (in Russian) | MR | Zbl

[49] V. V. Katrakhov, S. M. Sitnik, “Factorization method in the transmutation theory”, Memorial Digest to the Memory of Boris A. Bubnov: Nonclassical Equations and Mixed-Type Equations, Novosibirsk, 1990, 104–122 (in Russian) | Zbl

[50] V. V. Katrakhov, S. M. Sitnik, “Composition method for constructing $B$-elliptic, $B$-hyperbolic and $B$-parabolic transmutations”, Rep. Russ. Acad. Sci., 337:3 (1994), 307–311 (in Russian) | Zbl

[51] V. V. Katrakhov, S. M. Sitnik, “The transmutation method and boundary-value problems for singular elliptic equations”, Contemp. Math. Fundam. Directions, 64, no. 2, 2018, 211–426 (in Russian) | MR

[52] I. A. Kipriyanov, “On boundary-value problems for partial differential equations with Bessel differential operator”, Rep. Acad. Sci. USSR, 158:2 (1964), 275–278 (in Russian) | Zbl

[53] I. A. Kipriyanov, “Fourier–Bessel transforms and embedding theorems for weighted classes”, Proc. Math. Inst. Russ. Acad. Sci., 89, 1967, 130–213 (in Russian) | Zbl

[54] I. A. Kipriyanov, “Boundary–value problems for singular elliptic partial differential operators”, Rep. Acad. Sci. USSR, 195:1 (1970), 32–35 (in Russian) | Zbl

[55] I. A. Kipriyanov, “On a class of singular elliptic operators”, Differ. Equ., 7:11 (1971), 2065–2077 (in Russian) | MR

[56] I. A. Kipriyanov, “On a class of singular elliptic equations”, Siberian Math. J., 14:3 (1973), 560–568 (in Russian) | Zbl

[57] I. A. Kipriyanov, Singular Elliptic Boundary-Value Problems, Fizmatlit, M., 1997 (in Russian)

[58] I. A. Kipriyanov, Yu. V. Zasorin, “On fundamental solution of wave equation with several singularities and on Huygens's principle”, Differ. Equ., 28:3 (1992), 452–462 (in Russian) | MR | Zbl

[59] I. A. Kipriyanov, L. A. Ivanov, “On lacunas for some classes of singular equations”, Math. Digest, 110:2 (1979), 235–250 (in Russian) | MR | Zbl

[60] I. A. Kipriyanov, L. A. Ivanov, “Fundamental solutions of homogeneous $B$-hyperbolic equations”, Siberian Math. J., 21:4 (1980), 95–102 (in Russian) | MR | Zbl

[61] I. A. Kipriyanov, L. A. Ivanov, “Euler–Poisson–Darboux equation in the Riemann space”, Rep. Acad. Sci. USSR, 260:4 (1981), 790–794 (in Russian) | MR | Zbl

[62] I. A. Kipriyanov, L. A. Ivanov, “Derivation of fundamental solutions for homogeneous equations with singularities with respect to multiple variables”, Proc. S. L. Sobolev Semin., 1983, no. 1, 55–77 (in Russian) | MR | Zbl

[63] I. A. Kipriyanov, L. A. Ivanov, “Cauchy problem for the Euler–Poisson–Darboux equation in homogeneous symmetric Riemann space. I”, Proc. Math. Inst. Russ. Acad. Sci., 170, 1984, 139–147 (in Russian) | Zbl

[64] I. A. Kipriyanov, L. A. Ivanov, “Cauchy problem for the Euler–Poisson–Darboux equation in symmetric space”, Math. Digest, 124:1 (1984), 45–55 (in Russian) | MR

[65] I. A. Kipriyanov, L. A. Ivanov, “Reisz potentials in Lorentz spaces”, Math. Digest, 130:4 (1986), 465–474 (in Russian) | Zbl

[66] I. A. Kipriyanov, L. A. Ivanov, “To the theory of Reisz potentials in Lorentz spaces”, Proc. Math. Inst. Russ. Acad. Sci., 180, 1987, 134–135 (in Russian)

[67] I. A. Kipriyanov, L. A. Ivanov, “The D'Alambert representation and energy equipartition”, Differ. Equ., 26:3 (1990), 458–464 (in Russian) | MR | Zbl

[68] I. A. Kipriyanov, V. V. Katrakhov, “On a class of multidimensional singular pseudodifferential operators”, Math. Digest, 104:1 (1977), 49–68 (in Russian) | MR | Zbl

[69] I. A. Kipriyanov, V. V. Katrakhov, “Boundary–value problem for elliptic equations of the second order with singularities at isolated boundary points”, Rep. Acad. Sci. USSR, 276:2 (1984), 274–276 (in Russian) | MR | Zbl

[70] I. A. Kipriyanov, V. V. Katrakhov, On a singular elliptic boundary-value problem at the sphere domain, Preprint IPM DVO RAN, 1989 (in Russian)

[71] I. A. Kipriyanov, V. V. Katrakhov, Singular boundary-value problems for some elliptic higher order equations, Preprint IPM DVO RAN, 1989 (in Russian)

[72] I. A. Kipriyanov, V. V. Katrakhov, “On a boundary-value problem for elliptic equations of the second order at the sphere domain”, Rep. Acad. Sci. USSR, 313:3 (1990), 545–548 (in Russian) | MR | Zbl

[73] I. A. Kipriyanov, M. I. Klyuchantsev, “On Poisson kernels for boundary-value problems with Bessel differential operator”, Partial Differential Equations, M., 1970, 119–134 (in Russian) | Zbl

[74] I. A. Kipriyanov, V. I. Kononenko, “On fundamental solutions for partial differential equations with Bessel differential operator”, Rep. Acad. Sci. USSR, 170:2 (1966), 261–264 (in Russian) | Zbl

[75] I. A. Kipriyanov, V. I. Kononenko, “Fundamental solutions for $B$-elliptic equations”, Differ. Equ., 3:1 (1967), 114–129 (in Russian) | MR | Zbl

