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A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, using solution point constraints. Applying a transform method, we reduce the inverse problems to direct ones, which are studied for well-posedness in special weighted Sobolev spaces. This means that the inverse problem is said to be well posed in the sense of Tikhonov (or conditionally well posed). The main aim of this study is to develop a finite difference method for solution of the transformed hyperbolic problems with a non-local differential operator and initial conditions. Numerical test examples are also analyzed. [ABSTRACT FROM AUTHOR]

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin's-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul'yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach. [ABSTRACT FROM AUTHOR]

In this work, we consider Cauchy-type problems for Laplace's equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green's second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations. [ABSTRACT FROM AUTHOR]