A high accurate method for solving an inverse problem of the Laplace equation in detection of a Robin coefficient

This study This work deals with an inverse problem for the harmonic equation to recover a Robin coefficient on a non-accessible part of a circle from Cauchy data measured on an accessible part of that circle. By assuming that the available data has a Fourier expansion, we adopt the Modified Collocation Trefftz Method (MCTM) to solve this problem. We use the truncation regularization method in combination with the collocation technique to approximate the solution, and the conjugate gradient method to obtain the coefficients, thus completing the missing Cauchy data. We recommend the least squares method to achieve a better stability. Finally, we illustrate the feasibility of this method with numerical examples.

On initial inverse problem for nonlinear couple heat with Kirchhoff type

The main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach’s fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019].

Direct Sampling for Recovering Sound Soft Scatterers from Point Source Measurements

In this paper, we consider the inverse problem of recovering a sound soft scatterer from the measured scattered field. The scattered field is assumed to be induced by a point source on a curve/surface that is known. Here, we propose and analyze new direct sampling methods for this problem. The first method we consider uses a far-field transformation of the near-field data, which allows us to derive explicit bounds in the resolution analysis for the direct sampling method’s imaging functional. Two direct sampling methods are studied, using the far-field transformation. For these imaging functionals, we use the Funk–Hecke identities to study the resolution analysis. We also study a direct sampling method for the case of the given Cauchy data. Numerical examples are given to show the applicability of the new imaging functionals for recovering a sound soft scatterer with full and partial aperture data.

Inverse Boundary Value Problem for the Magnetohydrodynamics Equations

In this paper, we consider the stationary magnetohydrodynamics (MHD) equations in a bounded domain of ℝ d d = 2 , 3 with viscosity and magnetic diffusing. By the linearization technique, we prove that the uniqueness of viscosity function and magnetic diffusing function in the MHD equations is determined from the knowledge of the Cauchy data measured on the boundary.

Solving the Cauchy problem for the Helmholtz equation using cubic smoothing splines

We consider the Cauchy problem for the Helmholtz equation defined in a rectangular domain. The Cauchy data are prescribed on a part of the boundary and the aim is to find the solution in the entire domain. The problem occurs in applications related to acoustics and is illposed in the sense of Hadamard. In our work we consider regularizing the problem by introducing a bounded approximation of the second derivative by using Cubic smoothing splines. We derive a bound for the approximate derivative and show how to obtain stability estimates for the method. Numerical tests show that the method works well and can produce accurate results. We also demonstrate that the method can be extended to more complicated domains.

On Problem of Internal Boundary Control for String Vibration Equation

The article deals with the vibration control problem described by one dimensional wave equation with integral type boundary condition. As usual, the initial and final moments of time for arbitrary displacements and velocities of the wave are specified by points on a string (Cauchy data). It is shown that the minimum time for the realizable control is uniquely determined by the condition of correct solvability to the Cauchy problem involving data lying on disconnected manifold. This suggests that the internal boundary conditions does not affect the minimum time value. Necessary and sufficient conditions for the existence of the desired internal-boundary controls that move the process from the state initially specified to a predetermined final one are obtained and written out. The controls are presented in explicit analytical form. Moreover, it is shown that for the inner-boundary controls expressions, one should use not the representation of the solution to the Cauchy problem in the sought-for domain, but the formula for the general solution of the string oscillation equation (d’Alembert’s formula).