[76] I. A. Kipriyanov, V. I. Kononenko, “On fundamental solutions for some singular partial differential equations”, Differ. Equ., 5:8 (1969), 1470–1483 (in Russian) | Zbl

[77] I. A. Kipriyanov, A. A. Kulikov, “Fundamental solutions for $B$-hypoelliptic equations”, Differ. Equ., 27:8 (1991), 1387–1395 (in Russian) | MR | Zbl

[78] A. Kolmogorov N, S. V. Fomin, Elements of Function Theory and Functional Analysis, Nauka, M., 1981 (in Russian)

[79] D. P. Kostomarov, Cauchy Problems for Ultrahyperbolic Equations, Nauka, M., 2003 (in Russian)

[80] V. V. Kravchenko, E. L. Shishkina, S. N. Torba, “On presentation as a series for integral kernels of transmutation operators for perturbed Bessel equations”, Math. Notes, 104:4 (2018), 552–570 (in Russian) | DOI | MR | Zbl

[81] A. Kratzer, W. Franz, Transcendental Functions, IL, M., 1963 (in Russian)

[82] L. D. Kudryavtsev, “Direct and inverse embedding theorems. Applications to variational method of solution of elliptic equations”, Proc. Math. Inst. Russ. Acad. Sci., 55, 1959, 3–182 (in Russian)

[83] A. G. Kuz'min, Nonclassical Equations of Mixed Type and Their Applications in Gas Dynamics, LGU, L., 1990 (in Russian)

[84] R. Courant, Partial Differential Equations, Mir, M., 1979 (Russian translation)

[85] R. Courant, D. Hilbert, Methods of mathematical physics, v. 1, GITTL, M.–L., 1933 (Russian translation)

[86] R. Kurant, D. Gil'bert, Methods of mathematical physics, v. 2, GITTL, M.–L., 1945 (Russian translation)

[87] B. M. Levitan, “Some questions of the theory of almost periodic functions. I”, Progr. Math. Sci., 2:21 (1947), 133–192 (in Russian) | Zbl

[88] B. M. Levitan, “Some questions of the theory of almost periodic function. II”, Progr. Math. Sci., 2:22 (1947), 174–214 (in Russian)

[89] B. M. Levitan, “Application of operators of generalized translation to second-order linear differential equations”, Progr. Math. Sci., 4:29 (1949), 3–112 (in Russian) | Zbl

[90] B. M. Levitan, “Expansions in Bessel functions into series and Fourier integrals”, Progr. Math. Sci., 6:2 (1951), 102–143 (in Russian) | MR | Zbl

[91] P. I. Lizorkin, “Anisotropic Bessel potentials. Embedding theorems for the Sobolev space $L_p(r_1,\ldots,r_n)$ with fractional derivatives”, Rep. Acad. Sci. USSR, 170:3 (1966), 508–511 (in Russian) | Zbl

[92] P. I. Lizorkin, “Behavior of functions from Liouville classes at infinity. On Riesz potentials of arbitrary order”, Math. Inst. Russ. Acad. Sci., 150, 1979, 174–197 (in Russian) | Zbl

[93] P. I. Lizorkin, S. M. Nikol'skiy, “Coercive properties of an elliptic equation with strong degeneration (the case of generalized solutions)”, Rep. Acad. Sci. USSR, 259:1 (1981), 28–30 (in Russian) | MR | Zbl

[94] P. I. Lizorkin, S. M. Nikol'skiy, “An elliptic equation with degeneration. Variational approach”, Rep. Acad. Sci. USSR, 257:1 (1981), 42–45 (in Russian) | MR | Zbl

[95] P. I. Lizorkin, S. M. Nikol'skiy, “An elliptic equation with degeneration. Differential properties of solutions”, Rep. Acad. Sci. USSR, 257:2 (1981), 278–282 (in Russian) | MR | Zbl

[96] L. N. Lyakhov, “On one class of hypersingular integrals”, Rep. Acad. Sci. USSR, 315:2 (1990), 291–296 (in Russian) | Zbl

[97] L. N. Lyakhov, “Inversion of $B$-potentials”, Rep. Acad. Sci. USSR, 321:3 (1991), 466–469 (in Russian) | Zbl

[98] L. N. Lyakhov, “Spaces of Riesz $B$-potentials”, Rep. Acad. Sci. USSR, 334:3 (1994), 278–280 (in Russian) | Zbl

[99] L. N. Lyakhov, “Description of the Riesz $B$-potentials space $U^\gamma_\alpha(L_p^\gamma)$ by means of $B$-derivatives of order $2[\alpha/2]$”, Rep. Russ. Acad. Sci., 341:2 (1995), 161–165 (in Russian) | MR | Zbl

[100] L. N. Lyakhov, “On the symbol of an integral operator of $B$-potential type with single characteristic”, Rep. Russ. Acad. Sci., 351:2 (1996), 164–168 (in Russian) | MR | Zbl

[101] L. N. Lyakhov, Spherical Weight Functions and Riesz Potentials Generated by Generalized Translation, VGTA, Voronezh, 1997 (in Russian)

[102] L. N. Lyakhov, “Multipliers of mixed Fourier–Bessel transformation”, Proc. Math. Inst. Russ. Acad. Sci., 214, 1997, 234–249 (in Russian) | Zbl

[103] L. N. Lyakhov, $B$-Hypersingular Integrals and Their Applications to Description of the Kipriyanov Classes of Functions and to Integral Equations with $B$-Potential Kernels, LGPU, Lipetsk, 2007 (in Russian)

[104] L. N. Lyakhov, I. P. Polovinkin, E. L. Shishkina, “On one I. A. Kipriyanov's problem for a singular ultrahyperbolic equation”, Differ. Equ., 50:4 (2014), 516–528 (in Russian) | DOI | MR | Zbl

