The main content of this survey is to systematically expound the theory of transmutation operators and their applications to differential equations with singularities in coefficients. In particular, applications to differential equations with Bessel operators are considered. Also, we consider some problems for differential–difference equations. We provide a short historical background, necessary notation, definitions, and auxiliary facts. The next topics are considered in a survey style: Sonin and Poisson transmutations, Buschman–Erdélyi transmutations and some of their applications, a universal compositional method to construct transmutations, and a new class of transmutations with sharp estimates of kernels for perturbed differential equations with the Bessel operator.
Citation: Andrey Muravnik, Sergei Sitnik. Recent enhances in transmutation theory and integral transforms[J]. AIMS Mathematics, 2026, 11(10): 35005-35046. doi: 10.3934/math.20261363
The main content of this survey is to systematically expound the theory of transmutation operators and their applications to differential equations with singularities in coefficients. In particular, applications to differential equations with Bessel operators are considered. Also, we consider some problems for differential–difference equations. We provide a short historical background, necessary notation, definitions, and auxiliary facts. The next topics are considered in a survey style: Sonin and Poisson transmutations, Buschman–Erdélyi transmutations and some of their applications, a universal compositional method to construct transmutations, and a new class of transmutations with sharp estimates of kernels for perturbed differential equations with the Bessel operator.
| [1] | M. V. Keldysh, On some cases of degeneration of elliptic equations on the boundary of a domain, Doklady Acad. Proc. USSR Acad. Sci., , 77 (1951), 181–183. |
| [2] |
M. El Fatry, M. Mekkour, Y. Akdim, Renormalized solutions for some nonlinear degenerated elliptic problems in weighted Sobolev space with variable exponents, Filomat, 39 (2025), 9101–9112. https://doi.org/10.2298/fil2526101e doi: 10.2298/fil2526101e
|
| [3] |
Z. A. Khan, W. Afzal, An estimation of different kinds of integral inequalities for a generalized class of Godunova–Levin convex and preinvex functions via pseudo and standard order relations, J. Funct. Spaces, 2025 (2025), 3942793. https://doi.org/10.1155/jofs/3942793 doi: 10.1155/jofs/3942793
|
| [4] |
M. A. Ragusa, V. B. Shakhmurov, A Navier–Stokes-type problem with high-order elliptic operator and applications, Mathematics, 8 (2020), 2256. https://doi.org/10.3390/math8122256 doi: 10.3390/math8122256
|
| [5] | A. D. Myshkis, Linear differential equations with delayed argument (in Russian), Moscow–Leningrad: Gostekhizdat, 1951. |
| [6] | L. E. El'sgol'ts, Introduction to the theory of differential equations with deviating agrument (in Russian), Moscow: Nauka, 1964. |
| [7] | L. E. El'sgol'ts, S. B. Norkin, Introduction to the theory of differential equations with deviating agrument (in Russian), Moscow: Nauka, 1971. |
| [8] | A. L. Skubachevskii, Elliptic functional differential equations and applications, Birkhäuser, Basel, 1997. https://doi.org/10.1007/978-3-0348-9033-5 |
| [9] |
A. B. Muravnik, Functional differential parabolic equations: integral transformations and qualitative properties of solutions of the Cauchy problem, J. Math. Sci., 216 (2016), 345–496. https://doi.org/10.1007/s10958-016-2904-0 doi: 10.1007/s10958-016-2904-0
|
| [10] | G. S. Litvinchuk, Boundary value problems and singular integral equations with shift (in Russian), Moscow: Nauka, 1977. |
| [11] | I. A. Kipriyanov, Singular elliptic boundary-value problems (in Russian), Moscow: Nauka–Fizmatlit, 1997. |
| [12] | A. Weinstein, Selecta, London–San Francisco: Pitman, 1978. |
| [13] | E. Shishkina, S. Sitnik, Transmutations, singular and fractional differential equations with applications to mathematical physics, New York: Academic Press, 2020. https://doi.org/10.1016/C2019-0-00708-X |
| [14] | S. M. Sitnik, E. L. Shishkina, Method of transmutations for differential equations with Bessel operators (in Russian), Moscow: Fizmatlit, 2019. |
| [15] |
V. V. Katrakhov, S. M. Sitnik, The transmutation method and boundary-value problems for singular elliptic equations, Contemp. Math. Fundam. Dir., 64 (2018), 211–426. https://doi.org/10.22363/2413-3639-2018-64-2-211-426 doi: 10.22363/2413-3639-2018-64-2-211-426
|
| [16] | R. W. Carroll, Transmutation and operator differential equations, Amsterdam–New York–Oxford: North Holland, 1979. https://doi.org/10.1016/s0304-0208(08)x7090-2 |
