# First Characterization of a New Method for Numerically Solving the Dirichlet Problem of the Two-Dimensional Electrical Impedance Equation

Authors: Marco Pedro Ramirez-Tachiquin; Cesar Marco Antonio Robles Gonzalez; Rogelio Adrian Hernandez-Becerril; Ariana Guadalupe Bucio Ramirez
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 1. Introduction

The study of the forward Dirichlet boundary value problem for the electrical impedance equation in the plane,(1)div(σgradu)=0,
is fundamental for understanding its inverse problem, commonly known as electrical impedance tomography, first correctly posed in mathematical form by Calderon [1] in 1980. It is remarkable that, for more than twenty years after the problem was stated, the mathematical complexity of (1) could provoke that many experts considered impossible to obtain its general solution in analytic form [2], even for the simplest cases of σ, excluding the constant case.This perception changed when, independently, Kravchenko in 2005 [3] and Astala and Päivärinta [4] in 2006 noticed that the two-dimensional case of (1) was completely equivalent to a special class of Vekua equation [5].Many other important results were obtained soon after. Indeed, Kravchenko et al. published in 2007 what can be considered the first general solution of (1) in analytic form [6], when σ possesses a certain form, employing Taylor series in formal powers [7]. Moreover, when the conductivity is separable variables, the real parts of the formal powers conform a complete set for approaching solutions of the forward Dirichlet boundary value problem of (1) in the plane [8].The main objective of this work is to start a discussion about the application of the Pseudoanalytic function theory whenσ is not, in general, a separable-variables function, which would allow the study of conductivity cases more interesting in physics and engineering. As a matter of fact, even it is not clear yet how to extend the proof provided in [8], about the completeness of the set of formal powers for the cases when σ is piecewise separable variables, the numerical calculations will show that a variation of the technique, originally posed for purely mathematical problems, could well serve for analyzing more general cases, providing quite acceptable results with compared when classical methods.In other words, this work states that if the values of the electrical conductivity are known at every point within a bounded domainΩ 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. Nonetheless, on behalf of briefness, we will only study the unit circle. The reader will appreciate that most of the results will be valid for more domains.In order to prove the veracity of the last assessments, we will consider a set of conductivity examples, separable variables, and nonseparable variables, for which we can obtain exact solutions, to be imposed as boundary conditions. The effectiveness of the approaches will be estimated by comparing them with the boundary conditions, employing a standard Lebesgue measure.Finally, we will discuss how this numerical technique could contribute to the study of the inverse Dirichlet boundary value problem of (1) in the plane, also known as the Electrical Impedance Tomography problem.


