Recovering degenerate kernels in hyperbolic integro-differential equations. (English) Zbl 1009.45014
The authors first consider the problem of recovering degenerate kernels appearing in hyperbolic integrodifferential equations related to viscoelastic multi-dimensional materials under boundary conditions of traction type. Then existence, uniqueness and stability for the solutions of such problems are proved.
Under suitable assumptions, it is established that a sequence of solutions corresponding to degenerate kernels can converge to a solution corresponding to a non-degenerate one.
Section 8 of the paper contains an application of already proved abstract result of sections 3-7 to the problems in viscoelasticity.
Under suitable assumptions, it is established that a sequence of solutions corresponding to degenerate kernels can converge to a solution corresponding to a non-degenerate one.
Section 8 of the paper contains an application of already proved abstract result of sections 3-7 to the problems in viscoelasticity.
Reviewer: K.C.Gupta (Jaipur)
MSC:
| 45Q05 | Inverse problems for integral equations |
| 45K05 | Integro-partial differential equations |
| 45M10 | Stability theory for integral equations |
| 74D05 | Linear constitutive equations for materials with memory |
Keywords:
viscoelasticity; inverse problem; recovering degenerate kernels; hyperbolic integrodifferential equations; stabilityReferences:
| [1] | Bukhgeim, A. L., Dyatlov, G. V. and V. Isakov: Stability of memory reconstruc- tion from the Dirichlet-to-Neumann operator. Siberian Math. J. 38 (1997), 636 - 646. · Zbl 0880.45006 · doi:10.1007/BF02674571 |
| [2] | Bukhgeim, A. L. and G. V. Dyatlov: Inverse problems for equations with mem- ory. Inverse Problems and Related Topics. Chapman & Hall/CRC, Res. Notes 419 (2000), 19 - 35. · Zbl 0976.35095 |
| [3] | Cavaterra, C.: An inverse problem for a viscoelastic Timoshenko beam model. Z. Anal. Anw. 17 (1998), 67 - 87. · Zbl 0898.35112 · doi:10.4171/ZAA/809 |
| [4] | Cavaterra, C. and M. Grasselli: On an inverse problem for a model of linear viscoelastic Kirchhoff plate. J. Int. Equ. Appl. 9 (1997)3, 179 - 218. · Zbl 0901.73037 · doi:10.1216/jiea/1181076012 |
| [5] | Colombo, F. and A. Lorenzi: Identification of time- and space-dependent relax- ation kernels for materials with memory related to cylindrical domains. Part I: J. Math. Anal. Appl. 213 (1997), 32 - 62. · Zbl 0892.45006 · doi:10.1006/jmaa.1997.5364 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
