ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Дифференциальные Уравнения
и
Процессы Управления

Numerical Identification of the Dependence of the Right Side of the Wave Equation on the Spatial Variable

Автор(ы):

Khanlar Mehbali oglu Gamzaev

Azerbaijan State Oil and Industry University
Department of General and Applied Mathematics
Doctor of Technical Sciences, Professor
AZ 1010 Baku, Azadlig avenue 20, Azerbaijan

xan.h@rambler.ru

Аннотация:

The problem of identifying the multiplier of the right side of a one-dimensional wave equation depending on a spatial variable is considered. As additional information, the condition of the final redefinition is set. A discrete analogue of the inverse problem is constructed using the finite difference method. To solve the resulting difference problem, a special representation is proposed, with the help of which the difference problem splits into two independent difference problems. As a result, an explicit formula is obtained for determining the approximate value of the desired function for each discrete value of a spatial variable. The presented results of numerical experiments conducted for model problems demonstrate the effectiveness of the proposed computational algorithm.

Ключевые слова

Ссылки:

  1. Kabanikhin S. I. Inverse and ill-posed problems. Berlin: Walter de Gruyter, 2011. 475 p
  2. Isakov V. Inverse Problems for Partial Differential Equations. Berlin: Springer, 2017. 345 p
  3. Alifanov O. M., Artioukhine E. A., Rumyantsev S. V. Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems. Begell House, 1995. 306 p
  4. Hasanov Hasanoğ lu A., Vladimir G. R. Introduction to Inverse Problems for Differential Equations. Springer, 2021. 516 p
  5. Borukhov V. T., Zayats G. M. Identification of a time-dependent source term in nonlinear hyperbolic or parabolic heat equation. International Journal of Heat and Mass Transfer. 2015, 91, 1106-1113
  6. Vabishchevich P. N. Computational identification of the time dependence of the right-hand side of a hyperbolic equation. Computational Mathematics and Mathematical Physics. 2019, 59(9), 1475-1483
  7. Denisov A. M. Problems of determining the unknown source in parabolic and hyperbolic equations. Computational Mathematics and Mathematical Physics. 2015, 55(5), 829-833
  8. Yibin Ding, Xiang Xu. On convexity of the functional for inverse problems of hyperbolic equations. Applied Mathematics Letters. 2019, 94, 174-180
  9. Safiullova R. R. Inverse Problems for the Second Order Hyperbolic Equation with Unknown Time Depended Coefficient. Bulletin of the South Ural State University. Series Mathematical Modelling, Programming & Computer Software. 2013, 6(4), 73-86 (in Russian)
  10. Jiang D., Liu Y., Yamamoto M. Inverse source problem for the hyperbolic equation with a time-dependent principal part. Journal of Differential Equations. 2017, 262(1), 653-681
  11. Giuseppe Floridia, Hiroshi Takase. Inverse problems for first-order hyperbolic equations with time-dependent coefficients. Journal of Differential Equations. 2021, 305, 45-71
  12. Safiullova R. R. On solvability of the linear inverse problem with unknown composite right-hand side in hyperbolic equation. Bulletin of the South Ural State University. Series Mathematical Modelling, Programming & Computer Software, 2009, 37 (170), 93-105 (in Russian).
  13. Prilepko A. I., Orlovsky D. G. and Vasin I. A. Methods for Solving Inverse Problems in Mathematical Physics. New York: Marcel Dekker, 2000. 744 p
  14. Denisov A. M. Integro-functional equations in the inverse source problem for the wave equation. Differential Equations. 2006, 42(9), 1221-1232
  15. Samarskii A. A., Vabishchevich P. N. Numerical Methods for Solving Inverse Problems of Mathematical Physics. Berlin: Walter de Gruyter, 2008. 438 p
  16. Gamzaev Kh. M., Huseynzade S. O., Gasimov G. G. A numerical method for solving identification problem for the lower coefficient and the source in the equation convection-reaction. Cybernetics and Systems Analysis. 2018, 54(6), 971-976
  17. Gamzaev Kh. M. The problem of identifying the trajectory of a mobile point source in the convective transport equation. Bulletin of the South Ural State University. Series Mathematical Modelling, Programming & Computer Software. 2021, 14(2), 78-84

Полный текст (pdf)