The Problem of Finding the One-Dimensional Kernel of the Thermoviscoelasticity Equation

The problem of determining the kernel h(t), t ∈ [0, T], appearing in the system of integro-differential thermoviscoelasticity equations is considered. It is assumed that the coefficients of the equations depend only on one space variable. The inverse problem is replaced by the equivalent system of integral equations for unknown functions. The contraction mapping principle with weighted norms is applied to this system in the space of continuous functions. A global unique solvability theorem is proved and an estimate of the stability of the solution of the inverse problem is obtained.     

Question of well-posedness of a certain inverse problem for a hyperbolic integrodifferential equation

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 3, pp. 69–77, May–June, 1992.     

Problem of determining the speed of sound and the memory of an anisotropic medium

Theoretical and Mathematical Physics - We consider the inverse problems of simultaneously determining two unknowns: the wave propagation velocity and the memory of a layered medium. To find them,...     

Inverse Problem for A System of Integro-Differential Equations for SH Waves in A Visco-Elastic Porous Medium: Global Solvability

We consider a system of hyperbolic integro-differential equations for SH waves in a visco-elastic porous medium. The inverse problem is to recover a kernel (memory) in the integral term of this system. We reduce this problem to solving a system of integral equations for the unknown functions. We apply the principle of contraction mappings to this system in the space of continuous functions with a weight norm. We prove the global unique solvability of the inverse problem and obtain a stability estimate of a solution of the inverse problem.     

Inverse problem for the system of thermoelasticity equations for a vertically inhomogeneous cohesionless medium with memory

We consider the problem of finding the integrand h(t), t ∈ [0, T], occurring in the system of thermoelasticity equations for a cohesionless medium with memory. The medium density and the Lamé parameters are assumed to be functions of one variable. We prove the theorem on the local unique solvability of the problem and obtain an estimate for the conditional stability of the solution.     

The problem of determining a function of the memory of a medium and of the regular part of a pulsed source

Mathematical Notes - Consider the problem of finding two coefficients one of which is under the sign of the integral in the hyperbolic equation and represents the memory of a medium and the other...