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

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

Abstract

A method is proposed for correcting of test classical solutions of the general inhomogeneous factorized oscillation equation for a semibounded string in order that they have minimal (necessary) smoothness requirements on its righthand side. The idea of the method is to calculate the correction to some of its trial (test) classical solutions, which may require an overestimated smoothness from the right-hand side of the equation. To this end, the correcting Goursat problem for the canonical form of this oscillation equation of a string is formulated and solved. Then, in the resulting solution, the smoothness of the test solution is analyzed and, if necessary, it is corrected by the corresponding solution of the homogeneous oscillation equation of the string. We find new classical solutions and the previously unknown necessary smoothness of the right-hand side.

Author Biography

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

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

References

  1. Tikhonov A. N., Samarskiy A. A. [The equations of mathematical physics]. Moscow, 2004 (in Russ.).
  2. Lomautsau F. E. [The method of supporting mixed problems for a semi-infinite string]. Shestye Bogdanovskie chteniya po obyknovennym differentsialʼnym uravneniyam [The Sixth Bogdanov Readings on Ordinary Differential Equations] : mater. Int. math. conf. (Minsk, 7–10 Dec., 2015) : in 2 part. Minsk, 2015. Part 2. P. 74 –75 (in Russ.).
  3. Lomautsau F. E., Novikov E. N. [Necessary and sufficient conditions for vibrations of a bounded string under oblique derivatives in boundary conditions]. Differentsial’nye uravn. [Differ. Equ.]. 2014. Vol. 50, No. 1. P. 126 –129 (in Russ.).
  4. Lomautsau F. E. [Solution without continuation of the data of the mixed problem for the inhomogeneous oscillation equation of a string with boundary oblique derivatives]. Differentsial’nye uravn. [Differ. Equ.]. 2016. Vol. 52, No. 8. P. 1128–1132 (in Russ.).
  5. Moiseev E. I., Lomautsau F. E., Novikov E. N. [Non-homogeneous factorized second-order hyperbolic equation in a quarter plane under a semi-nonstationary second oblique derivative in the boundary condition]. Dokl. Akad. nauk. 2014. Vol. 459, No. 5. P. 544 –549 (in Russ.).
  6. Lomautsau F. E., Novikov E. N. [Classic solutions of factorized inhomogeneous second order hyperbolic equation in a quadrant at floor unsteady second directional derivative in the boundary condition]. Vesnіk Vitsebskaga dzyarz. unіversіteta іmya P. M. Masherava. 2015. No. 4 (88). P. 5–11 (in Russ.).
  7. Lomautsau F. E. [Discontinuities of the first and second partial derivatives of the total solution for factorized one-dimensional wave equation in the quarter of the plane]. Vestnik Polockogo gos. univ. 2016. No. 12. P. 117–124 (in Russ.).
  8. Vladimirov V. S. [The equations of mathematical physics]. Moscow : Nauka, 2003 (in Russ.).
  9. Brish N. I., Yurchuk N. I. [The Goursat problem for abstract linear second-order differential equations]. Differentsial’nye uravn. [Differ. Equ.]. 1971. Vol. 7, No. 6. P. 1020 (in Russ.).
  10. Sobolev S. L. [Application functional analysis in mathematical physics]. Moscow, 1988 (in Russ.).
  11. Yurchuk N. I., Novikov E. N. [Necessary conditions for the existence of classical solutions of a semi-infinite string vibration]. Vesti Nacyjanalʼaj Akadjemii navuk Belarusi. Ser. fiz.-mat. navuk. 2016. No. 4. P. 116 –120 (in Russ.).
  12. Lomautsau F. E., Novikov E. N. Duhamelʼs method for solving the inhomogeneous oscillation equation for a semibounded string with oblique derivative in an unsteady boundary condition. Vestnik BGU. Ser. 1, Fiz. Mat. Inform. 2012. No. 1. P. 83–86 (in Russ.).
  13. Lomautsau F. E., Novik Y. F. An initial boundary-value problem for the inhomogeneous oscillation equation for a bounded string of general form with the first non-characteristic oblique derivatives in non-stationary boundary conditions. Vestnik BGU. Ser. 1, Fiz. Mat. Inform. 2016. No. 1. P. 129 –135 (in Russ.).
  14. Lomautsau F. E., Novikov E. N. [Resolution of the mixed problem for the factorized oscillation equation of a bounded string under semi-nonstationary factorized second oblique derivatives in the boundary conditions]. Vesnіk Vitsebskaga dzyarz. unіversіteta іmya P. M. Masherava. 2015. No. 2/3 (86/87). P. 15–21 (in Russ.).
  15. Korzyuk V. I. [Equations of mathematical physics]. Minsk, 2011 (in Russ.).
  16. Kozhanov A. I. [Problems with integral type conditions for certain classes of non-stationary equations]. Dokl. Akad. nauk. 2014. Vol. 457, No. 2. P. 152–156 (in Russ.).
  17. Korzyuk V. I., Kozlovskaya I. S., Naumovets S. N. [Classical solution of the first mixed problem of the one-dimensional wave equation with the Cauchy type conditions]. Vesti Nacyjanalʼaj Akadjemii navuk Belarusi. Ser. fiz.-mat. navuk. 2015. No. 1. P. 7–21 (in Russ.).
  18. Kornev V. V., Khromov A. P. [Resolvent approach to the Fourier method in the mixed problem for the inhomogeneous wave equation]. Izv. Sarat. univ. Ser.: Mat. Meh. Inform. 2016. Vol. 16, issue 4. P. 403– 413 (in Russ.).
  19. Korzyuk V. I., Mandrik A. A. [Boundary problem for non-strictly hyperbolic third-order equation]. Differentsial’nye uravn. [Differ. Equ.]. 2016. Vol. 52, No. 2. P. 209 –219 (in Russ.).
  20. Korzyuk V. I., Mandrik A. A. [First mixed problem for non-strictly hyperbolic third-order equation in the limited area]. Differentsial’nye uravn. [Differ. Equ.]. 2016. Vol. 52, No. 6. P. 788–802 (in Russ.).
  21. Gavrilov V. S. [Existence and uniqueness of solutions of hyperbolic equations of divergence form with different boundary conditions on different parts of the border]. Differentsial’nye uravn. [Differ. Equ.]. 2016. Vol. 52, No. 8. P. 1050 –1061 (in Russ.).
  22. Korzyuk V. I., Stolyarchuk I. I. [Classical solution of the first mixed problem for a hyperbolic second-order equation in a curvilinear half-strip with variable coefficients]. Differentsial’nye uravn. [Differ. Equ.]. 2017. Vol. 53, No. 1. P. 77–88 (in Russ.).
Published
2018-02-14
Keywords: correction method of solutions, necessary smoothness, correcting Goursat problem, test solution, correction of the solution, corrective solution, corrected solution
How to Cite
Lomautsau, F. E. (2018). 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, 3, 38-52. Retrieved from https://journals.bsu.by/index.php/mathematics/article/view/756
Issue
No 3 (2017): Journal of the Belarusian State University. Mathematics and Informatics
Section
Mathematical Physics

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