## 2. Preliminaries

According to the Pseudoanalytic function theory posed by Bers [7], let the pair of complex-valued functions F and G fulfill the condition:
(2)Im(F¯G)>0,
where F¯ represents the complex conjugation of F: F¯=ReF-iImF and i denotes the standard imaginary unit i2=-1. This condition implies that the functions F and G are linearly independent, therefore any complex-valued function W can be expressed by their linear combination:
(3)W=ϕF+ψG,
where ϕ and ψ are purely real-valued functions. On the light of this idea, L. Bers introduced the concept of the (F,G)-derivative of W in the form
(4)∂(F,G)W=(∂zϕ)F+(∂zψ)G.
This derivative will exist if and only if the following equality is true:
(5)(∂z¯ϕ)F+(∂z¯ψ)G=0.
Hereafter, we will use the notations ∂z=∂x-i∂y and ∂z¯=∂x+i∂y. These pairs of partial differential operators are usually introduced with the factor 1/2, but in this work it will be somehow more convenient to work without it.Introducing the notations:(6)A(F,G)=F¯∂zG-G¯∂zFFG¯-GF¯,a(F,G)=-F¯∂z¯G-G¯∂z¯FFG¯-GF¯,B(F,G)=F∂zG-G∂zFFG¯-GF¯,b(F,G)=-G∂z¯F-F∂z¯GFG¯-GF¯,
the derivative presented in (4), that we will refer to as the (F,G)-derivative of W, can be rewritten as
(7)∂(F,G)W=∂zW-A(F,G)W-B(F,G)W¯,
whereas condition (5) will become
(8)∂z¯W-a(F,G)W-b(F,G)W¯=0.A pair of complex functions(F,G) satisfying condition (2) will be called a generating pair, and the functions introduced in (6) will be referred to as the characteristic coefficients of the generating pair (F,G). As a matter of fact, the expression (8) is known as the Vekua equation [5], and every function W, solution of (8), will be called (F,G)-pseudoanalytic.The following statements were originally posed in [7]. 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.Theorem 1.
The functionsF and G, constituting the generating pair of the form (2), are (F,G)-pseudoanalytic, and their (F,G)-derivatives vanish identically:
(9)∂(F,G)F≡∂(F,G)G≡0.Theorem 2 (see [7, 9]).
Letp be a nonvanishing function, defined within some domain Ω(ℝ2), and let
(10)F=p,G=ip.
It is easy to verify that F and G conform a generating pair (2), whose characteristic coefficients, according to (6), are
(11)A(F,G)=a(F,G)=0,B(F,G)=∂zpp,b(F,G)=∂z¯pp.
Therefore, for this special class of generating pairs, the corresponding Vekua equations will have the form
(12)∂z¯W-∂z¯ppW¯=0.Definition 3.
Let(F0,G0) and (F1,G1) be two generating pairs of the form (10), and let their characteristic coefficients fulfill the condition
(13)B(F0,G0)=-b(F1,G1).
Thus the generating pair (F1,G1) will be called a successor pair of (F0,G0), as well (F0,G0) will be called a predecessor of (F1,G1).Definition 4.
Let the elements of the set(14){(Fm,Gm)},m=0,±1,±2,…,
be all generating pairs, and let every (Fm+1,Gm+1) be a successor of (Fm,Gm). Hence, the set (14) will be called a generating sequence. If (F,G)=(F0,G0), we will say that (F,G) is embedded into (14). Moreover, if there exists a number k such that (Fm+k,Gm+k)=(Fm,Gm), we will say that the generating sequence (14) is periodic, with period k.L. Bers also introduced the concept of the(F,G)-integral of a complex function W. The reader can find the conditions for warranting its existence and a detailed description of its properties in [7, 9]. Since the functions employed in this work are, by definition, (F,G)-integrable, we will only present a certain set of those properties.Definition 5.
Let(F0,G0) be a generating pair with the form (10). Its adjoin pair (F0*,G0*) is defined as
(15)F0*=-iF0,G0*=-iG0.Definition 6.
The(F0,G0)-integral of a complex function W (when it exists) is defined according to the expression
(16)∫Λ‍Wd(F0,G0)z=F0Re∫Λ‍G0*Wdz+G0Re∫Λ‍F0*Wdz,
where Λ is a rectifiable curve going from z0 up to z1, in the complex plain. In particular, the (F0,G0)-integral of ∂(F0,G0)W reaches
(17)∫z0z‍∂(F0,G0)WdF0,G0z=W-ϕ(z0)F0-ψ(z0)G0.
But according to Theorem 1, the (F0,G0)-derivatives of F0 and G0 vanish identically, hence (17) can be considered the (F0,G0)-antiderivative of ∂(F0,G0)W.


### 2.1. Formal Powers

Definition 7.
The formal powerZm(0)(a0,z0;z) belonging to the generating pair (Fm,Gm), with formal exponent 0, complex coefficient a0, center at z0, and depending upon the complex variable z=x+iy, is defined by the expression
(18)Zm(0)(a0,z0;z)=λFm+μGm,
where λ and μ are constants that fulfill the condition
(19)λFm(z0)+μGm(z0)=a0.
The formal powers with higher formal exponents are defined according to the recursive formulas
(20)Zm(n)(an,z0;z)=n∫z0z‍Zm-1(n-1)(an,z0;z)d(Fm,Gm)z.
Notice the integral operators at the right hand side of the last equality are all (Fm,Gm)-antiderivatives.Remark 8.
The formal powers possess the following properties:(1)
Zm(n)(an,z0;z)→an(z-z0)n when z→z0.(2)
EveryZm(n)(an,z0;z) is (Fm,Gm)-pseudoanalytic.(3)
Ifan=an′+ian′′, where an′ and an′′ are real constants, we will have
(21)Zm(n)(an,z0;z)=an′Zm(n)(1,z0;z)+an′′Zm(n)(i,z0;z).Theorem 9.
Every complex-valued functionW, solution of the Vekua equation (8), can be expanded in terms of the so-called Taylor series in formal powers:
(22)W=∑n=0∞‍Z(n)(an,z0;z),
where the absence of the subindex “m” indicates that all formal powers belong to the same generating pair.Remark 10.
Since every complex-valued functionW, solution of (8), can be expressed in the form (22), it is possible to assert that (22) is an analytic representation of the general solution for the Vekua equation (8).


