The first mixed problem for the general telegraph equation with variable coefficients on the half-line

DOI: https://doi.org/10.33581/2520-6508-2021-1-18-38

Abstract

For the first time, an explicit formula is obtained for the unique and stable classical solution of the inhomogeneous model telegraph equation with variable velocity in the part of the first quarter of the plane, where the boundary and initial conditions are specified. The correctness of the first mixed problem for the general inhomogeneous telegraph equation in the first quarter of the plane is proved. The existence of a classical solution was established by the continuation method with respect to a parameter using theorems on increasing the smoothness of strong solutions. The uniqueness of this solution is derived from the energy inequality for strong solutions. The stability of the solution is established and necessary and sufficient smoothness conditions of the boundary and initial data and three their matching conditions with the right-hand side of the equation are derived. Sufficient smoothness requirements are indicated for the right-hand side of the equation.

Author Biography

Fiodar E. Lomautsau, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

doctor of science (physics and mathematics), full professor; professor at the department of mathematical cybernetics, faculty of mechanics and mathematics

References

  1. Baranovskaya SN. O klassicheskom reshenii pervoi smeshannoi zadachi dlya odnomernogo giperbolicheskogo uravneniya [On the classical solution of the first mixed problem for a one-dimensional hyperbolic equation] [dissertation]. Minsk: Belarusian State University; 1991. 59 p. Russian.
  2. Lomovtsev FE. [The method of auxiliary mixed problems for a semi-infinite string]. In: Krasovskii SG, editor. Shestye Bogdanovskie chteniya po obyknovennym differentsial’nym uravneniyam. Materialy Mezhdunarodnoi matematicheskoi konferentsii; 7–10 dekabrya 2015 g.; Minsk, Belarus’. Chast’ 2. Teoriya ustoichivosti i upravleniya dvizheniem. Stokhasticheskie differentsial’nye uravneniya. Differentsial’nye uravneniya v chastnykh proizvodnykh. Metodika prepodavaniya matematiki [The 6th Bogdanov readings on ordinary differential equations. Materials of the International mathematical conference; 2015 December 7–10; Minsk, Belarus. Part 2. The theory of stability and motion control. Stochastic differential equations. Partial differential equations. Methods of teaching mathematics]. Minsk: Institute of Mathematics, National Academy of Sciences of Belarus; 2015. p. 74–75. Russian.
  3. Moiseev EI, Yurchuk NI. [Classical and generalized solutions to problems for telegraph equations with the Dirac potential]. Differentsial’nye uravneniya. 2015;51(10):1338–1344. Russian. DOI: 10.1134/S0374064115100088.
  4. Baranovskaya SN, Novikov EN, Yurchuk NI. [The oblique derivative problem in the boundary condition for the telegraph equation with the Dirac potential]. Differentsial’nye uravneniya. 2018;54(9):1176–1183. Russian. DOI: 10.1134/S0374064118090030.
  5. Anikonov DS, Konovalova DS. [Generalized d’Alembert formula for the wave equation with discontinuous coefficients]. Differentsial’nye uravneniya. 2019;55(2):265–268. Russian. DOI: 10.1134/S0374064119020134.
  6. Khromov AP. [On the convergence of the formal Fourier solution of the wave equation with a summable potential]. Zhurnal vychislitel’noy matematiki i matematicheskoy fiziki. 2016;56(10):1795–1809. Russian. DOI: 10.7868/S0044466916100112.
  7. Kornev VV, Khromov AP. Resolvent approach to Fourier method in a mixed problem for non-homogeneous wave equation. Izvestiya Saratovskogo universiteta. Novaya seriya. Seriya: Matematika. Mekhanika. Informatika. 2016;16(4):403–413. Russian. DOI: 10.18500/1816-9791-2016-16-4-403-413.
  8. Khromov AP. [Mixed problem for wave equation with summable potential and nonzero initial velocity]. Doklady Akademii nauk. 2017;474(6):668–670. Russian.
  9. Khromov AP. On classic solution of the problem for a homogeneous wave equation with fixed end-points and zero initial velocity. Izvestiya Saratovskogo universiteta. Novaya seriya. Seriya: Matematika. Mekhanika. Informatika. 2019;19(3):280–288. Russian. DOI: 10.18500/1816-9791-2019-19-3-280-288.
  10. Khromov AP, Kornev VV. [Classical and generalized solutions of a mixed problem for a non-homogeneous wave equation]. Zhurnal vychislitel’noi matematiki i matematicheskoi fiziki. 2019;59(2):286–300. Russian. DOI: 10.1134/S0044466919020091.
  11. Khromov AP. [Necessary and sufficient conditions for the existence of a classical solution of a mixed problem for a homogeneous wave equation with an integrable potential]. Differentsial’nye uravneniya. 2019;55(5):717–731. Russian. DOI: 10.1134/S0374064119050121.