[105] L. N. Lyakhov, I. P. Polovinkin, E. L. Shishkina, “Formulas of solution of the Cauchy problem for a singular wave equation with Bessel operator with respect to time”, Rep. Russ. Acad. Sci., 459:5 (2014), 533–538 (in Russian) | DOI | Zbl

[106] L. N. Lyakhov, E. L. Shishkina, “Generalized Riesz $B$-potentials of mixed type”, Rep. Russ. Acad. Sci., 406:3 (2006), 303–307 (in Russian) | MR | Zbl

[107] L. N. Lyakhov, E. L. Shishkina, “General $B$-hypersingular operators with homogeneous characteristic”, Rep. Russ. Acad. Sci., 412:2 (2007), 162–166 (in Russian) | Zbl

[108] L. N. Lyakhov, E. L. Shishkina, “Inversion of general Riesz $B$-potentials with homogeneous characteristic in weighted spaces”, Rep. Russ. Acad. Sci., 426:4 (2009), 443–447 (in Russian) | Zbl

[109] V. A. Marchenko, “Generalized translation, transmutation operators and inverse problems”, Mathematical Events of KhKhth Century, Fazis, M., 2003, 209–226 (in Russian)

[110] M. I. Matiychuk, Parabolic Singular Boundary-Value Problems, Univ. Math. Nat. Acad. Sci. of Ukraine, Kiev, 1999 (in Ukrainian)

[111] M. I. Matiychuk, Parabolic and Elliptic Boundary-Value Problems with Singularities, Prut, Chernivtsi, 2003 (in Ukrainian)

[112] R. von Mises, Mathematical Theory of Compressible Fluid Flow, IL, M., 1961 (Russian translation)

[113] A. B. Muravnik, “Functional Differential Parabolic Equations: Integral Transformations and Qualitative Properties of the Cauchy Problem Solutions”, Contemp. Math. Fundam. Directions, 52, 2014, 3–141 (in Russian)

[114] S. M. Nikol'skiy, Approximation of Multivariate Functions and Embedding Theorems, Nauka, M., 1977 (in Russian)

[115] S. M. Nikol'skiy, P. I. Lizorkin, “On some inequalities for functions from weighted classes and boundary-value problems with strong degeneracy at the boundary”, Rep. Acad. Sci. USSR, 159:3 (1964), 512–515 (in Russian)

[116] V. A. Nogin, E. V. Sukhinin, Obrashchenie i opisanie giperbolicheskikh potentsialov s $L_p$-plotnostyami, Dep. v VINITI, No 2512-92, M., 1992 (in Russian) | Zbl

[117] V. A. Nogin, E. V. Sukhinin, “Inversion and description of hyperbolic potentials with $L_p$-densities”, Rep. Russ. Acad. Sci., 329:5 (1993), 550–552 (in Russian) | Zbl

[118] V. A. Nogin, K. S. Shevchenko, “Inversion of some Riesz potentials with oscillating characteristics in the nonelliptic case”, Bull. Higher Edu. Inst. Ser. Math., 1999, no. 10, 77–80 (in Russian) | Zbl

[119] M. N. Olevskiy, “Solution of the Dirichlet problem related to equation $\Delta u+\dfrac{p}{x_n}\dfrac{\partial u}{x_n}=\rho$ in a semishperic domain”, Rep. Acad. Sci. USSR, 64:6 (1949), 767–770 (in Russian)

[120] S. S. Platonov, “Generalized Bessel translations and some problems of function approximation theory in the $L_2$ metric. 1”, Proc. Petr. State Univ. Ser. Math., 7 (2000), 70–82 (in Russian)

[121] S. S. Platonov, “Generalized Bessel translations and some problems of function approximation theory in the $L_2$ metric. 2”, Proc. Petr. State Univ. Ser. Math., 8 (2001), 20–36 (in Russian) | Zbl

[122] S. S. Platonov, “Bessel harmonic analysis and approximation of functions on a semiaxis”, Bull. Russ. Acad. Sci. Ser. Math., 71:5 (2007), 149–196 (in Russian) | DOI | MR | Zbl

[123] S. S. Platonov, “Generalized Bessel translations and some inverse theorems of the function approximations theory on a semiaxis”, Proc. Petr. State Univ. Ser. Math., 14 (2007), 44–57 (in Russian)

[124] S. S. Platonov, “Generalized Bessel translations and some problems of the function approximations theory on a semiaxis”, Sib. mat. zh., 50:1 (2009), 154–174 (in Russian) | MR | Zbl

[125] I. P. Polovinkin, Mean-value theorems for wave equations and Euler–Poisson–Darboux equations, PhD Thesis, Voronezh, 1992 (in Russian)

[126] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, v. 1, Elementary Functions, Nauka, M., 1981 (in Russian) | MR

[127] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, v. 2, Special Functions, Nauka, M., 1983 (in Russian) | MR

[128] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, v. 3, Special Functions. Additional Chapters, Nauka, M., 2003 (in Russian) | MR

[129] S. P. Pul'kin, “Some boundary-value problems for the equation $u_{xx}\pm u_{yy}+\dfrac{p}{x}u_x$”, Sci. Notes Kuibyshev Ped. Univ., 21 (1958), 3–54 (in Russian)

[130] S. P. Pul'kin, Selected Works, Univers Grupp, Samara, 2007 (in Russian)

[131] B. Riemann, “The propagation of planar air waves of finite amplitude”, Works, OGIZ, M.–L., 1948, 376–395 (Russian translation)

[132] K. B. Sabitov, R. R. Il'yasov, “Solution of the Tricomi problem for a mixed-type equation with a singular coefficient by a spectral method”, Bull. Higher Edu. Inst. Ser. Math., 2004, no. 2, 64–71 (in Russian) | MR | Zbl

[133] S. G. Samko, “On main functions vanishing in a given set and on division by functions”, Math. Notes, 21:5 (1977), 677–689 (in Russian) | MR | Zbl

