Numerical method for restoring the initial condition for the wave equation
The inverse problem of restoring the initial condition for the time derivative for the one-dimensional wave equation is considered. As an additional condition, the solution of the wave equation at a finite time is given. First, the discretization of the derivative with respect to the spatial variable is carried out and the initial problem is reduced to a differential-difference problem with respect to functions depending on the time variable. To solve the resulting differential-difference problem, a special representation is proposed, with the help of which the problem splits into two independent differential-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 finite difference method is used for the numerical solution of the obtained differential-difference problems. The presented results of numerical experiments conducted for model problems demonstrate the effectiveness of the proposed computational algorithm.
Keywords
wave equation,
inverse problem,
recovery of the initial condition,
differential-difference problemAuthors
| Gamzaev Khanlar M. | Azerbaijan State Oil and Industry University; Western Caspian University | xan.h@rambler.ru |
Всего: 1
References
Kabanikhin S.I. (2011) Inverse and Ill-Posed Problems. Berlin: Walter de Gruyter.
Isakov V. (2017) Inverse Problems for Partial Differential Equations. Berlin: Springer.
Alifanov O.M., Artioukhine E.A., Rumyantsev S.V. (1995) Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems. Danbury: Begell House.
Prilepko A.I., Orlovsky D.G., Vasin I.A. (2000) Methods for Solving Inverse Problems in Mathematical Physics. New York: Marcel Dekker.
Hasanov A.H., Romanov V.G. (2021) Introduction to Inverse Problems for Differential Equa tions. Springer.
Borukhov V.T., Zayats G.M. (2015) Identification of a time-dependent source term in nonlinear hyperbolic or parabolic heat equation.International Journal of Heat and Mass Transfer. 91. pp. 1106-1113.
Vabishchevich P.N. (2019) Computational identification of the time dependence of the righthand side of a hyperbolic equation.Computational Mathematics and Mathematical Physics. 59(9). pp. 1475-1483.
Denisov A.M. (2015) Problems of determining the unknown source in parabolic and hyperbolic equations.Computational Mathematics and Mathematical Physics. 55(5). pp. 829-833.
Ismailov M.I., Tekin, I. (2016) Inverse coefficient problems for a first order hyperbolic system. Applied Numerical Mathematics. 106. pp. 98-115.
Liao W. (2011) A computational method to estimate the unknown coefficient in a wave equation using boundary measurements. Inverse Problems in Science and Engineering. 19(6). pp. 855-877.
Jiang D., Liu Y., Yamamoto M. (2017) Inverse source problem for the hyperbolic equation with a time-dependent principal part. Journal of Differential Equations. 262. pp. 653-681.
Safiullova R.R. (2013) Obratnaya zadacha dlya giperbolicheskogo uravneniya vtorogo poryadka s neizvestnym koeffitsiyentom, zavisyashchim ot vremeni [Inverse problem for the second order hyperbolic equation with an unknown time-dependent coefficient]. Vestnik Yuzho-Ural'skogo Gosudarstvennogo Universiteta. Seriya Matematicheskoye modelirovaniye i programmirovaniye - Bulletin of the South Ural State University. Mathematical Modelling, Programming & Computer Software. 6(4). pp. 73-86.
Kabanikhin S.I., Bektemesov M.A., Nurseitov D.B., Alimova A.N. (2011) Resheniye zadachi Dirikhle dlya dvumernogo volnovogo uravneniya metodom iteratsiy Landvebera [Solving the Dirichlet problem for a two-dimensional wave equation by the Landweber iteration method]. Vestnik Kazakhskogo natsional'nogo universiteta Seriya Matematika, Mekhanika, Informatika. - Journal of Mathematics, Mechanics and Computer Science. 69(2). pp. 102-110.
Kabanikhin S.I., Bektemesov M.A., Nurseitov D.B., Krivorotko O.I., Alimova A.N. (2012) An optimization method in the Dirichlet problem for the wave equation. Journal of Inverse and Ill-Posed Problems. 20(2). pp. 193-211.
Kabanikhin S.I., Krivorotko O.I. (2013) A numerical method for solving the Dirichlet problem for the wave equation. Journal of Applied and Industrial Mathematics. 7(2). pp. 187-198.
Vasilev V.I., Kardashevsky A.M., Popov V.V. (2017) Iteratsionnyy metod resheniya zadachi Dirikhle i yeye modifikatsiy [Iterative method for solving the Dirichlet problem and its modifications]. Matematicheskiye zametkiSVFU. 24(3). pp. 38-51.
Samarskii A.A., Vabishchevich P.N. (2008) Numerical Methods for Solving Inverse Problems of Mathematical Physics. Berlin: Walter de Gruyter.
Gamzaev Kh.M., Huseynzade S.O., Gasimov G.A. (2018). Numerical method to solve identification problem for the lower coefficient and the source in the convection-reaction equation. Cybernetics and Systems Analysis. 54(6). pp. 971-976.
