Transactions of the Japan Society for Industrial and Applied Mathematics

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This paper presents an interpolation method and a numerical multiple integration using the boundary integral equations and a polyharmonic function of volume distribution. In the method using B-spline, points must be assigned in a gridiron layout. In the presented method using the polyharmonic function of volume distribution, arbitrary points can be assigned instead of using a gridiron layout, therefore it becomes easy to interpolate. This method requires a boundary geometry of the region and arbitrary internal points. The values at arbitrary point and the integral value are calculated after solving the discretized boundary integral equations. In order to investigate the efficiency of this method, several examples are given. Especially, a use of the polyharmonic function of volume distribution is effective for the three-dimensional case. The advantages of using the polyharmonic function of volume distribution are the decrease of CPU time and the stability of the interpolation.

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In this paper, we discuss an inverse source problem for three-dimensional Helmholtz equation. The source term is assumed as several point sources. We propose an identification method of point sources for this inverse source problem based on weighted integral formulation. Our method does not need a priori information of the locations, magnitudes, and number of point sources. We also give practical estimates for errors of identified locations and magnitudes. The effectiveness of the method is shown by numerical examples.

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We consider a method to solve several shifted linear systems (A+σl)x = b with shift parameter σ. Krylov subspace for shifted linear systems is not depend on the parameter σ, therefore we can solve several shifted linear systems simultaneously without generating Krylov subspace for each parameter cr. In this paper, we show that shifted linear systems appear in an eigensolver using numerical integration. We applied Krylov subspace methods for shifted linear systems in this eigensolver. We have also presented some numerical examples illustrate the efficiency of the method.

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A transient intermittency which lasts a very long time interval for a wide parameter range is shown. The behavior occurs for a simple two-dimensional map system which has a chaotic set (A) in a one-dimensional subuianifold and has a periodic attractor (B) apart from the submanifold for a positive region of normal Lyapunov exponent for a typical orbit of the chaotic set A. In this case, the basin structure of the system changes quite gradually as the parameter varies. This property of the basin structure is the principal cause of occurrence of such long life transient intermittency.

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