[134] S. G. Samko, “On density of Lizorkin-type spaces $\Phi_V$ in $L_p(R_n)$”, Math. Notes, 31:6 (1982), 855–865 (in Russian) | MR | Zbl

[135] S. G. Samko, “On density of Lizorkin-type spaces $\Phi_V$ in spaces $L_p(R_n)$ with mixed metric”, Rep. Russ. Acad. Sci., 319:3 (1991), 567–569 (in Russian) | Zbl

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

[137] S. M. Sitnik, On unitary transmutations, Dep. VINITI, No 7770-V86, VGU, Voronezh, 1986 (in Russian)

[138] S. M. Sitnik, On the decay rate of solutions of some elliptic and ultraelliptic equations, Dep. VINITI, No 7771-B86, VGU, Voronezh, 1986 (in Russian)

[139] S. M. Sitnik, Transmutations for the Bessel differential expression, Dep. VINITI, No 535-B87, VGU, Voronezh, 1986 (in Russian)

[140] S. M. Sitnik, “On one pair of transmutations”, Boundary-Value Problems for Nonclassical Equations of Mathematical Physics, Novosibirsk, 1987, 168–173 (in Russian) | MR

[141] S. M. Sitnik, “On the decay rate of solutions of some elliptic and ultraelliptic equations”, Differ. Equ., 24:3 (1988), 538–539 (in Russian) | MR | Zbl

[142] S. M. Sitnik, “Transmutations for singular differential equations with the Bessel operator”, Boundary-Value Problems for Nonclassical Equations of Mathematical Physics, Novosibirsk, 1989, 179–185 (in Russian) | MR

[143] S. M. Sitnik, Unitarity and boundedness of the Buschman–Erdélyi operators of zero order of smoothness, Preprint Inst. Automat. Control Proc. RAS, Vladivostok, 1990 (in Russian)

[144] S. M. Sitnik, “Factorization and estimates of norms of the Buschman–Erdélyi operators in weighted Lebesgue spaces”, Rep. Acad. Sci. USSR, 320:6 (1991), 1326–1330 (in Russian) | Zbl

[145] S. M. Sitnik preprint Transmutation and Jost representation for an equation with singular potential, Preprint Inst. Automat. Control Proc. RAS, Vladivostok, 1993 (in Russian)

[146] S. M. Sitnik, Inequalities for full elliptic Legendre integrals, Preprint Inst. Automat. Control Proc. RAS, Vladivostok, 1994 (in Russian)

[147] S. M. Sitnik, “Inequalities for Bessel functions”, Rep. Russ. Acad. Sci., 340:1 (1995), 29–32 (in Russian) | MR | Zbl

[148] S. M. Sitnik, “Generalization of the Cauchy–Bunyakovskii inequalities by the means method and their applications”, Chernozem. Digest Sci. Study. Ser. Fundam. Math., 2005, no. 1, 3–42 (in Russian)

[149] S. M. Sitnik, “The method of factorization of transmutations in the theory of differential equations”, Bull. Samar. State Univ. Ser. Nat. Sci., 2008, no. 8/1(67), 237–248 (in Russian)

[150] S. M. Sitnik, “Transmutations and their applications”, Studies in Contemporary Analysis and Mathematical Modelling, Vladikavkaz. Nauch. Tsentr RAN i RSO-A, Vladikavkaz, 2008, 226–293 (in Russian)

[151] S. M. Sitnik, “Refinements and generalizations of classic inequalities”, Math. Anal., Totals Sci. Southern Fed. Distr. Ser. Math. Forum, 3, Yuzhnyy Mat. Inst. VNTS RAN i RSO Alaniya, Vladikavkaz, 2009, 221–266 (in Russian)

[152] S. M. Sitnik, “On integral representation of solutions of one differential equation with singularities in coefficietns”, Vladikavkaz. Math. J., 12:4 (2010), 73–78 (in Russian) | MR | Zbl

[153] S. M. Sitnik, “Transmutation of special form for a differential operator with singular at zero potential”, Nonclassical Equations of Mathematical Physics, Inst. Mat. im. S. L. Soboleva SO RAN, Novosibirsk, 2010, 264–278 (in Russian)

[154] S. M. Sitnik, “On explicit realizations of fractional powers of the Bessel differential operator and their applications to differential equations”, Rep. Adyg. (Cherkess.) Int. Acad. Sci., 12:2 (2010), 69–75 (in Russian) | MR

[155] S. M. Sitnik, “Survey of basic properties of Buschman–Erdélyi transmutations”, Chelyabinsk Phys. Math. J., 1:4 (2016), 63–93 (in Russian) | MR

[156] S. M. Sitnik, Application of Buschman–Erdélyi transmutations and their generalizations in the theory of differential equations with singularities in coefficients, Doctoral Thesis, Voronezh, 2016 (in Russian)

[157] S. M. Sitnik, D. B. Karp, Composition formulas for integral transforms with Bessel functions in kernels, Preprint Inst. Automat. Control Proc. RAS, Vladivostok, 1993 (in Russian)

[158] S. M. Sitnik, D. B. Karp, Fractional Hankel transform and its applications in mathematical physics, Preprint Inst. Automat. Control Proc. RAS, Vladivostok, 1994 (in Russian)

[159] S. M. Sitnik, G. V. Lyakhovetskiy, Composition formulas for Buschman–Erdélyi transmutations, Preprint Inst. Automat. Control Proc. RAS, Vladivostok, 1991 (in Russian)

[160] S. M. Sitnik, G. V. Lyakhovetskiy, Vekua–Erdélyi–Lowndes transmutations, Preprint Inst. Automat. Control Proc. RAS, Vladivostok, 1994 (in Russian)

[161] S. M. Sitnik, E. L. Shishkina, “On one identity for an iterated weighted spherical mean and its applications”, Siberian Electron Math. Bull., 13 (2016), 849–860 (in Russian) | Zbl

