Abstract
Based upon the elements of the modern pseudoanalytic function theory, we analyze a new method for numerically solving the forward Dirichlet boundary value problem corresponding to the two-dimensional electrical impedance equation. The analysis is performed by introducing interpolating piecewise separable-variables conductivity functions in the unit circle. To warrant the effectiveness of the posed method, we consider several examples of conductivity functions, whose boundary conditions are exact solutions of the electrical impedance equation, performing a brief comparison with the finite element method. Finally, we discuss the possible contributions of these results to the field of the electrical impedance tomography.
Highlights
The study of the forward Dirichlet boundary value problem for the electrical impedance equation in the plane, div (σ grad u) = 0, (1)is fundamental for understanding its inverse problem, commonly known as electrical impedance tomography, first correctly posed in mathematical form by Calderon [1] in 1980
This work states that if the values of the electrical conductivity are known at every point within a bounded domain Ω in the plane, it will be always possible to introduce a piecewise separable-variables function, such that we can use it to obtain a set of base functions for approaching solutions of the forward Dirichlet boundary value problem of (1), employing pseudoanalytic functions
We present them with certain modifications, in order to better analyze the special class of Vekua equation corresponding to the electrical impedance equation (1) in the plane
Summary
This work states that if the values of the electrical conductivity are known at every point within a bounded domain Ω in the plane, it will be always possible to introduce a piecewise separable-variables function, such that we can use it to obtain a set of base functions for approaching solutions of the forward Dirichlet boundary value problem of (1), employing pseudoanalytic functions. This would be true for a certain class of bounded domains, but this class will be wide enough to include many interesting cases for the applied sciences. We will discuss how this numerical technique could contribute to the study of the inverse Dirichlet boundary value problem of (1) in the plane, known as the Electrical Impedance Tomography problem
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