On an inverse boundary-value problem for the pseudohyperbolic equation with nonclassical boundary conditions
Öz
In this paper, we consider an inverse boundary-value problem for a fourth-order pseudohyperbolic equation with nonclassical boundary conditions. The primary purpose of the work is to study the existence and uniqueness of the classical solution of the considered inverse boundary-value problem. To investigate the solvability of the considered problem, we carried out a transformation from the original problem to some auxiliary equivalent problem with trivial boundary conditions. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness of the classical solution of the original problem are shown.
Anahtar Kelimeler
inverse problem pseudohyperbolic equation integral overdetermination condition classical solution existence uniqueness
Destekleyen Kurum
Academy of Public Administration under the President of the Republic of Azerbaijan
Kaynakça
- [1] A.N. Tikhonov, On stability of inverse problems, Doklady Akademii Nauk SSSR. 39(5), 195-198, 1943. (in Russian)
- [2] M.M. Lavrentiev, Inverse problems of mathematical physics, Utrecht: VSP, The Netherland, 2003.
- [3] V.K. Ivanov, V.V. Vasin and V.P. Tanana, Theory of Linear Ill-posed Problems and Its Applications, Moscow, 1978. (in Russian)
- [4] A.G. Ramm, Inverse Problems, Springer, 2005.
- [5] M.I. Ivanchov, Inverse Problem for Equations of Parabolic Type, Monograph Series, Lviv:VNTL Publishers, 2003.
- [6] V.G. Romanov, Investigation methods for Inverse Problems, Inverse and Ill-Posed Problems Series, De Gruyter, 2002.
- [7] A.I. Prilepko, D.G. Orlovsky and I.A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, 2000.
- [8] G.K. Namazov, Inverse Problems of the Theory of Equations of Mathematical Physics, Baku, 1984 (in Russian).
- [9] A.I. Kozhanov, Composite Type Equations and Inverse Problems, Inverse and ill-posed problems series, VSP, Utrecht, 1999.
- [10] D. Lesnic, Inverse Problems with Applications in Science and Engineering, Chapman and Hall/CRC, 2021.
- [11] S.S. Voyt, A distribution of the initial consolidations in the viscous gas, Scientists Notes of MSU. Ser.: Mechanics, 4 (2), 125142, 1954. (in Russian)
- [12] K. Longren, Experimental Study of Solitons in Nonlinear Transfer Problems with Dispersion, Solitons When in Use, Mir, Moscow, 1981. (in Russian)
- [13] H. Ikezi, Experimental Study of Solitons in Plasma, Solitons When in Use, Mir, Moscow, 1981. (in Russian)
- [14] J.S. Rao, Advanced Theory of Vibration: Nonlinear Vibration and One Dimensional Structures, John Wiley & Sons, Inc., 1992.
- [15] V.V. Bolotin, Vibrations in Technique: Handbook in 6 volumes, The vibrations of linear systems, Engineering Industry, Moscow, 1978. (in Russian)
- [16] A.B. Beylin and L.S. Pulkina, Task on longitudinal vibrations of a rod with dynamic boundary conditions, J. Samara State Univ., Natural Science Series, (3), 9-19, 2014. (in Russian)
- [17] A.B. Beylin, The problem of oscillations of an elastically fixed loaded rod, J. Samara State Tech. Univ., Ser. Phys. Math. Sci., 20 (2), 249-258, 2016. (in Russian)
- [18] S.V. Kirichenko, Nonlocal problems with integral conditions for the hyperbolic, pseudohyperbolic and mixed type equations, PhD thesis, Samara, 2014. (in Russian)
- [19] G.B. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New Jersey, 1974.
- [20] A.I. Kozhanov and R.R. Safiullova, On some class of the pseudohyperbolic equations with an unknown coefficient, Chelyab. Fiz.-Mat. Zh. 7 (2), 164180, 2022. (in Russian)
- [21] Ya.T. Megraliev, On solvability of the inverse problem for the fourth order pseudohyperbolic equation with additional integral condition, University proceedings. Volga region, Ser. Physical and mathematical sciences (1), 19-33, 2013. (in Russian)
- [22] G.V. Namsaraeva, Inverse problems of recovering external sources in the equation of longitudinal wave propagation, J. Appl. Ind. Math. 10(3), 386-396, 2016.