[162] S. M. Sitnik, E. L. Shishkina, The Transmutation Method for Differential Equations with Bessel Operators, Fizmatlit, M., 2018 (in Russian)

[163] S. M. Sitnik, E. L. Shishkina, “On fractional powers of the Bessel operator on a semiaxis”, Siberian Electron Math. Bull., 15 (2018), 1–10 (in Russian) | MR | Zbl

[164] A. L. Skubachevskiy, “Nonclassical boundary-value problems. I”, Contemp. Math. Fundam. Directions, 26, 2007, 3–132 (in Russian)

[165] A. L. Skubachevskiy, “Nonclassical boundary-value problems. II”, Contemp. Math. Fundam. Directions, 33, 2009, 3–179 (in Russian)

[166] M. M. Smirnov, Degenerating Hyperbolic Equations, Vysheysh. shkola, Minsk, 1977 (in Russian)

[167] V. V. Stashevskaya, “The transmutation method”, Rep. Acad. Sci. USSR, 113:3 (1953), 409–412 (in Russian) | MR

[168] V. V. Stashevskaya, “On an inverse problem of spectral analysis for differential operator with singularity at zero”, Sci. Notes Kharkov Math. Soc., 1957, no. 5, 49–86 (in Russian)

[169] E. Stein, Singular Integrals and Differentiability Properties of Functions, Mir, M., 1973 (in Russian)

[170] E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Mir, M., 1974 (Russian translation)

[171] S. A. Tersenov, “Introduction to Theory of Equations Degenerating at the Boundary”, NGU, 1973, Novosibirsk (in Russian)

[172] R. S. Khayrullin, “To the theory of the Euler–Poisson–Darboux equation”, Bull. Higher Edu. Inst. Ser. Math., 1993, no. 11, 69–76 (in Russian) | MR | Zbl

[173] Khe Kan Cher, “Mixed problem for generalized Euler–Poisson–Darboux equation in exceptional case”, Math. Notes, 40:1 (1986), 87–92 (in Russian) | MR

[174] Khe Kan Cher, “On explicit formulas for solutions of the Darboux and Cauchy–Goursat problems for degenerating hyperbolic equation”, Siberian Math. J., 40:3 (1999), 710–717 (in Russian) | MR | Zbl

[175] L. Hörmander, The Analysis of Linear Partial Differential Operators, v. I, Distribution Theory and Fourier Analysis, Mir, M., 1986 (Russian translation)

[176] G. L. Chernyshev, On the Cauchy problem with singular hyperbolic operator, Abstract of PhD Thesis, VGU, Voronezh, 1973 (in Russian)

[177] E. L. Shishkina, “Generalized weight function $r^\gamma$”, Bull. Voronezh State Univ. Ser. Phys. Math., 2006, no. 1, 215–221 (in Russian) | Zbl

[178] E. L. Shishkina, “An identity for iterated weighted spherical means generated by generalized shift”, Proc. LXVI Sci. Conf. “Gertsenovskie chteniya-2013”, RGPU im. A. I. Gertsena, Saint-Petersburg, 2013, 143–145 (in Russian)

[179] E. L. Shishkina, “Integral representation of kernel of the operator approximating the inverse operator to the hyperbolic Riesz $B$-potential”, Bull. Tambov Univ. Ser. Nat. Tech. Sci., 2016, no. 2, 450–458 (in Russian)

[180] E. L. Shishkina, “On properties of one averaging kernel in a weighted Lebesgue class”, Sci. Bull. Belgorod Univ. Ser. Math. Phys., 42:6 (2016), 12–19 (in Russian)

[181] E. L. Shishkina, “Weighted generalized functions corresponding to quadratic form with complex coefficients”, Chelyabinsk Phys. Math. J., 2:1 (2017), 88–98 (in Russian) | MR

[182] E. L. Shishkina, “Fractional Euler–Poisson–Darboux equation and stochastic walks”, Theory Probab. Appl., 62:4, Abstr. Second Int. Conf. on Stoch. Methods (2017), 837–838 (in Russian)

[183] E. L. Shishkina, Integral transforms composition method for singular differential equations with Bessel operator and its fractional powers, Doctoral Thesis, M., 2019 (in Russian)

[184] L. Asgeirsson, “Uber eine Mittelwertseigenschaft von Losungen homogener linearer partieller Differentialgleichungen 2. Ordnung mit konstanten Koeffizienten”, Math. Ann, 1937, 321–346 | DOI | MR

[185] B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens' Principle, Oxford University Press, New York, 1939 | MR

[186] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus: models and numerical methods, World Scientific etc., Jersey–London–Singapore, 2012 | MR | Zbl

[187] F. W. Bessel, “Untersuchung des Teils der planetarischen St\:orungen, welcher aus der Bewegung der Sonne entsteht”, Abhandlungen der Berliner Akademie, 1824, 1–52

[188] D. W. Bresters, “On the equation of Euler–Poisson–Darboux”, SIAM J. Math. Anal., 4:1 (1973), 31–41 | DOI | MR | Zbl

[189] D. W. Bresters, “On a generalized Euler–Poisson–Darboux equation”, SIAM J. Math. Anal., 9:5 (1978), 924–934 | DOI | MR | Zbl

[190] H. Campos, V. V. Kravchenko, S. M. Torba, “Transmutations, $L$-bases and complete families of solutions of the stationary Schrödinger equation in the plane”, J. Math. Anal. Appl., 389:2 (2012), 1222–1238 | DOI | MR | Zbl

[191] R. W. Carroll, Transmutation and operator differential equations, North Holland, Amsterdam–New York–Oxford, 1979 | MR | Zbl

[192] R. W. Carroll, R. E. Showalter, Singular and degenerate Cauchy problems, Academic Press, N. Y., 1976 | MR

[193] R. Castillo-Pérez, V. V. Kravchenko, S. M. Torba, “Spectral parameter power series for perturbed Bessel equations”, Appl. Math. Comput., 220 (2013), 676–694 | MR | Zbl