| [17] | R. W. Carroll, Transmutation, scattering theory and special functions, Amsterdam–New York–Oxford: North Holland, 1982. https://doi.org/10.1016/s0304-0208(08)x7117-8 |
| [18] | R. W. Carroll, Transmutation theory and applications, Amsterdam–New York: North Holland, 1985. https://doi.org/10.1016/s0304-0208(08)x7118-x |
| [19] | R. W. Carroll, R. E. Showalter, Singular and degenerate Cauchy problems, New York: Academic Press, 1976. https://doi.org/10.1016/s0076-5392(08)x6195-8 |
| [20] | V. A. Marchenko, Spectral theory of Sturm–Liouville operators (in Russian), Kiev: Naukova dumka, 1972. |
| [21] | V. A. Marchenko, Sturm–Liouville operators and their applications (in Russian), Kiev: Naukova dumka, 1977. |
| [22] | B. M. Levitan, Generalized translations and some applications (in Russian), Moscow: GIFML, 1962. |
| [23] | B. M. Levitan, Theory of generalized translation operators (in Russian), Moscow: Nauka, 1973. |
| [24] | B. M. Levitan, Inverse Sturm–Lioville problems (in Russian), Moscow: Nauka, 1984. |
| [25] | D. K. Fage, N. I. Nagnibida, Problem of equivalency of ordinary differential operators (in Russian), Novosibirsk: Nauka, 1977. |
| [26] | S. M. Sitnik, Analytical and computational methods in differential equations, special functions, transmutations and integral transforms, Basel: MDPI, 2023. https://doi.org/10.3390/books978-3-0365-8627-4 |
| [27] | V. V. Kravchenko, S. M. Sitnik, Transmutation operators and applications, Cham: Birkhauser, 2020. https://doi.org/10.1007/978-3-030-35914-0 |
| [28] |
E. L. Shishkina, S. M. Sitnik, I. Jebabli, Application of integral transforms composition method (ITCM) to obtaining transmutations via integral transforms with Bessel functions in kernels, Sib. Èlektron. Mat. Izv., 18 (2021), 884–900. https://doi.org/10.33048/semi.2021.18.067 doi: 10.33048/semi.2021.18.067
|
| [29] |
S. M. Sitnik, On one integral equation in the theory of transform operators, Comput. Math. Math. Phys., 60 (2020), 1381–1391. https://doi.org/10.1134/s096554252008014x doi: 10.1134/s096554252008014x
|
| [30] | M. Abramowitz, I. Stegun, Handbook of mathematical functions (in Russian), Moscow: Nauka, 1979. |
| [31] | H. Bateman, A. Erdélyi, Higher transcendental functions. V. 1 (in Russian), Moscow: Nauka, 1966. |
| [32] | H. Bateman, A. Erdélyi, Higher transcendental functions. V. 2 (in Russian), Moscow: Nauka, 1966. |
| [33] | H. Bateman, A. Erdélyi, Higher transcendental functions. V. 3 (in Russian), Moscow: Nauka, 1966. |
| [34] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Integrals and derivatives of fractional order and some their applications (in Russian), Minsk: Nauka i Tekhnika, 1987. |
| [35] | B. M. Levitan, Expansions in Fourier series and integrals with Bessel functions, Uspekhi Mat. Nauk, 6 (1951), 102–143. |
| [36] | V. V. Stashevskaya, The transmutation method, Dokl. AN SSSR, 113 (1953), 409–412. |
| [37] | A. B. Muravnik, On non-classical Cauchy problem for singular parabolic integrodifferential equations, Russ. J. Math. Phys., 9 (2002), 300–314. |
| [38] | I. A. Kipriyanov, V. V. Katrakhov, V. M. Lyapin, On boundary-value problems in domains of general type for singular parabolic systems of equations, Sov. Math., Dokl., 17 (1976), 1461–1464. |
| [39] | V. V. Krehivs'kiĭ, M. I. Matiĭčuk, Fundamental solutions and the Cauchy problem for linear parabolic systems with Bessel's operator, Dokl. Akad. Nauk SSSR, 181 (1968), 1320–1323. |
| [40] | M. I. Matiĭčuk, Fundamental solutions of parabolic systems with discontinuous coefficients, and their applications to boundary value problems. Ⅰ, Differ. Uravn., 10 (1974), 1463–1477. |
| [41] | M. I. Matiĭčuk, Fundamental solutions of parabolic systems with discontinuous coefficients, and their applications to boundary value problems. Ⅱ, Differ. Uravn., 11 (1975), 1293–1303. |
| [42] | M. I. Matiĭčuk, Fundamental solutions of parabolic systems with discontinuous coefficients, and their applications to boundary value problems. Ⅲ, Differ. Uravn., 14 (1978), 291–303. |
| [43] | M. I. Matiĭčuk, Fundamental solutions of parabolic systems with discontinuous coefficients, and their applications to boundary value problems. Ⅳ, Differ. Uravn., 14 (1978), 885–899. |
| [44] | M. I. Matiĭčuk, The Cauchy problem for a class of degenerate parabolic systems, Ukrain. Mat. Zh., 36 (1984), 321–327. |
Andrey Muravnik, Sergei Sitnik. Recent enhances in transmutation theory and integral transforms[J]. AIMS Mathematics, 2026, 11(10): 35005-35046. doi: 10.3934/math.20261363
DownLoad: