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In this article the unique solution of the Cauchy problem is founded by the Riemann method. Some relations for given here confluent hypergeometric functions of two and three variables are used.
This paper studies an inverse hyperbolic problem for the wave equation with dynamic boundary conditions. It consists of determining some forcing terms from the final overdetermination of the displacement. First, the Fr\'echet…
A nonparametric Bayesian approach is developed to determine quantum potentials from empirical data for quantum systems at finite temperature. The approach combines the likelihood model of quantum mechanics with a priori information over…
This article addresses the inverse source problem for a nonlocal heat equation involving the fractional Laplacian. The primary goal is to reconstruct the spatial component of the source term from partial observations of the system's state…
The paper considers an inverse source problem for a one-dimensional time-fractional heat equation with the generalized impedance boundary condition. The inverse problem is the time dependent source parameter identification together with the…
This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two…
We consider an inverse problem of reconstructing two spatially varying coefficients in an acoustic equation of hyperbolic type using interior data of solutions with suitable choices of initial condition. Using a Carleman estimate, we prove…
We consider an inverse initial-data problem for the compressible anisotropic Navier--Stokes equations, in which the goal is to reconstruct the initial velocity field from noisy lateral boundary observations. In the formulation studied here,…
This paper is concerned with an inverse wavenumber/frequency-dependent source problem for the Helmholtz equation. In two and three dimensions, the unknown source term is supposed to be compactly supported in spatial variables but…
This paper is concerned with the resolution of an inverse problem related to the recovery of a scalar (potential) function $V$ from the source to solution map, of the semi-linear equation $(\Box_{g}+V)u+u^3=0$ on a globally hyperbolic…
Regularization methods have been recently developed to construct stable approximate solutions to classical partial differential equations considered as final value problems. In this paper, we investigate the backward parabolic problem with…
The paper studies some inverse boundary value problem for simplest parabolic equations such that the homogenuous Cauchy condition is ill posed at initial time. Some regularity of the solution is established for a wide class of boundary…
In this paper we investigate the problem of identifying the source term in an elliptic system from a single noisy measurement couple of the Neumann and Dirichlet data. A variational method of Tikhonov-type regularization with specific…
Backward parabolic equations, such as the backward heat equation, are classical examples of ill-posed problems where solutions may not exist or depend continuously on the data. In this work, we study a least squares finite element method to…
In this paper, we study both the direct and inverse random source problems associated with the multi-term time-fractional diffusion-wave equation driven by a fractional Brownian motion. Regarding the direct problem, the well-posedness is…
In this paper, a nonlinear inverse boundary value problem for the second-order hyperbolic equation with nonlocal conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary…
We initiate the study of inverse source problems for quasilinear elliptic equations of the form \[ \left\{ \begin{array}{ll} \nabla \cdot (\gamma(x,u,\nabla u) \nabla u) = F & \text{in } \Omega, \\ u = f & \text{on } \partial\Omega,…
This paper introduces a novel numerical method for the inverse problem of electroencephalography(EEG). We pose the inverse EEG problem as an optimal control (OC) problem for Poisson's equation. The optimality conditions lead to a…
This paper is concerned with reconstruction issue of some typical inverse problems and consists of three parts. First a framework of the enclosure method for an inverse source problem governed by the Helmholtz equation at a fixed wave…
Through coupled physics, we study an early-warning inverse source problem for the elasto-gravitational equations. It consists of a mixed hyperbolic-elliptic system of partial differential equations describing elastic wave displacement and…