Recovery of the unknown coefficients in a two-dimensional hyperbolic equation

In this paper, an inverse boundary value problem for a two-dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. We then use the Fourier method to reduce such an equivalent problem to a system of integral equations. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness theorem for the classical solution of the original inverse problem is proved. Some discussions on the numerical solutions for this inverse problem are presented including some numerical examples.     

System of nonlinear integral equations for the unknown functions in a functional-differential hyperbolic equation

We consider the inverse problem for a functional-differential equation in which the delay function and a function occurring in the source are unknown. The values of the solution and its derivative at x = 0 are given as additional information. We derive a system of nonlinear integral equations for the unknown functions. This system is used to prove a uniqueness theorem for the inverse problem.     

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.     

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.     

Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator

We consider a one-dimensional perturbation of the convolution operator. We study the inverse reconstruction problem for the convolution component using the characteristic numbers under the assumption that the perturbation summand is known a priori. The problem is reduced to the solution of the so-called basic nonlinear integral equation with singularity. We prove the global solvability of this nonlinear equation. On the basis of these results, we prove a uniqueness theorem and obtain necessary and sufficient conditions for the solvability of the inverse problem.     

Inverse problems for a quasilinear hyperbolic equation in the case of moving observation point

We consider two inverse coefficient problems for a quasilinear hyperbolic equation, where the additional information used for finding the coefficients is the values of the solution on some curve. (This corresponds to measurements performed at a moving observation point.) The unknown coefficient depends on the space variable in the first inverse problem and on the solution of the equation in the second inverse problem. We prove theorems of uniqueness of solution to the inverse problems.     

Inverse scattering problem for a wave equation with absorption

We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1580–1584, November, 2007.     

Inverse dirichlet problem for an equation of elliptic type

We study the inverse problem of finding the source in an abstract second-order elliptic equation on a finite interval. The additional information given is the value of the solution at an interior point of the interval. We prove existence, uniqueness, and Fredholm property theorems for the inverse problem.     

A Finite Horizon Linear Quadratic Optimal Stochastic Control Problem Driven by Both Brownian Motion and L\'{e}vy Processes

We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

Inverse spectral problem for differential operator pencils on noncompact spatial networks

We study boundary value problems on noncompact cycle-free graphs (i.e., trees) for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of the spectrum and analyze the inverse problem of reconstructing the coefficients of a differential equation on the basis of the so-called Weyl functions. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mapping.     

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??We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

On the uniqueness of solutions of inverse multidimensional spectral problems in dual asymptotic models

The solution of an elliptic equation in an unbounded region is approximated by the solution of an ordinary differential equation. To obtain a new inverse problem we prove a uniqueness theorem and exhibit a solution algorithm. Bibliography: 14 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 132–140.     

On a boundary value problem with shift for an equation of mixed type in an unbounded domain

We study the unique solvability of a problem with shift for an equation of mixed type in an unbounded domain. We prove the uniqueness theorem under inequality-type constraints for known functions for various orders of the fractional differentiation operators in the boundary condition. The existence of a solution is proved by reduction to a Fredholm equation of the second kind, whose unconditional solvability follows from the uniqueness of the solution of the problem.     

Recovery of differential equations with nonlinear dependence on the spectral parameter

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line is studied. We give a formulation of the inverse problem, prove the uniqueness theorem and provided a procedure for constructing the solution of the inverse problem. We also establishe connections with inverse problems for partial differential equations.     

An inverse problem for an integro‐differential operator on a star‐shaped graph

A partial inverse problem for an integro‐differential Sturm‐Liouville operator on a star‐shaped graph is studied. We suppose that the convolution kernels are known on all the edges of the graph except one and recover the kernel on the remaining edge from a part of the spectrum. We prove the uniqueness theorem for this problem and develop a constructive algorithm for its solution, based on the reduction of the inverse problem on the graph to the inverse problem on the interval by using the Riesz basis property of the special system of functions.     

Nonclassical boundary value problem for a pseudodifferential equation of variable order

We prove an existence and uniqueness theorem for the solution of a nonclassical boundary value problem of Egorov-Kondrat’ev type for a pseudodifferential equation of variable order.     

Inverse problem for an equation of parabolic-hyperbolic type with a nonlocal boundary condition

For an equation of the parabolic-hyperbolic type, we consider an inverse problem with a nonlocal condition relating solution derivatives that belong to different types of the equation in question. We justify a uniqueness criterion and prove the existence of a solution of the problem by the spectral analysis method. We prove the stability of the solution with respect to the nonlocal boundary condition.     

A problem with operators of fractional differentiation in boundary condition for mixed-type equation

We consider a question on unique solvability of a boundary-value problem with fractional derivatives for a mixed-type equation of second order. We prove first a uniqueness theorem. The existence theorem is proved by means of reduction to Fredholm equation of the second kind, and its unconditional solvability follows from the uniqueness of solution.     

Inverse problem for differential pencils on a hedgehog graph

We study boundary value problems on a hedgehog graph for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of spectral characteristics and consider the inverse spectral problem of reconstructing the coefficients of a differential pencil on the basis of spectral data. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing its solution.     

Inverse spectral problems for differential pencils on the half-line with turning points

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.