Dirichlet problem for a third-order equation of mixed type in a rectangular domain

For a third-order differential equation of parabolic-hyperbolic type, we suggest a method for studying the first boundary value problem by solving an inverse problem for a second-order equation of mixed type with unknown right-hand side. We obtain a uniqueness criterion for the solution of the inverse problem. The solution of the inverse problem and the Dirichlet problem for the original equation is constructed in the form of the sum of a Fourier series.     

Iterative method for solving an inverse coefficient problem for a hyperbolic equation

For a hyperbolic equation, we consider an inverse coefficient problem in which the unknown coefficient occurs in both the equation and the initial condition. The solution values on a given curve serve as additional information for determining the unknown coefficient. We suggest an iterative method for solving the inverse problem based on reduction to a nonlinear operator equation for the unknown coefficient and prove the uniform convergence of the iterations to a function that is a solution of the inverse problem.     

An inverse coefficient problem for a nonlinear parabolic variational inequality

This study is related to inverse coefficient problems for a nonlinear parabolic variational inequality with an unknown leading coefficient in the equation for the gradient of the solution. An inverse method, involving minimization of a least-squares cost functional, is developed to identify the unknown coefficient. It is proved that the solution of the corresponding direct problem depends continuously on the coefficient. On the basis of this, the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.     

An inverse problem for a parabolic equation in a free-boundary domain degenerating at the initial time moment

We consider an inverse problem for a one-dimensional parabolic equation with unknown time-dependent major coefficient in a domain whose unknown boundary weakly degenerates at the initial time moment. The conditions for existence and uniqueness of the classical solution of the problem are established.     

Numerical Solution of Inverse Problems for a Hyperbolic Equation with a Small Parameter Multiplying the Highest Derivative

We consider two inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative. The inverse problems are reduced to systems of linear Volterra integral equations of the second kind for the unknown functions. These systems are used to prove the existence and uniqueness of the solution of the inverse problems and numerically solve them. The applicability of the methods developed here to the approximate solution of the problem on an unknown source in the heat equation is studied numerically.     

Inverse problem of simultaneously determining the right-hand side and the coefficient of a lower order derivative for a parabolic equation on the plane

We obtain existence and uniqueness theorems for the solution of the inverse problem of simultaneously determining the right-hand side and the coefficient of a lower-order derivative in a parabolic equation under an integral observation condition. We give explicit estimates for the maximum absolute value of the unknown right-hand side and the unknown coefficient of the equation with constants expressed via the input data of the problem. We present a nontrivial example of an inverse problem to which our theorems apply.     

The inverse eigenvalue problem for a weighted helmholtz equation

Given the m lowest eigenvalues, we seek to recover an approximation to the density function ρ in the weighted Helmholtz equation -Δ=λρu on a rectangle with Dirchlet boundary conditions. The density ρ is assumed to be symmetric with respect to the midlines of the rectangle. Projection of the boundary value problem and the unknown density function onto appropriate vector spaces leads to a matrix inverse problem. Solutions of the matrix inverse problem exist provided that the reciprocals of the prescribed eigenvalues are close to the reciprocals of the simple eigenvalues of the base problem with ρ = 1. The matrix inverse problem is solved by a fixed—point iterative method and a density function ρ^* is constructed which has the same m lowest eigenvalues as the unknown ρ. The algorithm can be modified when multiple base eigenvalues arise, although the success of the modification depends on the symmetry properties of the base eigenfunctions.     

Analytical solutions of a class of inverse coefficient problems

This paper is devoted to some class of inverse coefficient problems. By using a well-known transformation, the inverse problem is transformed to a new problem without the unknown time dependent coefficient. Therefore, the new inverse problem can be solved easily. To show the efficiency of the present method, some examples are presented.     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

The Inverse Problem for Numerical Determination of the Nonlinear Right-Hand Side in a Parabolic Equation

The article considers the inverse problem of determining the nonlinear right-hand side of a quasi-linear parabolic equation and proves a uniqueness theorem. A method is proposed for numerical solution of the inverse problem based on parametric representation of the sought coefficient. The inverse problem thus reduces to finding the error-minimizing vector of unknown coefficients of the parametric representation of the sought function.     

Integral-Functional Equation Arising in the Study of an Inverse Problem for a Quasilinear Hyperbolic Equation

A problem with data on the characteristics is considered for a quasilinear hyperbolic equation. The inverse problem of determining two unknown coefficients of the equation from some additional information about the solution is posed. One of the unknown coefficients depends on the independent variable, and the other, on the solution of the equation. Uniqueness theorems are proved for the solution of the inverse problem. The proof is based on the derivation of the integro-functional equation and the analysis of the uniqueness of its solution.     

一非线性发展方程的反问题

研究一非线性发展方程的未知源项的反演问题.首先,把所考虑的初边值问题化成一等价非线性发展方程的Cauchy问题;然后,利用半群理论,论证反问题解的存在性和唯一性;最后,利用压缩映射不动点方法,得到反问题的可解性.     

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 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.     

The inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain

In this paper with the help of the spectral method we obtain a criterion for the unique solvability of the inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain. This problem implies the search of the unknown right-hand side.     

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.     

Integral equations related to the study of an inverse coefficient problem for a system of partial differential equations

We consider an inverse coefficient problem for a linear system of partial differential equations. The values of one solution component on a given curve are used as additional information for determining the unknown coefficient. The proof of the uniqueness of the solution of the inverse problem is based on the analysis of the unique solvability of a homogeneous integral equation of the first kind. The existence of a solution of the inverse problem is proved by reduction to a system of nonlinear integral equations.     

Coefficient inverse problem for a parabolic equation in a domain with free boundary

We establish conditions for the local existence and uniqueness of a solution of an inverse problem for a parabolic equation with unknown minor coefficients in a domain with free boundary.     

Iterative method for solving a three-dimensional electrical impedance tomography problem in the case of piecewise constant conductivity and one measurement on the boundary

The problem of electrical impedance tomography in a bounded three-dimensional domain with a piecewise constant electrical conductivity is considered. The boundary of the inhomogeneity is assumed to be unknown. The inverse problem is to determine the surface that is the boundary of the inhomogeneity from given measurements of the potential and its normal derivative on the outer boundary of the domain. An iterative method for solving the inverse problem is proposed, and numerical results are presented.     

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.