- [23] A.K. Kurmanbaeva, Inverse problems for pseudohyperbolic equations, PhD thesis, Bishkek, 2002. (in Russian)
- [24] E.I. Azizbayov, A time non-local inverse coefficient problem for the longitudinal wave propagation equation, News of Baku University, Series of Physico-Mathematical Sciences, (4), 39-51, 2019.
- [25] E.I. Azizbayov and Y.T. Mehraliyev, Inverse boundary-value problem for the equation of longitudinal wave propagation with non-self-adjoint boundary conditions, Filomat, 33 (16), 5259-5271, 2019.
- [26] A. Das, M. Rabbani, S.A. Mohiuddine and B.C. Deuri, Iterative algorithm and theoretical treatment of existence of solution for (k, z)-RiemannLiouville fractional integral equations, J. Pseudo-Differ. Oper. Appl., 13, (3), Article no. 39, 2022.
- [27] S.A. Gabov and B.B. Orazov, The equation $\frac{{\partial ^2 }}{{\partial t^2 }}[u_{xx} - u] + u_{xx} = 0$ and several problems associated with it, Comput. Math. Math. Phys. 26(1), 58-64, 1986. (in Russian)
- [28] Yu.D. Pletner, The mathematical structure of the solution of the equation $\frac{{\partial ^2 }}{{\partial t^2 }}[u_{xx} - \beta^2 u] + \omega_0^2u_{xx} = 0$ and problems connected with it, Comput. Math. Math. Phys. 30 (3), 161-171, 1990.
- [29] N.Yu. Kapustin and E.I. Moiseev, Convergence of spectral expansions for functions of the Holder class for two problems with a spectral parameter in the boundary condition, Differ. Equ. 36 (8), 10691074, 2000. (in Russian)
- [30] K.I. Khudaverdiyev and A.A. Veliyev, Investigation of One-Dimensional Mixed Problem for a Class of Pseudohyperbolic Equations of Third Order with Nonlinear Operator Right Side, Chashyoghly, Baku, 2010. (in Russian)
Öz
Anahtar Kelimeler
Inverse problem pseudohyperbolic equation integral overdetermination condition classical solution existence uniqueness
Kaynakça
- [1] A.N. Tikhonov, On stability of inverse problems, Doklady Akademii Nauk SSSR. 39(5), 195-198, 1943. (in Russian)
- [2] M.M. Lavrentiev, Inverse problems of mathematical physics, Utrecht: VSP, The Netherland, 2003.
- [3] V.K. Ivanov, V.V. Vasin and V.P. Tanana, Theory of Linear Ill-posed Problems and Its Applications, Moscow, 1978. (in Russian)
- [4] A.G. Ramm, Inverse Problems, Springer, 2005.
- [5] M.I. Ivanchov, Inverse Problem for Equations of Parabolic Type, Monograph Series, Lviv:VNTL Publishers, 2003.
- [6] V.G. Romanov, Investigation methods for Inverse Problems, Inverse and Ill-Posed Problems Series, De Gruyter, 2002.
- [7] A.I. Prilepko, D.G. Orlovsky and I.A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, 2000.
- [8] G.K. Namazov, Inverse Problems of the Theory of Equations of Mathematical Physics, Baku, 1984 (in Russian).
- [9] A.I. Kozhanov, Composite Type Equations and Inverse Problems, Inverse and ill-posed problems series, VSP, Utrecht, 1999.
- [10] D. Lesnic, Inverse Problems with Applications in Science and Engineering, Chapman and Hall/CRC, 2021.
- [11] S.S. Voyt, A distribution of the initial consolidations in the viscous gas, Scientists Notes of MSU. Ser.: Mechanics, 4 (2), 125142, 1954. (in Russian)
- [12] K. Longren, Experimental Study of Solitons in Nonlinear Transfer Problems with Dispersion, Solitons When in Use, Mir, Moscow, 1981. (in Russian)
- [13] H. Ikezi, Experimental Study of Solitons in Plasma, Solitons When in Use, Mir, Moscow, 1981. (in Russian)
- [14] J.S. Rao, Advanced Theory of Vibration: Nonlinear Vibration and One Dimensional Structures, John Wiley & Sons, Inc., 1992.