[194] E. T. Copson, “Some applications of Marcel Riesz's integrals of fractional order”, Proc. Roy. Soc. Edinburgh Sect. A, 61 (1943), 260–272 | MR | Zbl

[195] W. Craig, S. Weinstein, “On determinism and well-posedness in multiple time dimensions”, Proc. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci., 465:2110 (2009), 3023–3046 | DOI | MR | Zbl

[196] G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, v. 2, Gauthier-Villars, Paris, 1915 | MR

[197] J. Delsarte, “Sur une extension de la formule de Taylor”, J. Math. Pures Appl., 17 (1938), 217–230

[198] J. Delsarte, “Une extension nouvelle de la theorie des fonctions presque-periodiques de Bohr”, Acta Math., 69 (1938), 259–317 | DOI | MR

[199] J. Delsarte, “Hypergroupes et operateurs de permutation et de transmutation”, Colloques Internat. Centre Nat. Rech. Sci., 71 (1956), 29–45 | MR | Zbl

[200] J. Delsarte, J. L. Lions, “Transmutations d'operateurs differentiels dans le domaine complexe”, C. R. Acad. Sci. Paris, 244 (1957), 832–834 | MR | Zbl

[201] I. Dimovski, “Foundations of operational calculi for the Bessel-type differential operators”, Serdica, 1:1 (1975), 51–63 | MR

[202] I. Dimovski, V. Kiryakova, “Transmutations, convolutions and fractional powers of Bessel-type operators via Meijer's G-function”, Proc. Int. Conf. Complex Anal. and Appl. (Varna, 1983), Sofia, 1985, 45–66 | MR

[203] I. Dimovski, V. Kiryakova, “The Obrechkoff integral transform: properties and relation to a generalized fractional calculus”, Numer. Funct. Anal. Optim., 21:1-2 (2007), 121–144 | MR

[204] S. Elouadih, R. Daher, “Generalization of Titchmarsh's theorem for the Dunkl transform in the space $L^p(\mathbb{R}^d,\omega_l(x)dx)$”, Int. J. Math. Model. Comput., 6:4 (2016), 261–267 | MR

[205] L. Euler, “Tentamen de sono campanarum”, Novi Comm. Acad. Petrop, X (1764), 261

[206] L. Euler, “Institutiones calculi integralis”, Opera Omnia, 1:13 (1914), 212–230

[207] H. Exton, “On the system of partial differential equations associated with Appell's function $F_4$”, J. Phys. A. Math. Gen., 28 (1995), 631–641 | DOI | MR | Zbl

[208] A. Fitouhi, I. Jebabli, E. Shishkina, S. M. Sitnik, “Applications of integral transforms composition method to wave-type singular differential equations and index shift transmutations”, Electron. J. Differ. Equ., 2018:130 (2018), 1–27 | MR

[209] J. Fourier, Théorie analytique de la chaleur, Firmin Didot, Paris, 1822 | MR

[210] D. N. Fox, “The solution and Huygens' principle for a singular Cauchy problem”, J. Math. Mech., 8 (1959), 197–219 | MR | Zbl

[211] A. D. Gadjiev, V. S. Guliyev, A. Serbetci, E. V. Guliyev, “The Stein–Weiss type inequalities for the $B$-Riesz potentials”, J. Math. Inequal., 5:1 (2011), 87–106 | DOI | MR | Zbl

[212] V. S. Guliev, “Sobolev theorems for $B$-Riesz potentials”, Dokl. Math., 57:1 (1998), 72–73 | MR | Zbl

[213] V. S. Guliev, “Some properties of the anisotropic Riesz–Bessel potential”, Anal. Math., 26:2 (2000), 99–118 | DOI | MR | Zbl

[214] V. S. Guliev, “On maximal function and fractional integral, associated with the Bessel differential operator”, Math. Inequal. Appl., 6:2 (2003), 317–330 | MR | Zbl

[215] V. S. Guliev, “Weighted inequality for fractional maximal functions and fractional integrals, associated with the Laplace–Bessel differential operator”, Trans. Natl. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci., 26:1 (2006), 71–80 | MR

[216] V. S. Guliev, J. J. Hasanov, “Sobolev–Morrey type inequality for Riesz potentials, associated with the Laplace–Bessel differential operator”, Fract. Calc. Appl. Anal., 9:1 (2006), 17–32 | MR

[217] V. S. Guliev, A. Miloud, “On maximal function on the Laguerre hypergroup”, Fract. Calc. Appl. Anal., 9:3 (2006), 1–12 | MR

[218] M. E. Hamma, R. Daher, “Estimate of $K$-functionals and modulus of smoothness constructed by generalized spherical mean operator”, Proc. Indian Acad. Sci. Math. Sci, 124:2 (2014), 235–242 | DOI | MR

[219] S. Helgason, Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Academic Press, Orlando etc., 1984 | MR | Zbl

[220] L. Hörmander, The analysis of linear partial differential operators, v. I-II, Springer, Berlin, 1983 | MR

[221] E. M. Jager, Applications of distributions in mathematical physics, Mathematisch Centrum, Amsterdam, 1964 | MR | Zbl

[222] A. John, “The Ultrahyperbolic differential equation with four independent variables”, Duke Math. J., 4:2 (1938), 300–322 | DOI | MR

[223] S. T. Karimov, “Multidimensional generalized Erdélyi–Kober operator and its application to solving Cauchy problems for differential equations with singular coefficients”, Fract. Calc. Appl. Anal., 18:4 (2015), 845–861 | DOI | MR | Zbl

[224] S. T. Karimov, “On some generalizations of properties of the Lowndes operator and their applications to partial differential equations of high order”, Filomat, 32:3 (2018), 873–883 | DOI | MR

[225] I. Karoui, On the Bessel–Wright harmonic analysis, PhD Thesis, Université de Carthage, 2017

[226] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier etc., Amsterdam, 2006 | MR | Zbl

