Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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

For citation:

Khromov A. P. On Classic Solution of the Problem for a Homogeneous Wave Equation with Fixed End-Points and Zero Initial Velocity. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 3, pp. 280-288. DOI: 10.18500/1816-9791-2019-19-3-280-288, EDN: OWGYTU

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 243)
Article type: 

On Classic Solution of the Problem for a Homogeneous Wave Equation with Fixed End-Points and Zero Initial Velocity

Khromov August Petrovich, Saratov State University

The paper gives necessary and sufficient conditions of classic solution for a homogeneous wave equation with a summable potential, fixed end-point, and zero initial velocity. With the use of Fourier method and Krylov method of improving series rate convergence an analogue of d’Alembert formula is derived in the form of exponentially convergent series. The paper essentially supports and extends the results of our work carried out in 2016. The suggested new method, based on the use of divergent (in Euler’s sense) series, is very economical in using well-known mathematical facts. It opens a perspective of considerable advancement in studying other boundary problems for partial differential equations.

  1. Khromov A. P. On the convergence of the formal Fourier solution of the wave equation with a summable potential. Comput. Math. and Math. Phys., 2016, vol. 56, iss. 10, pp. 1778–1792. DOI: https://doi.org/10.1134/S0965542516100110
  2. Chernyatin V. A. Obosnovanie metoda Fur’e v smeshannoi zadache dlya uravnenii v chastnykh proizvodnykh [Justification of the Fourier Method in a Mixed Problem for Partial Differential Equations]. Moscow, Moscow Univ. Press, 1991. 112 p. (in Russian).
  3. Krylov A. N. O nekotorykh differentsial’nykh uravneniyakh matematicheskoj fiziki, imeyushchikh prilozheniya v tekhnicheskikh voprosakh [On Some Differential Equations of Mathematical Physics Having Applications in Engineering]. Moscow, Leningrad, GIT- TL, 1950. 368 p. (in Russian).
  4. Burlutskaya M. S., Khromov A. P. Rezolventny approach in the Fourier method. Dokl. Math., 2014, vol. 90, iss. 2, pp. 545–548. DOI:https://doi.org/10.1134/S1064562414060076
  5. Burlutskaya M. S., Khromov A. P. The resolvent approach for the wave equation. Comput. Math. and Math. Phys., 2015, vol. 55, iss. 2, pp. 227–239. DOI: https://doi.org/10.1134/S0965542515020050
  6. Euler L. Differencial’noe ischislenie [Differential calculus]. Moscow, Leningrad, GITTL, 1949. 280 p. (in Russian).
  7. Khromov A. P. Divergent series and functional equations related to analogues of a geometric progression. In: Proc. Intern. Conf. “Pontryaginskie chteniya – XXX”. Voronezh, Izdatel’skij dom VGU, 2019, pp. 291–300 (in Russian).
  8. Kornev V. V., Khromov A. P. Classical and Generalized Solutions of a Mixed Problem for a Nonhomogeneous Wave Equation. Comput. Math. and Math. Phys., 2019, vol. 59, iss. 2, pp. 275–289. DOI: https://doi.org/10.1134/S096554251902009X
  9. Khromov A. P. Necessary and Sufficient Conditions for the Existence of a Classical Solution of the Mixed Problem for the Homogeneous Wave Equation with an In-tegrable Potential. Differential Equations, 2019, vol. 55, iss. 5, pp. 703–717. DOI: https://doi.org/10.1134/S0012266119050112