scholarly journals The enclosure method for inverse obstacle scattering over a finite time interval: V. Using time-reversal invariance

2019 ◽  
Vol 27 (1) ◽  
pp. 133-149 ◽  
Author(s):  
Masaru Ikehata
Keyword(s):  
Wave Equation ◽  
Time Reversal ◽  
Finite Time ◽  
Single Point ◽  
Arbitrary Point ◽  
Time Interval ◽  
Finite Time Interval ◽  
Reversal Invariance ◽  
Enclosure Method ◽  
Space Wave

Abstract The wave equation is time-reversal invariant. The enclosure method, using a Neumann data generated by this invariance, is introduced. The method yields the minimum ball that is centered at a given arbitrary point and encloses an unknown obstacle embedded in a known bounded domain from a single point on the graph of the so-called response operator on the boundary of the domain over a finite time interval. The occurrence of the lacuna in the solution of the free space wave equation is positively used.

scholarly journals The enclosure method for the heat equation using time-reversal invariance for a wave equation

2020 ◽  
Vol 28 (1) ◽  
pp. 93-104
Author(s):  
Masaru Ikehata
Keyword(s):  
Heat Flux ◽  
Wave Equation ◽  
Heat Equation ◽  
Time Reversal ◽  
The Body ◽  
Time Interval ◽  
Time Reversal Invariance ◽  
Large Parameter ◽  
Reversal Invariance ◽  
Enclosure Method

AbstractThe heat equation does not have time-reversal invariance. However, using a solution of an associated wave equation which has time-reversal invariance, one can establish an explicit extraction formula of the minimum sphere that is centered at an arbitrary given point and encloses an unknown cavity inside a heat conductive body. The data employed in the formula consist of a special heat flux depending on a large parameter prescribed on the surface of the body over an arbitrary fixed finite time interval and the corresponding temperature field. The heat flux never blows up as the parameter tends to infinity. This is different from a previous formula for the heat equation which also yields the minimum sphere. In this sense, the prescribed heat flux is moderate.

scholarly journals The enclosure method for inverse obstacle scattering over a finite time interval: IV. Extraction from a single point on the graph of the response operator

2017 ◽  
Vol 25 (6) ◽  
Author(s):  
Masaru Ikehata
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Finite Time ◽  
Single Point ◽  
Obstacle Problems ◽  
Time Interval ◽  
Finite Time Interval ◽  
Partial Differential ◽  
Response Operator ◽  
Enclosure Method

AbstractA final and maybe the simplest formulation of the enclosure method applied to inverse obstacle problems governed by partial differential equations in a

Identification of Forces in Vibrating Plates by Pointwise and Line Observations: Uniqueness and Stability

1995 ◽  
Author(s):  
Masahiro Yamamoto
Keyword(s):  
Finite Time ◽  
Large Time ◽  
Single Point ◽  
Time Interval ◽  
Finite Time Interval ◽  
Line Parallel ◽  
Vibrating Plates ◽  
External Forces ◽  
Spatially Varying

Abstract We consider determination of spatially varying external forces in a rectangle vibrating plate from displacement observed along a line parallel to a side of the plate over a finite time interval. For a suitable choice of the line and a sufficient large time interval, we prove the uniqueness of external forces and estimate them by appropriate norm of displacement. Moreover we discuss determination of external forces from displacement observed at a single point over a time interval.

A finite-time ruin probability formula for continuous claim severities

2004 ◽  
Vol 41 (2) ◽  
pp. 570-578 ◽  
Author(s):  
Zvetan G. Ignatov ◽  
Vladimir K. Kaishev
Keyword(s):  
Real Function ◽  
Finite Time ◽  
Ruin Probability ◽  
Joint Distribution ◽  
Time Interval ◽  
Insurance Company ◽  
Finite Time Interval ◽  
Appell Polynomials ◽  
Premium Income ◽  
Inverted Dirichlet

An explicit formula for the probability of nonruin of an insurance company in a finite time interval is derived, assuming Poisson claim arrivals, any continuous joint distribution of the claim amounts and any nonnegative, increasing real function representing its premium income. The formula is compact and expresses the nonruin probability in terms of Appell polynomials. An example, illustrating its numerical convenience, is also given in the case of inverted Dirichlet-distributed claims and a linearly increasing premium-income function.

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