### 2.2. The Electrical Impedance Equation in the Plane

As it has been previously posed in several works (see, e.g., [3, 9, 10]), when the conductivity function σ can be expressed in terms of a separable-variables function
(23)σ=σ1(x)σ2(y),
by introducing the notations
(24)W=σ∂xu-iσ∂yu,p=σ2σ1,
the two-dimensional electrical impedance equation (1) can be rewritten precisely as a Vekua equation of the form (12). Moreover, its corresponding generating pair
(25)F0=p,G0=ip
is embedded into a periodic generating sequence, with period k=2, such that
(26)Fm=σ2σ1,Gm=iσ1σ2,
when m is an even number, and
(27)Fm=σ1σ2,Gm=iσ1σ2,
when m is odd.Therefore, based upon the statements posed in Definition7, possessing a generating sequence will allow us to approach a set of formal powers:
(28){Z0(n)(1,0;z),Z0(n)(i,0;z)}n=0N,
within a bounded domain Ω, and, subsequently by virtue of Remark 8, we will be able to approach any formal power Z0(n)(an,0;z), at any point z∈Ω.Because the present work intends to contribute to the construction of a novel theory for the electrical impedance tomography problem, we will focus our attention into a classic domainΩ, the unit disk with center at z0=0.


### 2.3. A Complete Orthonormal System

In [8] Campos et al. posed a very important property of the formal powers.Theorem 11 (see [8]).
The set of real parts of the formal powers, with coefficients1 and i, valued at the boundary Γ of a bounded domain Ω(29){ReZ(n)(1,0;z)|Γ,ReZ(n)(i,0;z)|Γ}n=0∞,
constitutes a complete system for approaching solutions of the forward Dirichlet boundary value problem for the electrical impedance equation (1) in the plane.That is, any boundary conditionu|Γ can be approached asymptotically by the linear combination of the elements belonging to (29):
(30)limN→∞u|Γ-(∑n=0N‍cn(1)ReZ(n)(1,0;z)|Γ+cn(i)ReZ(n)(i,0;z)|Γ∑n=0N‍)=0,
where the coefficients cn(1) and cn(i) are all real constants.Since the elements of the set (29) are, by definition, linearly independent [7], it is possible to perform a standard Gram-Schmidt orthonormalization process in order to obtain the set of functions
(31){un}n=02N-1,
defined on the boundary Γ, such that when imposing a boundary condition u|Γ, we will have
(32)u|Γ~∑n=02N-1‍αnun.Because the set{uk}k=02N-1 is orthonormal, the calculation of the coefficients αn can be performed by several classical methods. Particularly, we will employ the scalar product
(33)αn=〈un,u|Γ〉.


