Abstract: In this paper, we study a three-dimensional model Volterra type integral equation with boundary weakly special, special and strongly special kernels in the domain \(\Omega=\{(x,y,z):\ 0\leq a < x < \infty,\ 0\leq b < y < b_{0},\ 0\leq c < z < c_{0}\}\), which we will call a rectangular pipe. In the case when the coefficients of the equation are interconnected, the solution of the equation is sought in the class of continuous functions in \(\Omega\) vanishing with a certain asymptotic behavior on special domains. It is proved that, under certain conditions, the problem of finding a solution to a three-dimensional integral equation of the Volterra type with boundary weakly special, special and strongly special kernels is reduced to solving one-dimensional integral equations of the Volterra type with special boundary kernels. Note that when solving this integral equation, connections of these equations with first-order differential equations with weakly singular, singular and strongly singular coefficients are used. Then it is established that there is no need to require differentiability from the obtained solution and the right-hand side, it is sufficient that the right-hand side of the three-dimensional integral equation with boundary special, weakly special, and strongly special kernels is continuous and vanishes with certain asymptotics on special domains. It is proved that, depending on the sign of the coefficients of the equation, the explicit solution of a three-dimensional Volterra-type model integral equation with special kernels can contain from one to three arbitrary functions of two variables, and the case is also determined when the solution of the integral equation is unique.
Keywords: model integral equation, three-dimensional integral equation, boundary singular kernels, arbitrary function
For citation: Radjabova, L. N. and Khushvakhtzoda, M. B. Model Three-Dimensional Volterra Type Integral Equations with Boundary Singular, Weak Singular and Strong Singular Kernels, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp. 103-112 (in Russian).
DOI 10.46698/y7151-5493-5096-h
1. Gakhov, F. D. Boundary Value Problems, Moscow, Nauka, 1977, 640 p. (in Russian).
2. Muskhelishvili, N. I. Singular Integral Equations, Publ. 3, Moscow, 1968, 512 p.
3. Urbanovich, T. M. and Soldatov, A. P. A Characteristic Singular Integral
Equation with a Cauchy Kernel in the Exceptional Case, Scientific
Bulletin of the Belarusian State University. Ser. Mathematics. Physics, 2011,
no. 17(112), issue 24, pp. 1-7 (in Russian).
4. Abapolova, E. A. and Soldatov, A. P. On the Theory of Singular Integral Equations on a Smooth Contour,
Scientific Bulletin of the Belarusian State University. Ser. Mathematics. Physics, 2010, no. 5(76), issue 18, pp. 6-20 (in Russian).
5. Sabitov, K. B. Functional, Differential and Integral Equations, Moscow, Higher School, 2005, 671 p. (in Russian).
6. Mikhlin, C. G. Multidimensional Singular Integrals and Integral Equations,
Moscow, Fizmatgiz, 1962, 254 p. (in Russian).
7. Dovgiy, C. A. and Lifanov, I. K. Methods for Solving Integral Equations, Kiev,
Publishing House "Naukova Dumka", 2002, 343 p. (in Russian).
8. Antipina, E. D. Inversion Formulas for the Three-Dimensional Volterra
Integral Equation of the First Kind with a Prehistory, Proceedings of Irkutsk State University. Ser. Mathematics,
2022, vol. 41, pp. 69-84 (in Russian).
9. Pleshchinsky, N. Singular Integral Equations with a Complex Singularity in the
Core, Kazan, 2018, 160 p. (in Russian).
10. Rasolko, G. A. Numerical Solution of Some Singular Integral Equations with Cauchy
Kernel by Orthogonal Polynomial Method, Minsk, BSU, 2007, 293 p. (in Russian).
11. Rajabov, N. Integral Equations of Volterra types with Fixed Boundary and
Internal Singular and Super-Singular Kernels and their
Applications, Dushanbe, 2007, 221 p. (in Russian).
12. Rajabov, N. and Rajabova, L. N. Introduction to the Theory of Multidimensional Integral Equations of the Type
Volterra with Fixed Singular and Super-Singular Kernels and their
Application, LAP Lambert Academic Publ., 2011, 502 p. (in Russian).
13. Rajabova, L. N. and Rajabov, N. On the Theory of One Class of a Two-Dimensional Weakly Singular
Integral Equation of Volterra Type on the First Quadrant, Reports of the Academy of Sciences of Tajikistan,
2014, vol. 57, pp. 443-451 (in Russian).
14. Radjabova, L. N. and Khushvakhtov, M. B. On the Theory of Special Two-Dimensional Integral Equations of the Type
Volterra with a Special and Weakly-Special Line on the Strip in
the Case when the Parameters of the Equation are not Related,
Reports of the Academy of Sciences of Tajikistan,
2018, vol. 61, no. 4, pp. 331-337 (in Russian).
15. Radjabova, L. N. and Khushvakhtov, M. B. On Some Cases of Non-Model Two-Dimensional
Volterra-Type Integral Equations with a Special and Weakly Special
Line on the Strip, Reports of the Academy of Sciences of
Tajikistan, 2019, vol. 62, no. 9-10, pp. 533-540 (in Russian).
16. Rajabova, L. N. and Khushvakhtov, M. B. To the Theory of Non-Model Two-Dimensional Integral Equations of
Volterra Type with a Strongly Singular and Weakly Singular Line on
a Strip, Bulletin of L. N. Gumilyov Eurasian National University,
Mathematics. Computer Science. Mechanics Series, 2019, vol. 129,
no. 4, pp. 67-72 (in English).
17. Khushvakhtov, M. B. On Some Cases of Two-Dimensional Volterra-Type Integral Equations with a Special and Weakly
Special Line on the Strip, Bulletin of the Tajik National University. Series of Natural Sciences,
2019, no. 1, pp. 44-49 (in Russian).
18. Khushvakhtov M. B. On Some Cases of Non-Model Two-Dimensional Volterra-Type Integral
Equations with a Strongly-Singular and Weakly-Singular Line on the
Strip, International Scientific Journal "Young Scientist",
2019, vol. 287, no. 49, pp. 1-4 (in Russian).
