About Global Solvability of a Multidimensional Inverse Problem for an Equation with Memory

Siberian Mathematical Journal - Under study is the multidimensional inverse problem of determining the convolutional kernel of the integral term in an integro-differential wave equation....     

Inverse Problem of Determining an Unknown Coefficient in the Beam Vibration Equation

Differential Equations - We consider the direct initial–boundary value problem for the equation of transverse vibrations of a homogeneous beam freely supported at the ends and study the...     

A problem of determining a special spatial part of 3D memory kernel in an integro‐differential hyperbolic equation

The inverse problem of determining 2D spatial part of integral member kernel in integro‐differential wave equation is considered. It is supposed that the unknown function is a trigonometric polynomial with respect to the spatial variable y with coefficients continuous with respect to the variable x. Herein, the direct problem is represented by the initial‐boundary value problem for the half‐space x>0 with the zero initial Cauchy data and Neumann boundary condition as Dirac delta function concentrated on the boundary of the domain . Local existence and uniqueness theorem for the solution to the inverse problem is obtained.     

Global solvability of an inverse problem for an integro-differential equation of electrodynamics

We consider the problem of reconstructing the time history of the electric field from the electrodynamic equation. As additional information, the magnetic field intensity vector at the point x = 0 is given. For this problem, the existence and uniqueness theorem is proved for an arbitrary given interval.     

Problem of determining the nonstationary potential in a hyperbolic-type equation

We solve the problem of determining the hyperbolic equation coefficient depending on two variables. Some additional information is given by the trace of the direct problem solution on the hyperplane x = 0. We estimate the stability of the solution of the inverse problem under study and prove the uniqueness theorem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 220–225, August, 2008.     

A multidimensional inverse problem for an equation with memory

Translated from Sibirskii Zhurnal, Vol. 35, No. 3, pp. 574–582, May–June, 1994.     

A 2D kernel determination problem in a visco-elastic porous medium with a weakly horizontally inhomogeneity

We consider a system of hyperbolic integro-differential equations of SH waves in a visco-elastic porous medium. In this work, it is assumed that the visco-elastic porous medium has weakly horizontally inhomogeneity. The direct problem is the initial-boundary problem: the initial data is equal to zero, and the Neumann-type boundary condition is specified at the half-plane boundary and is an impulse function. As additional information, the oscillation mode of the half-plane line is given. It is assumed that the unknown kernel has the form K(x,t)=K₀(t)+ϵxK₁(t)+…, where ϵ is a small parameter. In this work, we construct a method for finding K₀,K₁ up to a correction of the order of O(ϵ²).     

Inverse Problem for an Integro-Differential Equation of Acoustics

We consider the hyperbolic integro-differential equation of acoustics. The direct problem is to determine the acoustic pressure created by a concentrated excitation source located at the boundary of a spatial domain from the initial boundary-value problem for this equation. For this direct problem, we study the inverse problem, which consists in determining the onedimensional kernel of the integral term from the known solution of the direct problem at the point x = 0 for t > 0. This problem reduces to solving a system of integral equations in unknown functions. The latter is solved by using the principle of contraction mapping in the space of continuous functions. The local unique solvability of the posed problem is proved.     

The Problem of Finding the Kernels in the System of Integro-Differential Maxwell’s Equations

Journal of Applied and Industrial Mathematics - We pose the direct and inverse problems of finding the electromagnetic field and the diagonal memory matrix for the reduced canonical system of...     

Inverse problem of determining the kernel in an integro-differential equation of parabolic type

We study the inverse problem of determining the multidimensional kernel of the integral term in a parabolic equation of second order. As additional information, the solution of the direct problem is given on the hyperplane x _n = 0. We prove a local existence and uniqueness theorem for the inverse problem.