### 2.4. A Basic Numerical Approach

Remembering thatΓ coincides with the perimeter of the unit circle, we can employ the numerical methods detailed in [11] for obtaining a system of 2N+1 formal powers (notice that, according to (25), ReZ0(0)(i,z0;z)=0), defined at the boundary Γ:
(34){ReZ0(n)(1,0;z)|Γ,ReZ0(n)(i,0;z)|Γ}n=0N.More precisely, let us consider the formal powersZ0(n)(1,0;z). Taking into account that the integral expressions introduced in (17) are path independent [7], we can choose the rectifiable curve Λ to be a straight line segment, going from z0=0 until some point of the unit circle. Thus, we can allocate P points equidistantly distributed on such line, obtaining the set of complex numbers
(35){z[p]=pPcosθ[r]+ipPsinθ[r]}p=0P-1,
where θ[r] is some angle associated to the radius. Given this set of points, and considering that the generating sequence corresponding to the pair (25) is periodic with period k=2, we can numerically approach the formal powers employing, for example, a variation of the trapezoidal integration method according to the expressions
(36)Z0(n)(z[p])=(n-1)F0(z[p])×Re∑q=0p-1‍(Z1(n-1)(z[q+1])G0*(z[q+1]))dz[q]+(n-1)F0(z[p])×Re∑q=0p‍(Z1(n-1)(z[q])G0*(z[q]))dz[q]+(n-1)G0(z[p])×Re∑q=0p-1‍(Z1(n-1)(z[q+1])F0*(z[q+1]))dz[q]+(n-1)G0(z[p])×Re∑q=0p‍(Z1(n-1)(z[q])F0*(z[q]))dz[q],
where p,q=0,1,…,P-1, and
(37)dz[p]=z[p+1]-z[p].Besides, we can introduce a set ofR angles θ[r], to be employed in (35), according to the formula
(38){θ[r]=2πrR}r=0R-1
for the iterative expressions (36) that can be performed at each angle θ[r]. When the full procedure is complete, we will possess a set of N+1 formal powers Z(n)(1,0;z), approached for R radii, and with P points per radius. Thus, by collecting the real parts of Z(n)(1,0;z) valued at the points z[P-1], which are precisely those located at the boundary Γ, we will obtain a numerical approximation of the set
(39){ReZ(n)(1,0;z)|Γ}n=0N.
An identical procedure can be performed for obtaining the elements of
(40){ReZ(n)(i,0;z)|Γ}n=0N.
Since the numerical approaching of the set (34) is complete, a standard Gram-Schmidt method will reach the orthonormal system (31) for approaching the boundary condition u|Γ.The effectiveness of this numerical approach has been analyzed in several works (see [8, 10]). Here we will only analyze two particular examples of separable-variables conductivity functions σ, before studying those that are not separable variables.


## 2.1. Formal Powers

Definition 7.
The formal powerZm(0)(a0,z0;z) belonging to the generating pair (Fm,Gm), with formal exponent 0, complex coefficient a0, center at z0, and depending upon the complex variable z=x+iy, is defined by the expression
(18)Zm(0)(a0,z0;z)=λFm+μGm,
where λ and μ are constants that fulfill the condition
(19)λFm(z0)+μGm(z0)=a0.
The formal powers with higher formal exponents are defined according to the recursive formulas
(20)Zm(n)(an,z0;z)=n∫z0z‍Zm-1(n-1)(an,z0;z)d(Fm,Gm)z.
Notice the integral operators at the right hand side of the last equality are all (Fm,Gm)-antiderivatives.Remark 8.
The formal powers possess the following properties:(1)
Zm(n)(an,z0;z)→an(z-z0)n when z→z0.(2)
EveryZm(n)(an,z0;z) is (Fm,Gm)-pseudoanalytic.(3)
Ifan=an′+ian′′, where an′ and an′′ are real constants, we will have
(21)Zm(n)(an,z0;z)=an′Zm(n)(1,z0;z)+an′′Zm(n)(i,z0;z).Theorem 9.
Every complex-valued functionW, solution of the Vekua equation (8), can be expanded in terms of the so-called Taylor series in formal powers:
(22)W=∑n=0∞‍Z(n)(an,z0;z),
where the absence of the subindex “m” indicates that all formal powers belong to the same generating pair.Remark 10.
Since every complex-valued functionW, solution of (8), can be expressed in the form (22), it is possible to assert that (22) is an analytic representation of the general solution for the Vekua equation (8).


