Regularization of initial inverse problem for strongly damped wave equation

Abstract

In this paper, we consider the problem of finding the function , from the final data and

where is a linear, unbounded, self-adjoint and positive definite operator. This problem is known as the inverse initial problem for non-linear strongly damped wave and is ill-posed in the sense of Hadamard. In order to obtain a stable numerical solution, we propose new quasi-boundary value method to solve the non-linear problem, i.e. for replacing by

with the operator will be defined later and satisfies (1.8). Moreover, we show that the regularized solutions converge to the exact solution strongly with respect to under a priori assumption on the exact solution in Gevrey space.

Keywords:

• Regularization method
• final value problem
• strongly damped wave
• error estimate

AMS Subject Classifications:

• 35K05
• 35K99
• 47J06
• 47H10

Acknowledgements

The authors also desire to thank the handling editor and anonymous referees for their helpful comments on this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper was supported by Institute for Computational Science and Technology Ho Chi Minh City under project named ‘The regularization of inverse problem for hyperbolic equation and pseudohyperbolic equation’..