Inverse Problems Of Mathematical Physics Research Articles

Infrared scanning thermography for a quantitative detection of cavities in a plane slab and a rectangular prism

An approach for treating nondestructive testing as the solution of inverse problems in mathematical physics has been used for the detection of cavities. The approach is developed based on the use of an additional boundary condition of scanned temperature on the surface to solve for the cavity geometry. For the present study, the condition at the cavity side is taken to be that of a specified temperature, and the experiment is carried out to meet this condition. Two specimens are tested in this paper, a plane slab and a rectangular prism. In both bodies the cavity is rectangular in shape. For the testing of the plane slab, the method is able to detect the cavity wall with high accuracy, whereas the cavity depth error is larger (6%). The detection of the cavity position in the rectangular prism has an error ranging from −9.7 to 7.7%. Errors in the experiment are attributed to the uncertainties in the measurements of temperature and the Biot number. The former is read off from the analog data output of the infrared scanner. The latter is not measured separately, but is computed from the scanned data and thus becomes a portion of the total nondestructive testing output. A final note is also made in this paper to relate how the presented method can be used in actual practice.

Inverse boundary value problems and a theorem of Gel'fand and Levitan

This report concerns two so-called inverse problems of mathematical physics. These are: (i) the problem of determining a second-order differential operator (in a normal form) on the half-axis from its spectral function; and, (ii) the problem of determining a hyperbolic boundary value problem of a special form in a (non-characteristic) half-plane from its response on the boundary to a unit impulse at some reference time t = 0 (boundary value of the Riemann function). We solve problem (ii) by a natural approach, and then indicate how the solution of problem (i) follows from the solution of problem (ii). Our solution of problem (ii) is constructive, and we obtain stability of the solution under perturbation of the data, in a well-defined sense. For problem (i), we obtain the well-known result of Gel'fand and Levitan, in the sharp formulation given by Levitan and Gasymov ([6]).

Experiment planning in inverse problems of mathematical physics

Experiment planning in inverse problems of mathematical physics