A generalized integral problem for a system of hyperbolic equations and its applications

Year 2023, , 1513 - 1532, 03.11.2023

Anar Assanova

https://doi.org/10.15672/hujms.1094454

Cited By: 1

Abstract

A nonlocal boundary value problem for a system of hyperbolic equations of second order with generalized integral condition is considered. By method of introduction of functional parameters the investigated problem is transformed to the inverse problem for the system of hyperbolic equations with unknown parameters and additional functional relations. Algorithms of finding solution to the inverse problem for the system of hyperbolic equations are constructed, and their convergence is proved. The conditions for existence of unique solution to the inverse problem for the system of hyperbolic equations are obtained in the terms of initial data. The coefficient conditions for unique solvability of nonlocal boundary value problem for the system of hyperbolic equations with generalized integral condition are established. The results are illustrated by numerical examples.

Keywords

hyperbolic equation, inverse nonlocal problem, generalized integral condition, algorithm, solvability

References

• [1] A.T. Assanova and D.S. Dzhumabaev, Unique solvability of the boundary value problem for systems of hyperbolic equations with data on the characteristics, Comput. Math. Math. Phys. 42 (11), 1609-1621, 2002.
• [2] A.T. Assanova and D.S. Dzhumabaev, Unique solvability of nonlocal boundary value problems for systems of hyperbolic equations, Differ. Equ. 39 (10), 1414-1427, 2003.
• [3] A.T. Assanova and D.S. Dzhumabaev, Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations, J. Math. Anal. Appl. 402 (1), 167-178, 2013.
• [4] A.T. Assanova, On the solvability of a nonlocal problem for the system of Sobolev-type differential equations with integral condition, Georgian Math. J. 28 (1), 49-57, 2021.
• [5] A.T. Assanova, S.S. Kabdrakhova, Modification of the Euler polygonal method for solving a semi-periodic boundary value problem for pseudo-parabolic equation of special type, Mediterr. J. Math. 17 (4), Art.no. 109, 2020.
• [6] A.T. Assanova, R.E. Uteshova, A singular boundary value problem for evolution equations of hyperbolic type, Chaos Solitons Fractals 143 (2), Art. no. 110517, 2021.
• [7] Y. Bai, N.S. Papageorgiou and S. Zeng, A singular eigenvalue problem for the Dirichlet (p,q)-Laplacian, Math. Z., 300 (2), 325345, 2022.
• [8] L. Byszewski, Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation $u_{xt}=F(x,t,u,u_x)$, J. Appl. math. stoch. anal. 3 (3), 163-168, 1990.
• [9] J.R. Cannon, The solution of the heat equation subject to the specification of energy, Q. Appl. Math. 21 (2), 155-160, 1963.
• [10] J. Cen, A. A Khan, D. Motreanu and S. Zeng, Inverse problems for generalized quasi- variational inequalities with application to elliptic mixed boundary value systems, Inverse Probl. 38, Art. no. 065006, 2022.
• [11] A.M. Denisov, Determination of a nonlinear coefficient in a hyperbolic equation for the Goursat problem, J. Inverse Ill-Posed Probl. 6 (4), 327-334, 1998.
• [12] A.M. Denisov, Elements of the Theory of Inverse Problems, VSP Utrecht Netherlands, 1999.
• [13] A.M. Denisov, An inverse problem for a hyperbolic equation, Differ. Equ. 36 (10), 1427-1429, 2000.
• [14] A.M. Denisov, Solvability of the inverse problem for a quasilinear hyperbolic equation, Differ. Equ. 38 (9), 1155-1164, 2002.
• [15] A.M. Denisov, E.Yu. Shirkova, Inverse problem for a quasilinear hyperbolic equation with a nonlocal boundary condition containing a delay argument, Differ. Equ. 49 (9), 1053-1061, 2013.
• [16] D.S. Dzhumabaev, On one approach to solve the linear boundary value problems for Fredholm integro-differential equations, J. Comput. Appl. Math. 294 (1), 342-357, 2016.
• [17] N.D. Golubeva and L.S. Pul’kina, A nonlocal problem with integral conditions, Math. Notes 59 (3), 326-328, 1996.
• [18] V. Isakov, Inverse Problems for Partial Differential Equations, second ed. Springer, New York, 2006.
• [19] S.I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications, De Gruyter, Germany, 2011.
• [20] S.I. Kabanikhin and A. Lorenzi, Identification Problems for Wave Phenomena, VSP Utrecht Netherlands, 1999.
• [21] T. Kiguradze, Some boundary value problems for systems of linear partial differential equations of hyperbolic type, Mem. Differ. Equ. Math. Phys. 1, 1-144, 1994.
• [22] A.Yu. Kolesov, E.F. Mishchenko and N.Kh. Rozov, Asymptotic methods of investigation of periodic solutions of nonlinear hyperbolic equations, 1-191, Tr. MIAN, 222, Nauka, Moscow, 1998. (in Russian)
• [23] A.I. Kozhanov, On the solvability of spatially nonlocal problems with conditions of integral form for some classes of nonstationary equations, Differ. Equ. 51 (8), 1043- 1050, 2015.
• [24] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem, Nonlinear Anal. Real World Appl. 12 (8), 1562-1590, 2011.
• [25] Yu.A. Mitropol’skii, G.P. Khoma and M.I. Gromyak, Asymptotical methods of research quasi-wave equations of hyperbolic type, Naukova Dumka, Kiev, Ukraine, 1991. (in Russian)
• [26] Z.A.Nakhusheva, On one nonlocal problem for partial differential equations, Differ- ents. uravnenia. 22 (1), 171-174, 1986. (in Russian)
• [27] A.M. Nakhushev, Problems with replacement for partial differential equations, Nauka, Moscow, 2006. (in Russian)
• [28] B.I. Ptashnyk, Ill-posed boundary value problems for partial differential equations, Naukova Dumka, Kiev, Ukraine, 1984. (in Russian)
• [29] V.G. Romanov, Inverse Problems of Mathematical Physics, VSP Utrecht Netherlands, 1987.
• [30] A.M. Samoilenko and B.P. Tkach, Numerical-analytical methods in the theory periodical solutions of equations with partial derivatives, Naukova Dumka, Kiev, Ukraine, 1992. (in Russian)
• [31] B.P. Tkach and L.B. Urmancheva, Numerical-analytical method for finding solutions of systems with distributed parameters and integral condition, Nonlinear Oscil. 12 (1), 110-119, 2009.
• [32] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl. 78 (1), 65-98, 1999.
• [33] S. Zeng, Y.Bai, L. Gasiski, P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calc. Var. PDEs. 59, Art. no. 176, 2020.
• [34] S.Zeng, S. Migórski, and Z. Liu, Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities, SIAM J. Optim. 31, 2829-2862, 2021.
• [35] S.Zeng, V.D. Rdulescu and P. Winkert, Double phase implicit obstacle problems with convection and multivalued mixed boundary value conditions, SIAM J. Math. Anal. 54, 1898-1926, 2022.