Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Malyshev K. Y. Representation of Green’s functions of the wave equation on a segment in finite terms. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 4, pp. 430-446. DOI: 10.18500/1816-9791-2022-22-4-430-446, EDN: UIUDUP

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
30.11.2022
Full text:
(downloads: 1029)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
517.98
EDN: 
UIUDUP

Representation of Green’s functions of the wave equation on a segment in finite terms

Autors: 
Malyshev Ksaverii Yurievich, Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics (SINP MSU)
Abstract: 

Solutions of initial-boundary value problems on the excitation of oscillations of a finite segment by an instantaneous point sourse are investigated. Solutions to these problems, called Green's functions of the equation of oscillations on a segment, are known in the form of infinite Fourier series or series in terms of Heaviside functions. A. N. Krylov's method of accelerating the convergence of Fourier series for several types of boundary conditions not only accelerates the convergence, but allows one to compose expressions for Green's functions in finite terms. In this paper, finite expressions of Green's functions are given in the form of elementary functions of a real variable. Four different formulations of boundary conditions are considered, including the periodicity conditions.

Acknowledgments: 
This work was supported by the RUDN University Strategic Academic Leadership Program. The author thanks Prof. M. D. Malykh (RUDN), for constant attention to the work, Prof. A. N. Bogolyubov (Faculty of Physics, Lomonosov Moscow State University), Prof. L. A. Sevastyanov (RUDN), M. V. Alekseev (HSE) for valuable discussions.
References: 
  1. Kurant R., Gilbert D. Metody matematicheskoy fiziki [Methods of Mathematical Physics]. Vol. 1. Moscow, Leningrad, GTTI, 1933. 525 p. (in Russian).
  2. Strutt J. W. The Theory of Sound. Vol. 1. New York, Dover Poblications, 1945. 520 p.
  3. Sveshnikov A. G., Bogolyubov A. N., Kravtsov A. V. Lektsii po matematicheskoy fizike [Lectures on Mathematical Physics]. Moscow, Nauka, 2004. 416 p. (in Russian).
  4. Dolya P. G. Periodic continuation of functions and solution of the equation of string vibrations in systems of symbolic mathematics. Vestnik Khar’kovskogo natsional’nogo universiteta. Seriya: Matematicheskoye modelirovaniye. Informatsionnyye tekhnologii. Avtomatizirovannyye sistemy upravleniya [Bulletin of Kharkov National University. Series: Mathematical Modeling. Information Technology. Automated Control Systems], 2006, iss. 733, pp. 106–116 (in Russian).
  5. Dolya P. G. Solution to the homogeneous boundary value problems of free vibrations of a finite string. Zhurnal Matematicheskoi Fiziki, Analiza, Geometrii [Journal of Mathematical Physics, Analysis, Geometry], 2008, vol. 4, iss. 2, pp. 237–251.
  6. Larin A. A. The origin of mathematical physics and the theory of oscillations of continuum systems in the “Dispute about the string”. Vestnik Natsional’nogo tekhnicheskogo universiteta “Khar’kovskiy politekhnicheskiy institut”. Istoriya nauki i tekhniki [Bulletin of the National Technical University “Kharkov Polytechnic Institute”. History of Science and Technology], 2008, iss. 8, pp. 89–97 (in Russian).
  7. Gavrilov V. S., Denisova N. A. Metod kharakteristik dlya odnomernogo volnovogo uravneniya [Method of Characteristics for One-Dimensional Wave Equation]. Nizhny Novgorod, Lobachevsky State University of Nizhny Novgorod Publ., 2014. 72 p. (in Russian).
  8. Markushevich A. I. Elementy teorii analiticheskikh funktsiy [Elements of the Theory of Analytic Functions]. Moscow, Uchpedgiz, 1944. 545 p. (in Russian).
  9. Bronstein M. Symbolic Integration I. Transcendental Functions. Second Edition. Springer, 2005. 325 p.
  10. Pavlov D. I. Symbolic integration. Komp’yuternyye instrumenty v obrazovanii [Computer Tools in Education], 2010, iss. 2, pp. 38–43 (in Russian). EDN: MQIBUR
  11. Liouville J. Memoire sur l’integration d’une classe de fonctions transcendantes. Journal fur die reine und angewandte Mathematik , 1835, vol. 13, iss. 2, pp. 93–118 (in German). https://doi.org/10.1515/crll.1835.13.93
  12. Il’in V. A., Poznyak E. G. Osnovy matematicheskogo analiza [Fundamentals of Mathematical Analysis]. Pt. 1. Moscow, Fizmatlit, 2005. 648 p. (in Russian).
  13. Krylov A. N. O nekotorykh differentsial’nykh uravneniyakh matematicheskoy fiziki [On Some Differential Equations of Mathematical Physics]. Moscow, Leningrad, GITTL, 1950. 368 p. (in Russian).
  14. Khromov A. P., Burlutskaya M. S. Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2014, vol. 14, iss. 2, pp. 171–198 (in Russian). https://doi.org/10.18500/1816-9791-2014-14-2-171-198
  15. Kantorovich L. V., Krylov V. I. Priblizhonnye metody vysshego analiza [Approximate Methods of Higher Analysis]. Moscow, Leningrad, GITTL, 1950. 696 p. (in Russian).
