Свойства обобщенного интегрального оператора Бесселя целого порядка
DOI:
https://doi.org/10.56143/ujmcs.v2i3s.4Ключевые слова:
higher-order ordinary differential equation; generalized Bessel differential operator; generalized Erdélyi–Kober operator; integer-order generalized Bessel integral operatorАннотация
В данной статье исследуются свойства решений, полученных для задачи Коши, ассоциированной с обыкновенными дифференциальными уравнениями, которые содержат сингулярный дифференциальный оператор Бесселя высшего порядка наряду с младшим членом. В рамках исследования выведено интегральное выражение для единственного решения задачи Коши, включающее как гипергеометрическую функцию Гаусса, так и гипергеометрическую функцию Гумберта второго рода. Строго доказано, что полученное решение полностью удовлетворяет заданным начальным условиям и является единственным. Кроме того, на основе этого интегрального представления введен обобщенный интегральный оператор Бесселя целого порядка. С помощью серии лемм установлен ряд его фундаментальных свойств, включая существование обратного оператора, формулы интегрирования по частям и полугрупповое свойство оператора.
Библиографические ссылки
[1] Sitnik, S., Karimov, Sh., & Boynazarov, A. (2025). Solving the Cauchy problem for an ordinary differential equation with an integer power of the Bessel operator using transmutation operators. Journal of Mathematical Analysis and Applications. Received: 30 July 2025; Accepted: 7 October 2025. https://doi.org/10.1007/s10958-025-xxxx-x
[2] Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1987). Fractional integrals and derivatives: Theory and applications. Minsk: Nauka i Tekhnika. 688 p.
[3] Urinov, A. K., Sitnik, S. M., Shishkina, E. L., & Karimov, Sh. T. (2022). Fractional integrals and derivatives (Generalizations and applications). [In Russian]. Fergana: Publishing House “Farg’ona”. 192 p.
[4] Urinov, A. K., & Karimov, Sh. T. (2021). Erdélyi–Kober operators and applications to partial differential equations. Fergana: Publishing House “Farg’ona”. 202 p.
[5] Sprinkhuizen-Kuyper, I. G. (1979). A fractional integral operator corresponding to negative powers of a certain second-order differential operator. Journal of Mathematical Analysis and Applications, 72(2), 674–702. https://doi.org/10.1016/0022-247X(79)90278-9
[6] Lowndes, J. S. (1970). A generalization of the Erdélyi–Kober operators. Proceedings of the Edinburgh Mathematical Society, 17(2), 139–148. https://doi.org/10.1017/S001309150000856X
[7] Kiryakova, V. (1986). Applications of the generalized Poisson transformation for solving hyper-Bessel differential equations. Godishnik VUZ. Applied Mathematics, 22(4), 129–140.
[8] Kiryakova, V. (1994). Generalized fractional calculus and applications. Harlow: Longman Scientific & Technical. 360 p.
[9] Kiryakova, V. (2008). Transmutation method for solving hyper-Bessel differential equations based on the Poisson–Dimovski transformation. Fractional Calculus and Applied Analysis, 11(3), 299–316.
[10] Kiryakova, V. S. (1994). Generalized fractional calculus and applications. New York: John Wiley & Sons. 360 p.
[11] Urinov, A. K. (2012). Special functions and special operators. [In Uzbek]. Fergana: “Farg’ona” nashriyoti. 112 p.
[12] Lebedev, N. N. (1972). Special functions and their applications. New York: Dover Publications. 358 p.
[13] Sitnik, S. M., & Skoromnik, O. V. (2020). Transmutation operators that generalize the Sonine and Poisson transforms. Journal of Mathematical Sciences, 249(4), 543–558. https://doi.org/10.1007/s10958-020-04987-2
[14] Luchko, Y., & Gorenflo, R. (1998). Fractional derivatives of Erdélyi–Kober type and some of their applications. Fractional Calculus and Applied Analysis, 1(1), 63–78.
[15] Kiryakova, V. (2021). The interplay of generalized fractional calculus and integral transforms. Mathematics, 9(15), 1789. https://doi.org/10.3390/math9151789
[16] Karimov, Sh. T. (2024). On a method for solving the Cauchy problem for a high-order equation with a Bessel operator. Izvestiya Vysshikh Uchebnykh Zavedeniy. Matematika.
[17] Hasanov, A. S., & Ergashev, T. G. (2023). New properties of the Humbert functions and their applications. Integral Transforms and Special Functions, 34(5), 412–428. https://doi.org/10.1080/10652469.2022.2156789
[18] Shishkina, E., & Sitnik, S. M. (2017). On fractional powers of the Bessel operator on a semiaxis. arXiv preprint arXiv:1706.01928. https://doi.org/10.48550/arXiv.1706.01928
[19] Sneddon, I. N. (1972). The use of integral transforms. New York: McGraw-Hill. 540 p.
[20] Bateman, H., & Erdélyi, A. (1954). Tables of integral transforms (Vol. 1). New York: McGraw-Hill. 391 p.
[21] Erdélyi, A., Magnus, W., Oberhettinger, F., & Tricomi, F. G. (1953). Higher transcendental functions (Vol. I). New York: McGraw-Hill. 302 p.
[22] Comtet, L. (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel Publishing Company, Dordrecht.
[23] Sh. T. Karimov, “The Cauchy Problem for the Iterated Klein–Gordon Equation with the Bessel Operator,” Lobachevskii Journal of Mathematics, vol. 41, no. 5, pp. 772–784, 2020.
[24] Sh. T. Karimov and Sh. A. Oripov, “On a Method for Constructing the Riemann Function for Partial Differential Equations with a Singular Bessel Operator,” Lobachevskii Journal of Mathematics, vol. 41, no. 6, pp. 1087–1093, 2020.
[25] Sh. T. Karimov and E. L. Shishkina, “Some methods of solution to the Cauchy problem for an inhomogeneous equation of hyperbolic type with a Bessel operator,” Journal of Physics: Conference Series, vol. 1203, no. 1, art. no. 012096, 2019.
[26] Sh. T. Karimov, “On One Method for the Solution of an Analog of the Cauchy Problem for a Polycaloric Equation with Singular Bessel Operator,” Ukrainian Mathematical Journal, vol. 69, no. 10, pp. 1593–1606, 2018.
[27] M. S. Salokhiddinov and G. N. Nasriddinov, Oddiy differensial tenglamalar [Ordinary Differential Equations], Tashkent: Ozbekiston, 1994 (in Uzbek).
[28] Dunaev A.S., Shlychkov V.I. Gipergeometricheskie Funktsii (Hypergeometric Functions).Educational Electronic Text Resource. Yekaterinburg, 2017.