[227] V. Kiryakova, “Applications of the generalized Poisson transformation for solving hyper-Bessel differential equations”, Godishnik VUZ. Appl. Math., 22:4 (1986), 129–140 | MR

[228] V. Kiryakova, Generalized fractional calculus and applications, Longman, Harlow, 1994 | MR | Zbl

[229] V. Kiryakova, “Transmutation method for solving hyper-Bessel differential equations based on the Poisson–Dimovski transformation”, Fract. Calc. Appl. Anal., 11:3 (2008), 299–316 | MR | Zbl

[230] V. Kiryakova, B. Al-Saqabi, “Explicit solutions to hyper-Bessel integral equations of second kind”, Comput. Math. Appl., 37 (1999), 75–86 | DOI | MR | Zbl

[231] V. V. Kravchenko, Applied pseudoanalytic function theory, Birkhäuser, Basel, 2009 | MR | Zbl

[232] V. V. Kravchenko, “Construction of a transmutation for the one-dimensional Schrödinger operator and a representation for solutions”, Appl. Math. Comput., 328 (2018), 75–81 | DOI | MR | Zbl

[233] V. V. Kravchenko, L. J. Navarro, S. M. Torba, “Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions”, Appl. Math. Comput., 314:1 (2017), 173–192 | MR | Zbl

[234] V. V. Kravchenko, J. A. Otero, S. M. Torba, “Analytic approximation of solutions of parabolic partial differential equations with variable coefficients”, Adv. Math. Phys., 2017 (2017), 2947275 | DOI | MR | Zbl

[235] V. V. Kravchenko, S. M. Torba, “Transmutations for Darboux transformed operators with applications”, J. Phys. A. Math. Theor., 45:7 (2012), 075201 | DOI | MR | Zbl

[236] V. V. Kravchenko, S. M. Torba, “Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems”, J. Comput. Appl. Math., 275 (2015), 1–26 | DOI | MR | Zbl

[237] V. V. Kravchenko, S. M. Torba, “Construction of transmutation operators and hyperbolic pseudoanalytic functions”, Complex Anal. Oper. Theory, 9:2 (2015), 379–429 | DOI | MR | Zbl

[238] V. V. Kravchenko, S. M. Torba, “Asymptotics with respect to the spectral parameter and Neumann series of Bessel functions for solutions of the one-dimensional Schrödinger equation”, J. Math. Phys., 58:12 (2017), 122107 | DOI | MR | Zbl

[239] Kravchenko V. V., Torba S. M., Khmelnytskaya K. V., “Transmutation operators: construction and applications”, Proc. 17th Int. Conf. on Comput. and Math. Methods in Sci. and Engin. (Cadiz, Andalucia, España, Jul. 4–8, 2017), 1198–1206 | MR

[240] V. V. Kravchenko, S. M. Torba, “A Neumann series of Bessel functions representation for solutions of Sturm–Liouville equations”, Calcolo, 55:1 (2018), 11 | DOI | MR | Zbl

[241] V. V. Kravchenko, S. M. Torba, R. Castillo-Pérez, “A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations”, Appl. Anal., 97:5 (2018), 677–704 | DOI | MR | Zbl

[242] J. L. Lagrange, “Sur le problème de Képler”, Mém. l'Académie R. Sci. Bell. Lett. Berlin, XXV (1771), 113–138

[243] L. N. Lyakhov, M. V. Polovinkina, E. L. Shishkina, “Accompanying distributions of singular differential operators”, J. Math. Sci., 219:2 (2016), 184–189 | DOI | MR | Zbl

[244] L. N. Lyakhov, E. L. Shishkina, “Inversion of general Riesz $B$-potentials”, Proc. Int. Conf. Analytic methods of analysis and differential equations, AMADE 2012, Cambridge Scientific Publishers, Cottenham, 2013, 115–126

[245] L. N. Lyakhov, E. L. Shishkina, “Weighted mixed spherical means and singular ultrahyperbolic equation”, Analysis (Munich), 36:2 (2016), 65–70 | MR | Zbl

[246] J. L. McGregor, Generalized translation operators, PhD Thesis, California Institute of Technology, Pasadena, 1954 | MR

[247] A. B. Muravnik, “On weighted norm estimates for the mixed Fourier–Bessel transforms on non-negative functions”, Integral methods in science and engineering, v. 1, Analytic methods, Longman, Harlow, 119–123 | MR | Zbl

[248] A. B. Muravnik, “Fourier–Bessel transformation of measures and singular differential operators”, Paul Erdős and his mathematics, János Bolyai Math. Soc., Budapest, 1999, 182–184 | MR | Zbl

[249] D. I. Nikolayev, H. Schaeben, “Characteristics of the ultrahyperbolic differential equation governing pole density functions”, Inverse Problems, 15 (1999), 1603–1619 | DOI | MR | Zbl

[250] N. Obrechkoff, “On certain integral representation of real functions on the real semi-axis”, Izvestia Mat. Inst. Sofia, 3 (1958), 2–28 | MR

[251] M. D. Ortigueira, Fractional calculus for scientists and engineers, Springer, Dordrecht, 2011 | MR | Zbl

[252] O. G. Owens, “Uniqueness of solutions of ultrahyperbolic partial differential equations”, Am. J. Math., 69:1 (1947), 184–188 | DOI | MR | Zbl

[253] O. G. Owens, “An ultrahyperbolic equation with an integral condition”, Am. J. Math., 82:4 (1960), 799–811 | DOI | MR | Zbl

[254] S. D. Poisson, “Mémoire sur l'intégration des équations linéaires aux différences partielles”, J. Éc. Roy. Polytech. Ser. 1, 19:12 (1823), 215–248

[255] J. Radzikowski, “On the uniqueness of the limit problem for the ultrahyperbolic equation”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys., 8:4 (1960), 203–207 | MR | Zbl

