Volume 27, Issue 5

In this paper, an inverse problem to identify the initial value for high dimension time fractional diffusion equation on spherically symmetric domain is considered. This problem is ill-posed in the sense of Hadamard, so the quasi-boundary regularization method is proposed to solve the problem. The convergence estimates between the regularization solution and the exact solution are presented under the a priori and a posteriori regularization parameter choice rules. Numerical examples are provided to show the effectiveness and stability of the proposed method.

The theoretical and numerical determination of a space-dependent load distribution in a simply supported non-homogeneous Euler–Bernoulli beam and Kirchhoff–Love plate is investigated. The uniqueness of a solution to this inverse source problem is proved, whilst counterexamples are constructed to discuss the conditions under which uniqueness holds. A convergent and stable iterative algorithm is proposed for the recovery of the unknown load source and a stopping criterion is also given. Several one-dimensional numerical experiments are considered to investigate the properties of the proposed iterative procedure.

The inverse scattering problem of determining a crack outside a known penetrable cavity from the measurements and point sources placed in a closed curve Λ inside the cavity is considered. The factorization method is used to reconstruct the crack on which the mixed-type boundary conditions are posed. To this end, we need to overcome two key difficulties. One of them is to find the relationship between the two data-to-data operators, which map the boundary data on the crack to the values of scattered fields on the closed curve Λ with outgoing radiation condition and incoming radiation condition, respectively. In addition, it is hard to decompose the near field operator in the usual way owing to the mixed-type boundary conditions. By the aid of an auxiliary operator, we can obtain the desired decomposition form in accordance with the theoretical framework of the factorization method.

A method based on least squares support vector machines (LS-SVM) is proposed to solve the source inverse problem of wave equations. Contrary to the most existing methods, the proposed method provides a closed form approximate solution which satisfies the boundary conditions and the initial conditions. The proposed method can recover the unknown source term with the given additional conditions. Furthermore, it has reasonable robustness to noise. Numerical results show the proposed method can be used to solve the source inverse problem of wave equations.

Consider the interior transmission problem arising in inverse boundary value problems for the diffusion equation with discontinuous diffusion coefficients. We prove the unique solvability of the interior transmission problem by constructing its Green function. First, we construct a local parametrix for the interior transmission problem near the boundary in the Laplace domain, by using the theory of pseudo-differential operators with a large parameter. Second, by carefully analyzing the analyticity of the local parametrix in the Laplace domain and estimating it there, a local parametrix for the original parabolic interior transmission problem is obtained via the inverse Laplace transform. Finally, using a partition of unity, we patch all the local parametrices and the fundamental solution of the diffusion equation to generate a global parametrix for the parabolic interior transmission problem and then compensate it to get the Green function by the Levi method. The uniqueness of the Green function is justified by using the duality argument, and then the unique solvability of the interior transmission problem is concluded. We would like to emphasize that the Green function for the parabolic interior transmission problem is constructed for the first time in this paper. It can be applied for active thermography and diffuse optical tomography modeled by diffusion equations to identify an unknown inclusion and its physical property.

Three methods for the estimation of blood or breath alcohol concentration (BAC/BrAC) from biosensor measured transdermal alcohol concentration (TAC) are evaluated and compared. Specifically, we consider a system identification/quasi-blind deconvolution scheme based on a distributed parameter model with unbounded input and output for ethanol transport in the skin and compare it to two more conventional system identification and filtering/deconvolution techniques for ill-posed inverse problems, one based on frequency domain methods and the other on a time series approach using an ARMA input/output model. Our basis for comparison are five statistical measures of interest to alcohol researchers and clinicians: peak BAC/BrAC, time of peak BAC/BrAC, the ascending and descending slopes of the BAC/BrAC curve, and the area underneath the BAC/BrAC curve.

The paper deals with the theoretical analysis of a non-Newtonian lubrication problem in a dynamic regime in a three-dimensional thin domain Ωε{\Omega^{\varepsilon}} with Tresca friction law. The problem statement and variational formulation of the problem are formulated. Then the problem is reformulated in a fixed domain, in which case the estimates on velocity and pressure are proved. These estimates are useful in order to give a specific Reynolds equation associated with variational inequalities and prove the uniqueness.

What happens when one prescribes a heat flux which is proportional to the Neumann data of a solution of the wave equation in the whole space on the surface of a heat conductive body? It is shown that there is a difference in the asymptotic behavior of the indicator function in the most recent version of the time domain enclosure method , which aims at extracting information about an unknown cavity embedded in the body.

In this paper, a new asymptotic-numerical approach to solving an inverse boundary value problem for a nonlinear singularly perturbed parabolic equation with time-periodic coefficients is proposed. An unknown boundary condition is reconstructed by using known additional information about the location of a moving front. An asymptotic analysis of the direct problem allows us to reduce the original inverse problem to that with a simpler numerical solution. Numerical examples demonstrate the efficiency of the method.