Solution of the inverse problem for an evolution equation in a Banach space

For the equationdy (t)/dt=Ay(t) + p, t [0, ), where A is a closed linear operator on a Banach space , p is an unknown parameter from , we consider the problem

On an inverse problem for the transport equation

Siberian Mathematical Journal -     

An inverse problem for an ordinary second-order differential equation

An inverse problem is solved for an ordinary second-order differential equation.Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 163–166, August, 1972.     

A multidimensional inverse problem for an equation with memory

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

An inverse problem for an equation of parabolic type

In this paper we consider an inverse problem for the differential equationu _t =u _xx +q(x, t) u; the problem amounts to finding the coefficient q(x, t) from the solution of a series of Cauchy problems for this equation, the solution being specified on some manifold. Our main result is a proof of a uniqueness theorem.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 595–600, April, 1976.     

Nonlocal inverse problem for an equation of elliptic-hyperbolic type

This paper proves the existence and uniqueness of solutions of the nonlocal inverse problem for a mixed-type equation.     

Uniqueness in an integral geometry problem and an inverse problem for the kinetic equation

In this paper, we discuss the uniqueness in an integral geometry problem along the straight lines in a strongly convex domain. Our problem is related with the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation and then the uniqueness theorem is proved using the tools of Fourier analysis.     

An inverse problem with final time additional condition for an abstract evolution equation in an ordered Banach space

Moscow Institute for Physical Engineering. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 1, pp. 81–83, January–March, 1993.     

An inverse problem for an elliptic equation with an affine term

We consider the following elliptic boundary value problem: on , u = 0 on where is a smooth bounded planar domain. We show that for a large class of domains and for any such that is not identically constant there exist at most finitely many different pairs of coefficients such that the problem has a solution with the normal flux on . Received: 4 February 1999     

Uniqueness and stability in an inverse problem for a Poisson's equation

Consider the Poisson's equation(?)″(x)=-e~(v-(?)) e~((?)-v)-N(x)with the Diriehlet boundary data,and we mainly investigate the inverse problem of determining the unknown function N(x)from a parameter function family.Some uniqueness and stability results in the inverse problem are obtained.     

A two-dimensional inverse problem for an integro-differential equation of electrodynamics

An integro-differential equation corresponding to a two-dimensional problem of electrodynamics with dispersion is considered. It is assumed that the electrodynamic properties of a nonconducting medium with a constant magnetic permeability and the external current are independent of the x ₃ coordinate. In this case, the third component of the electric field vector satisfies a second-order scalar integro-differential equation with a variable permittivity of the medium. For this equation, we study the problem of finding the spatial part of the kernel entering the integral term. This corresponds to finding the part of the permittivity that depends on the electromagnetic frequency. It is assumed that the permittivity support is contained in some compact domain Ω ? ?². To find this coefficient inside Ω, we use information on the solution of the corresponding direct problem on the boundary of Ω on a finite time interval. An estimate for the conditional stability of the solution of the inverse problem is established under the assumption that the time interval is sufficiently large.     

Numerical analysis of an inverse problem for the eikonal equation

We are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface. This is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so-called Soner boundary condition. The state equation is discretised by a suitable finite difference scheme for which we obtain existence, uniqueness and an error bound. We set up an approximate optimisation problem and show that a subsequence of the discrete mimina converges to a solution of the continuous optimisation problem as the mesh size goes to zero. The derivative of the discrete functional is calculated with the help of an adjoint equation which can be solved efficiently by using fast marching techniques. Finally we describe some numerical results.     

An inverse problem for an inhomogeneous conformal Killing field equation

Let be a Riemannian metric defined on a bounded domain with boundary and let be a vector field on satisfying . We show that if is a gradient field of a solution to the equation on , then both inner products and are uniquely determined by the restriction of the tensor to the gradient field , where is the Lie derivative of the metric tensor under the vector field and . This work solves a problem related to an inverse boundary value problem for nonlinear elliptic equations.