  12. Khromov AP, Kornev VV. Classical and generalized of a mixed problem – solutions for a non-homogeneous wave equation. Doklady Akademii nauk. 2019;484(1):18–20. Russian. DOI: 10.31857/S0869-5652484118-20.
  13. Kornev VV, Khromov AP. Divergent series and generalized solution of a mixed problem for wave equation. In: Sovremennye metody teorii kraevykh zadach. Materialy mezhdunarodnoi konferentsii. Voronezhskaya vesennyaya matematicheskaya shkola «Pontryaginskie chteniya – XXXI»; 3–9 maya 2020 g.; Voronezh, Rossiya [Modern methods of the theory of boundary value problems. Materials of the International conference. Voronezh spring mathematical school «Pontryagin readings – XXXI»; 2020 May 3–9; Voronezh, Russia]. Voronezh: Nauka-Yunipress; 2020. p. 99–102. Russian.
  14. Lomov IS. d’Alembert generalized formula for the telegraph equation in case of a substantially non-self-adjoint operator. In: Sovremennye metody teorii kraevykh zadach. Materialy mezhdunarodnoi konferentsii. Voronezhskaya vesennyaya matematicheskaya shkola «Pontryaginskie chteniya – XXXI»; 3–9 maya 2020 g.; Voronezh, Rossiya [Modern methods of the theory of boundary value problems. Materials of the International conference. Voronezh spring mathematical school «Pontryagin readings – XXXI»; 2020 May 3–9; Voronezh, Russia]. Voronezh: Nauka-Yunipress; 2020. p. 124–126. Russian.
  15. Courant R, Hilbert D. Methods of mathematical physics. Volume II. Partial differential equations. New York: Wiley; 1962. XXII, 830 p. Russian edition: Courant R. Uravneniya s chastnymi proizvodnymi. Venttsel’ TD, translator; Oleinik OA, editor. Moscow: Mir; 1964. 830 p.
  16. Lomovtsev FE, Lysenko VV. Non-characteristic mixed problem for a one-dimensional wave equation in the first quarter of the plane with non-stationary boundary second derivatives. Vesnik Vicebskaga dzjarzhawnaga wniversitjeta. 2019;3:5–17. Russian.
  17. Lomautsau FE. Correction method of test solutions of the general wave equation in the first quarter of the plane for minimal smoothness of its right-hand side. Journal of the Belarusian State University. Mathematics and Informatics. 2017;3:38–52. Russian.
  18. Schauder J. Über lineare elliptische Differentialgleichungen zweiter Ordnung. Mathematische Zeitschrift. 1934;38:257–282. DOI: 10.1007/BF01170635.
  19. Ladyzhenskaya OA. Kraevye zadachi matematicheskoi fiziki [Boundary-value problems of mathematical physics]. Moscow: Nauka; 1973. 408 p. Russian.
  20. Yurchuk NI. [Partially characteristic boundary value problem for one kind of partial differential equations. II]. Differentsial’nye uravneniya. 1969;5(3):531–542. Russian.
  21. Lomautsau FE. [On necessary and sufficient conditions for the unique solvability of the Cauchy problem for second-order hyperbolic differential equations with a variable domain of definition of operator coefficients]. Differentsial’nye uravneniya. 1992;28(5):873–886. Russian.
  22. Lomautsau FE. [Smoothness of strong solutions of complete hyperbolic second-order differential equations with variable domains of operator coefficients]. Differentsial’nye uravneniya. 2001;37(2):276–278. Russian.
  23. Lions J-L, Magenes E. Problèmes aux limites non homogènes et applications. Volume 1. Paris: Dunod; 1968. XIX, 372 p. Russian edition: Lions J-L, Magenes E. Neodnorodnye granichnye zadachi i ikh prilozheniya. Frank LS, translator; Grushin VV, editor. Moscow: Mir; 1971. 371 p.
  24. Yosida K. Functional analysis. Berlin: Springer-Verlag; 1965. XI, 458 p. Russian edition: Yosida K. Funktsional’nyi analiz. Volosov VM, translator. Moscow: Mir; 1967. 623 p.
Published
2021-04-12
Keywords: general telegraph equation, variable coefficients, implicit characteristics, critical characteristic, mixed problem, classical solution
How to Cite
Lomautsau, F. E. (2021). The first mixed problem for the general telegraph equation with variable coefficients on the half-line. Journal of the Belarusian State University. Mathematics and Informatics, 1, 18-38. https://doi.org/10.33581/2520-6508-2021-1-18-38
Issue
No 1 (2021): Journal of the Belarusian State University. Mathematics and Informatics
Section
Differential Equations and Optimal Control

Copyright (c) 2021 Journal of the Belarusian State University. Mathematics and Informatics

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

The authors who are published in this journal agree to the following:

  1. The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
  2. The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
  3. The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)