- [15] V.V. Bolotin, Vibrations in Technique: Handbook in 6 volumes, The vibrations of linear systems, Engineering Industry, Moscow, 1978. (in Russian)
- [16] A.B. Beylin and L.S. Pulkina, Task on longitudinal vibrations of a rod with dynamic boundary conditions, J. Samara State Univ., Natural Science Series, (3), 9-19, 2014. (in Russian)
- [17] A.B. Beylin, The problem of oscillations of an elastically fixed loaded rod, J. Samara State Tech. Univ., Ser. Phys. Math. Sci., 20 (2), 249-258, 2016. (in Russian)
- [18] S.V. Kirichenko, Nonlocal problems with integral conditions for the hyperbolic, pseudohyperbolic and mixed type equations, PhD thesis, Samara, 2014. (in Russian)
- [19] G.B. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New Jersey, 1974.
- [20] A.I. Kozhanov and R.R. Safiullova, On some class of the pseudohyperbolic equations with an unknown coefficient, Chelyab. Fiz.-Mat. Zh. 7 (2), 164180, 2022. (in Russian)
- [21] Ya.T. Megraliev, On solvability of the inverse problem for the fourth order pseudohyperbolic equation with additional integral condition, University proceedings. Volga region, Ser. Physical and mathematical sciences (1), 19-33, 2013. (in Russian)
- [22] G.V. Namsaraeva, Inverse problems of recovering external sources in the equation of longitudinal wave propagation, J. Appl. Ind. Math. 10(3), 386-396, 2016.
- [23] A.K. Kurmanbaeva, Inverse problems for pseudohyperbolic equations, PhD thesis, Bishkek, 2002. (in Russian)
- [24] E.I. Azizbayov, A time non-local inverse coefficient problem for the longitudinal wave propagation equation, News of Baku University, Series of Physico-Mathematical Sciences, (4), 39-51, 2019.
- [25] E.I. Azizbayov and Y.T. Mehraliyev, Inverse boundary-value problem for the equation of longitudinal wave propagation with non-self-adjoint boundary conditions, Filomat, 33 (16), 5259-5271, 2019.
- [26] A. Das, M. Rabbani, S.A. Mohiuddine and B.C. Deuri, Iterative algorithm and theoretical treatment of existence of solution for (k, z)-RiemannLiouville fractional integral equations, J. Pseudo-Differ. Oper. Appl., 13, (3), Article no. 39, 2022.
- [27] S.A. Gabov and B.B. Orazov, The equation $\frac{{\partial ^2 }}{{\partial t^2 }}[u_{xx} - u] + u_{xx} = 0$ and several problems associated with it, Comput. Math. Math. Phys. 26(1), 58-64, 1986. (in Russian)
- [28] Yu.D. Pletner, The mathematical structure of the solution of the equation $\frac{{\partial ^2 }}{{\partial t^2 }}[u_{xx} - \beta^2 u] + \omega_0^2u_{xx} = 0$ and problems connected with it, Comput. Math. Math. Phys. 30 (3), 161-171, 1990.
- [29] N.Yu. Kapustin and E.I. Moiseev, Convergence of spectral expansions for functions of the Holder class for two problems with a spectral parameter in the boundary condition, Differ. Equ. 36 (8), 10691074, 2000. (in Russian)
- [30] K.I. Khudaverdiyev and A.A. Veliyev, Investigation of One-Dimensional Mixed Problem for a Class of Pseudohyperbolic Equations of Third Order with Nonlinear Operator Right Side, Chashyoghly, Baku, 2010. (in Russian)
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Matematik |
| Yazarlar | |
| Erken Görünüm Tarihi | 14 Nisan 2024 |
| Yayımlanma Tarihi | 28 Şubat 2025 |
| Yayımlandığı Sayı | Yıl 2025 Cilt: 54 Sayı: 1 |