[256] M. Riesz, “Intégrale de Riemann-Liouville et solution invariantive du probléme de Cauchy pour l'équation de sondes”, Comptes Rendus du Congres International des Mathematiciens, 2 (1936), 44–45

[257] M. Riesz, “L'intégrale de Riemann–Liouville et le probleme de Cauchy”, Acta Math., 81:1-2 (1949), 1–223 | DOI | MR | Zbl

[258] B. Rubin, Fractional integrals and potentials, Addison Wesley Longman, Harlow, 1996 | MR | Zbl

[259] A. Sajǧlam, H. Yıldırım, M. Z. Sarıkaya, “On the product of the ultra-hyperbolic Bessel operator related to the elastic waves”, Selçuk J. Appl. Math., 10:1 (2009), 85–93 | MR

[260] M. Z. Sarıkaya, H. Yıldırım, Ö. Akin, “On generalized Riesz type potential with Lorentz distance”, Lobachevskii J. Math., 28 (2008), 24–31 | MR

[261] L. Schwartz, Théorie des distributions, Hermann, Paris, 1966 | MR

[262] E. L. Shishkina, “Inversion of integral of $B$-potential type with density from $\Phi_\gamma$”, J. Math. Sci., 160:1 (2009), 95–102 | DOI | MR | Zbl

[263] E. L. Shishkina, “On the boundedness of hyperbolic Riesz $B$-potential”, Lith. Math. J., 56:4 (2016), 540–551 | DOI | MR | Zbl

[264] E. L. Shishkina, “On weighted generalized functions associated with quadratic forms”, Probl. Anal. Issues Anal., 5:2 (2016), 52–68 | DOI | MR | Zbl

[265] E. L. Shishkina, “Inversion of the mixed Riesz hyperbolic $B$-potentials”, Int. J. Appl. Math., 30:6 (2017), 487–500 | DOI | MR

[266] E. L. Shishkina, “Generalized Euler–Poisson–Darboux equation and singular Klein–Gordon equation”, J. Phys. Conf. Ser., 973 (2018), 1–21 | DOI

[267] E. L. Shishkina, “Properties of mixed hyperbolic $B$-potential”, Progr. Fract. Differ. Appl., 4:2 (2018), 83–98 | DOI

[268] E. L. Shishkina, “Singular Cauchy problem for the general Euler–Poisson–Darboux equation”, Open Math. J., 16 (2018), 23–31 | DOI | MR | Zbl

[269] E. L. Shishkina, “Solution of the singular Cauchy problem for a general inhomogeneous Euler–Poisson–Darboux equation”, Carpathian J. Math., 34:2 (2018), 255–267 | MR

[270] E. L. Shishkina, S. Abbas, “Method of Riesz potentials applied to solution to nonhomogeneous singular wave equations”, Math. Notes SVFU, 25:3 (2018), 68–91 | Zbl

[271] E. L. Shishkina, M. Karabacak, “Singular Cauchy problem for generalized homogeneous Euler–Poisson–Darboux equation”, Math. Notes SVFU, 25:2 (2018), 85–96 | MR | Zbl

[272] E. L. Shishkina, S. M. Sitnik, “General form of the Euler–Poisson–Darboux equation and application of the transmutation method”, Electron. J. Differ. Equ., 177 (2017), 1–20 | MR

[273] E. L. Shishkina, S. M. Sitnik, “On fractional powers of Bessel operators”, J. Inequal. Spec. Funct., 8:1 (2017), 49–67 | MR

[274] S. M. Sitnik, Transmutations and applications: a survey, arXiv: 1012.3741 [math.CA]

[275] S. M. Sitnik, “A short survey of recent results on Buschman–Erdélyi transmutations”, J. Inequal. Spec. Funct., 8:1 (2017), 140–157 | MR

[276] S. M. Sitnik, “Buschman–Erdélyi transmutations and applications”, Abstr. 8th Int. Conf. «Transform Methods and Special Functions» (Bulgaria, Sofia, Aug. 27–31, 2017), Inst. Math. Inf. Bulg. Acad. Sci., 2017, 59

[277] H. M. Srivastava, P. W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood, Chichester, 1985 | MR | Zbl

[278] K. L. Stellmacher, “Eine Klasse Huygenscher Differentialgleichungen und ihre Integration”, Math. Ann., 130 (1955), 219–233 | DOI | MR | Zbl

[279] S. R. Umarov, Introduction to fractional and pseudo-differential equations with singular symbols, Springer, Cham, 2015 | MR | Zbl

[280] A. K. Urinov, S. T. Karimov, “Solution of the Cauchy problem for generalized Euler–Poisson–Darboux equation by the method of fractional integrals”, Progress in Partial Differential Equations, Springer, Heidelberg, 2013, 321–337 | MR | Zbl

[281] A. Weinstein, “Discontinuous integrals and generalized theory of potential”, Trans. Am. Math. Soc., 63:2 (1948), 342–354 | DOI | MR | Zbl

[282] A. Weinstein, “Generalized axially symmetric potential theory”, Bull. Am. Math. Soc., 59 (1953), 20–38 | DOI | MR | Zbl

[283] A. Weinstein, “On the wave equation and the equation of Euler–Poisson”, Proc. Symp. Appl. Math., v. V, Wave motion and vibration theory, McGraw-Hill, New York–Toronto–London, 1954, 137–147 | DOI | MR

[284] A. Weinstein, “The generalized radiation problem and the Euler–Poisson–Darboux equation”, Summa Brasil. Math., 3 (1955), 125–147 | MR

[285] A. Weinstein, “Spherical means in spaces of constant curvature”, Ann. Mat. Pura Appl. (4), 4:60 (1962), 87–91 | DOI | MR | Zbl

[286] A. Weinstein, “"Some applications of generalized axially symmetric potential theory to continuum mechanics”, Prilozheniya teorii funktsii v mekhanike sploshnykh sred, v. 2, Mekhanika zhidkosti i gaza, matematicheskie metody, Nauka, M., 1965, 440–453