Numerical recovery of Robin boundary from boundary measurements for the Laplace equation

Abstract

MSC: 65R32 65N21 35J05 45A05 Keywords: Corrosion detection Robin inverse problem Boundary integral equation a b s t r a c t Based on an integral equation formulation, we present numerical methods for the inverse problem of recovering part of the domain boundary from boundary measurements of solutions to the Laplace equation on an accessible part of the boundary.

Simultaneous numerical determination of a corroded boundary and its admittance

“…Then, clearly one set of Cauchy boundary measurements (2.1) and (2.3) is not sufficient to simultaneously recover the boundary Γ 2 affected by a corrosion attack and its corrosion coefficient α. Even when α is known, one set of Cauchy data (2.1) and (2.3) may not be enough to determine uniquely the corroded boundary Γ 2 , as shown by the counterexamples given in [7,9,35] and some thorough numerical investigation reported in [15]. However, it turns out that two linearly independent boundary data f 1 and f 2 , one of which is positive, inducing, via (2.3), two corresponding flux measurements g 1 and g 2 , are sufficient to provide a unique solution for the pair (Γ 2 , α), [1,34,35].…”

“…Alternatively, cubic B-splines can also be used, [15], in place of the trigonometric approximation (3.10). , ℓ = 1, 2, in (3.1), the 2K +1 coefficients in (3.10), the 2L + 1 coefficients in (3.11) and the dilation coefficient η in (3.9).…”

Solving a class of Robin problems in simply connected regions via integral equations with a generalized Neumann kernel

“…There are many applications of Laplacian differential operator related to physical geodesy, electromagnetic, measurement 1,2 , and to specific boundary problems such as Dirichlet problem and Neumann problem 3 . The applications of the mixed boundary value problem in potential theory can be found in Ref.…”

Computing Robin Problem on Unbounded Simply Connected Domain via an Integral Equation with the Generalized Neumann Kernel

“…Many applications of the Laplacian differential operator are related to physical geodesy, measurement [1][2], while the application of mixed boundary value problem has been developed only during recent century [3]. In this study the mixed boundary value problem in the literature is the mixed D-N boundary value problem BVP.…”