We discuss an identification problem of a single point source for a threedimensional scalar wave equation. In this problem, the location and magnitude of the point source are assumed to be unknown. The location moves in a compact region, and the magnitude changes with time. As given data, we use the observed values of the retarded potential and its derivatives obtained at a single observation point and time. For this problem, we propose a direct identification method for the location and magnitude of the source. Our method consists only of the methods of linear algebra, though the problem is formulated by a partial differential equation. Numerical examples are also given to illustrate the effectiveness of our method.

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This paper discusses a reconstruction of point sources in a three-dimensional scalar wave equation from boundary measurements. We assume that the number, locations and magnitudes of point sources are unknown. Under these assumptions, we propose a real-time reconstruction method of these point sources based on the concept of the reciprocity gap functional. In our method, the number, locations and magnitudes of point sources can be identified within small delay. The effectiveness of the proposed method is shown by numerical examples.

We consider the Cauchy problem for the Laplace equation in the neighborhood of the circle. The charge simulation method is applied to the problem, and a theoretical analysis for the numerical solution is given. The analysis for the numerical solution of the charge simulation method can be found in some papers, but the approach in these papers is only for well-posed problems such as the Dirichlet problem. Since our problem is ill-posed, a different approach is required to analyze the convergence of the numerical solution. In this paper, we prove the unique existence of the numerical solution and its exponential convergence to the exact solution. Our result agrees well with numerical experiments.

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