phone +7 (3412) 91 60 92

Archive of Issues

Kazakhstan Karaganda
Section Mathematics
Title On the singular Volterra integral equation of the boundary value problem for heat conduction in a degenerating domain
Author(-s) Ramazanov M.I.a, Gulmanov N.K.a
Affiliations Karaganda State Universitya
Abstract In this paper, we consider a singular Volterra type integral equation of the second kind, to which some boundary value problems of heat conduction in domains with a boundary varying with time are reduced by the method of thermal potentials. The peculiarity of such problems is that the domain degenerates into a point at the initial moment of time. Accordingly, a distinctive feature of the integral equation under study is that the integral of the kernel, as the upper limit of integration tends to the lower one, is not equal to zero. This circumstance does not allow solving this equation by the method of successive approximations. We constructed the general solution of the corresponding characteristic equation and found the solution of the complete integral equation by the Carleman–Vekua method of equivalent regularization. It is shown that the corresponding homogeneous integral equation has a nonzero solution.
Keywords integral equation, a singular Volterra type integral equation of the second kind, Carleman–Vekua regularization method
UDC 517.968
MSC 45D05, 45E10
DOI 10.35634/vm210206
Received 20 April 2021
Language Russian
Citation Ramazanov M.I., Gulmanov N.K. On the singular Volterra integral equation of the boundary value problem for heat conduction in a degenerating domain, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 2, pp. 241-252.
  1. Kheloufi A., Sadallah B.-K. On the regularity of the heat equation solution in non-cylindrical domains: Two approaches, Applied Mathematics and Computation, 2011, vol. 218, issue 5, pp. 1623-1633.
  2. Kheloufi A. Existence and uniqueness results for parabolic equations with Robin type boundary conditions in a non-regular domain of $\mathbb R$$3$ , Applied Mathematics and Computation, 2013, vol. 220, pp. 756-769.
  3. Cherfaoui S., Kessab A., Kheloufi A. Well-posedness and regularity results for a $2m$-th order parabolic equation in symmetric conical domains of $\mathbb{R}$${N+1}$ , Mathematical Methods in the Applied Sciences, 2017, vol. 40, issue 16, pp. 6035-6047.
  4. Kheloufi A. On a fourth order parabolic equation in a nonregular domain of $\mathbb{R}$$3$ , Mediterranean Journal of Mathematics, 2014, vol. 12, issue 3, pp. 803-820.
  5. Kheloufi A., Sadallah B.-K. Study of the heat equation in a symmetric conical type domain of $\mathbb{R}$${N+1}$ , Mathematical Methods in the Applied Sciences, 2014, vol. 37, issue 12, pp. 1807-1818.
  6. Chapko R., Johansson B.T., Vavrychuk V. Numerical solution of parabolic Cauchy problems in planar corner domains, Mathematics and Computers in Simulation, 2014, vol. 101, pp. 1-12.
  7. Wang Y., Huang J., Wen X. Two-dimensional Euler polynomials solutions of two-dimensional Volterra integral equations of fractional order, Applied Numerical Mathematics, 2021, vol. 163, pp. 77-95.
  8. Dehbozorgi R., Nedaiasl K. Numerical solution of nonlinear weakly singular Volterra integral equations of the first kind: An hp-version collocation approach, Applied Numerical Mathematics, 2021, vol. 161, pp. 111-135.
  9. Kavokin A.A., Kulakhmetova A.T., Shpadi Y.R. Application of thermal potentials to the solution of the problem of heat conduction in a region degenerates at the initial moment, Filomat, 2018, vol. 32, issue 3, pp. 825-836.
  10. Amangalieva M.M., Akhmanova D.M., Dzhenaliev M.T., Ramazanov M.I. Boundary value problems for a spectrally loaded heat operator with load line approaching the time axis at zero or infinity, Differential Equations, 2011, vol. 47, issue 2, pp. 231-243.
  11. Jenaliyev M., Amangaliyeva M., Kosmakova M., Ramazanov M. On a Volterra equation of the second kind with “incompressible” kernel, Advances in Difference Equations, 2015, vol. 2015, issue 1, article number 71.
  12. Amangaliyeva M.M., Dzhenaliev M.T., Kosmakova M.T., Ramazanov M.I. On one homogeneous problem for the heat equation in an infinite angular domain, Siberian Mathematical Journal, 2015, vol. 56, issue 6, pp. 982-995.
  13. Jenaliyev M., Ramazanov M. On a homogeneous parabolic problem in an infinite corner domain, Filomat, 2018, vol. 32, issue 3, pp. 965-974.
  14. Baderko E.A. Parabolic problems and boundary integral equations, Mathematical Methods in the Applied Sciences, 1997, vol. 20, issue 5, pp. 449-459.
  15. Polyanin A.D., Manzhirov A.V. Handbook of integral equations, Boca Raton: CRC Press, 2008.
  16. Ditkin V.A., Prudnikov A.P. Spravochnik po operatsionnomu ischisleniyu (Operational calculus handbook), Moscow: Vysshaya shkola, 1965.
  17. Gradshteyn I.S., Ryzhik I.M. Table of integrals, series, and products, Academic Press, 2014.
  18. Lavrent'ev M.A., Shabat B.V. Metody teorii funktsii kompleksnogo peremennogo (Methods of the theory of function of complex variable), Moscow: Nauka, 1973.
Full text
<< Previous article
Next article >>