  16. Prudnikov A. P., Brychkov Yu. A., Marichev O. I. Integraly i ryady [Integrals and Series]. Vol. 1. Moscow, Fizmatlit, 2002. 632 p. (in Russian).
  17. Polyanin A. D. Spravochnik po lineynym uravneniyam matematicheskoy fiziki [Handbook of Linear Equations of Mathematical Physics]. Moscow, Fizmatlit, 2001. 576 p. (in Russian).
  18. Budak B. M., Samarskiy A. A., Tikhonov A. N. Sbornik zadach po matematicheskoy fizike [Collection of Problems in Mathematical Physics]. Moscow, Nauka, 1972. 688 p. (in Russian).
  19. Lasy P. G., Meleshko I. N. Approximate solution of one problem on electrical oscillations in wires with the use of polylogarithms. ENERGETIKA. Proceedings of CIS higher education institutions and power engineering associations, 2017, vol. 60, iss. 4, pp. 334–340 (in Russian). https://doi.org/10.21122/1029-7448-2017-60-4-334-340
  20. Lasy P. G., Meleshko I. N. Application of Polylogarithms to the Approximate Solution of the Inhomogeneous Telegraph Equation for the Distortionless Line. ENERGETIKA. Proceedings of CIS higher education institutions and power engineering associations, 2019, vol. 62, iss. 5, pp. 413–421 (in Russian). https://doi.org/10.21122/1029-7448-2019-62-5-413-421
  21. Kadyrova V. D., Nasyrov F. S., Suchkova D. A. A probability representation of solutions of wave equations, and the function of Greene. Vestnik USATU, 2017, vol. 21, iss. 4 (78), pp. 129–135 (in Russian). EDN: ZWSQOT
  22. Tikhonov A. N., Samarsky A. A. Uravneniya matematicheskoy fiziki [Equations of Mathematical Physics]. Moscow, Nauka, 2004. 798 p. (in Russian).
  23. Vladimirov V. S. Uravneniya matematicheskoy fiziki [Equations of Mathematical Physics]. Moscow, Nauka, 1981. 512 p. (in Russian).
  24. Zwibach B. Nachal’nyy kurs teorii strun [An Introductory Course in String Theory]. Moscow, Editorial URSS, 2011. 784 p. (in Russian).
  25. Il’in V. A., Poznyak E. G. Osnovy matematicheskogo analiza [Fundamentals of Mathematical Analysis]. Pt. 2. Moscow, Fizmatlit, 2002. 464 p. (in Russian).
  26. Butuzov V. F. Chislovye ryady. Funktsional’nye posledovatel’nosti i ryady [Number Series. Functional Sequences and Series]. Moscow, Faculty of Physics, Moscow State University, 2015. 40 p. (in Russian).
  27. Nikishin E. M. Rearrangements of function series. Sbornik: Mathematics, 1971, vol. 14, iss. 2, pp. 267–280. http://dx.doi.org/10.1070/SM1971v014n02ABEH002617
  28. Fikhtengolts G. M. Kurs differentsial’nogo i integral’nogo ischisleniya [Course of Differential and Integral Calculus]. Moscow, Nauka, 1966. 656 p. (in Russian).
  29. Jolley L. B. W. Summation of Series. New York, Dover Publications, inc, 1961. 278 p.
  30. Gradshteyn I. S., Ryzhik I. M. Tablitsy integralov, summ, ryadov i proizvedeniy [Tables of Integrals, Sums, Series and Products]. Moscow, Nauka, 1963. 1110 p. (in Russian).
  31. Grinberg G. A. Izbrannyye voprosy teorii elektricheskikh i magnitnykh yavleniy [Selected Questions of the Theory of Electrical and Magnetic Phenomena]. Moscow, AN USSR Publ., 1948. 730 p. (in Russian).
  32. Bogolyubov A. N., Levashova N. T., Mogilevsky I. E., Mukhartova Yu. V., Shapkina N. E. Funktsiia Grina operatora Laplasa [Green’s Function of the Laplace Operator]. Moscow, Faculty of Physics, Moscow State University, 2018. 188 p. (in Russian).
  33. Malaschonok G. I. MathPartner computer algebra. Programming and Computer Software, 2017, vol. 43, iss. 2, pp. 112–118. https://doi.org/10.1134/S0361768817020086
  34. Vasiliev S. A., Edneral V. F., Malykh M. D., Sevastyanov L. A. Matematicheskii analiz. Riady s MS Mathematics [Mathematical Analysis. Series with MS Mathematics]. Moscow, RUDN University Publ., 2016. 119 p. (in Russian).
  35. Tikhomirov V. M. Abel and his great theorem. Kvant, 2003, iss. 1, pp. 11–15 (in Russian).
  36. Lobachevsky N. I. Polnoe sobranie sochinenii [Complete Works]. Vol. 5. Moscow, Leningrad, GITTL, 1951. 500 p. (in Russian).
  37. Pak I. N. On the sums of trigonometric series. Russian Mathematical Surveys, 1980, vol. 35, iss. 2, pp. 105–168. http://dx.doi.org/10.1070/RM1980v035n02ABEH001631
  38. Telyakovskii S. A. On the properties of blocks of terms of the series $ \sum {\frac{1}{k}\sin k\,x}$. Ukrainian Mathematical Journal, 2012, vol. 64, iss. 5, pp. 816–822. https://doi.org/10.1007/s11253-012-0680-7
  39. Knut D., Graham R., Patashnik O. Konkretnaia matematika. Matematicheskie osnovy informatiki [Concrete Mathematics. Mathematical Foundations of Informatics], Moscow, Mir, 1998. 703 p. (in Russian).
  40. Kolokolov V. V., Lebedev I. V. Izbrannye glavy matematicheskoi fiziki [Selected Chapters of Mathematical Physics]. Moscow, ITF im. Landau Publ., 2018. 53 p. (in Russian).
  41. Zorich V. A. Matematicheskiy analiz [Mathematical Analysis]. Pt. 2. Moscow, MTsNMO, 2019. 676 p. (in Russian).
Received: 
17.06.2022
Accepted: 
05.08.2022
Published: 
30.11.2022