Abstract
In this paper, we present a survey of the inverse eigenvalue problem for a Laplacian equation based on available Cauchy data on a known part Γ0 and a homogeneous Dirichlet condition on an unknown part Γ0 of the boundary of a bounded domain, Ω⊂ℝN. We consider variations in the eigenvalues and propose a conformal mapping tool to reconstruct a part of the boundary curve of the two-dimensional bounded domain based on the Cauchy data of a holomorphic function that maps the unit disk onto the unknown domain. The boundary values of this holomorphic function are obtained by solving a nonlocal differential Bessel equation. Then, the unknown boundary is obtained as the image of the boundary of the unit disk by solving an ill-posed Cauchy problem for holomorphic functions via a regularized power expansion. The Cauchy data were restricted to a nonvanishing function and to the normal derivatives without zeros. We prove the existence and uniqueness of the holomorphic function being considered and use the fixed-point method to numerically analyze the results of convergence. We’ll calculate the eigenvalues and compare the result with the shape obtained via minimization functional method, as developed in a previous study. Further, we’ll observe via simulations the shape of Γ and if it preserves its properties with varying the eigenvalues.
Highlights
We consider the mathematical modeling employed in electrostatic imaging during nondestructive testing
This analysis engenders an inverse boundary value problem for a Laplace equation. This problem entails the identification of an unknown domain, Γ, within a conducting host medium with constant conductivity based on known Cauchy data on the boundary of the medium, Γ0
This paper is primarily aimed at presenting iterative methods to solve the inverse problem ð1Þ − ð2Þ − ð3Þ by using ideas from conformal mapping theory that have been developed over the last decade
Summary
We consider the mathematical modeling employed in electrostatic imaging during nondestructive testing. This analysis engenders an inverse boundary value problem for a Laplace equation. This problem entails the identification of an unknown domain, Γ, within a conducting host medium with constant conductivity based on known Cauchy data on the boundary of the medium, Γ0. We suppose that Ω is a connected domain in RN with C2 smooth boundaries ∂Ω and ∂Ω = Γ0oΓ: ð1Þ. The inverse problem we are concerned with is to determine the unknown part, Γ, of the boundary of the bounded domain Ω ⊂ RN , based on available Cauchy data ðg, hÞ on The boundary Γ is assessed by imposing a voltage pattern at a number of electrodes attached to the boundary Γ0 and measuring the resultant current through the electrodes.
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