## 2.2. The Electrical Impedance Equation in the Plane

As it has been previously posed in several works (see, e.g., [3, 9, 10]), when the conductivity function σ can be expressed in terms of a separable-variables function
(23)σ=σ1(x)σ2(y),
by introducing the notations
(24)W=σ∂xu-iσ∂yu,p=σ2σ1,
the two-dimensional electrical impedance equation (1) can be rewritten precisely as a Vekua equation of the form (12). Moreover, its corresponding generating pair
(25)F0=p,G0=ip
is embedded into a periodic generating sequence, with period k=2, such that
(26)Fm=σ2σ1,Gm=iσ1σ2,
when m is an even number, and
(27)Fm=σ1σ2,Gm=iσ1σ2,
when m is odd.Therefore, based upon the statements posed in Definition7, possessing a generating sequence will allow us to approach a set of formal powers:
(28){Z0(n)(1,0;z),Z0(n)(i,0;z)}n=0N,
within a bounded domain Ω, and, subsequently by virtue of Remark 8, we will be able to approach any formal power Z0(n)(an,0;z), at any point z∈Ω.Because the present work intends to contribute to the construction of a novel theory for the electrical impedance tomography problem, we will focus our attention into a classic domainΩ, the unit disk with center at z0=0.


## 2.3. A Complete Orthonormal System

In [8] Campos et al. posed a very important property of the formal powers.Theorem 11 (see [8]).
The set of real parts of the formal powers, with coefficients1 and i, valued at the boundary Γ of a bounded domain Ω(29){ReZ(n)(1,0;z)|Γ,ReZ(n)(i,0;z)|Γ}n=0∞,
constitutes a complete system for approaching solutions of the forward Dirichlet boundary value problem for the electrical impedance equation (1) in the plane.That is, any boundary conditionu|Γ can be approached asymptotically by the linear combination of the elements belonging to (29):
(30)limN→∞u|Γ-(∑n=0N‍cn(1)ReZ(n)(1,0;z)|Γ+cn(i)ReZ(n)(i,0;z)|Γ∑n=0N‍)=0,
where the coefficients cn(1) and cn(i) are all real constants.Since the elements of the set (29) are, by definition, linearly independent [7], it is possible to perform a standard Gram-Schmidt orthonormalization process in order to obtain the set of functions
(31){un}n=02N-1,
defined on the boundary Γ, such that when imposing a boundary condition u|Γ, we will have
(32)u|Γ~∑n=02N-1‍αnun.Because the set{uk}k=02N-1 is orthonormal, the calculation of the coefficients αn can be performed by several classical methods. Particularly, we will employ the scalar product
(33)αn=〈un,u|Γ〉.


## 2.4. A Basic Numerical Approach

Remembering thatΓ coincides with the perimeter of the unit circle, we can employ the numerical methods detailed in [11] for obtaining a system of 2N+1 formal powers (notice that, according to (25), ReZ0(0)(i,z0;z)=0), defined at the boundary Γ:
(34){ReZ0(n)(1,0;z)|Γ,ReZ0(n)(i,0;z)|Γ}n=0N.More precisely, let us consider the formal powersZ0(n)(1,0;z). Taking into account that the integral expressions introduced in (17) are path independent [7], we can choose the rectifiable curve Λ to be a straight line segment, going from z0=0 until some point of the unit circle. Thus, we can allocate P points equidistantly distributed on such line, obtaining the set of complex numbers
(35){z[p]=pPcosθ[r]+ipPsinθ[r]}p=0P-1,
where θ[r] is some angle associated to the radius. Given this set of points, and considering that the generating sequence corresponding to the pair (25) is periodic with period k=2, we can numerically approach the formal powers employing, for example, a variation of the trapezoidal integration method according to the expressions
(36)Z0(n)(z[p])=(n-1)F0(z[p])×Re∑q=0p-1‍(Z1(n-1)(z[q+1])G0*(z[q+1]))dz[q]+(n-1)F0(z[p])×Re∑q=0p‍(Z1(n-1)(z[q])G0*(z[q]))dz[q]+(n-1)G0(z[p])×Re∑q=0p-1‍(Z1(n-1)(z[q+1])F0*(z[q+1]))dz[q]+(n-1)G0(z[p])×Re∑q=0p‍(Z1(n-1)(z[q])F0*(z[q]))dz[q],
where p,q=0,1,…,P-1, and
(37)dz[p]=z[p+1]-z[p].Besides, we can introduce a set ofR angles θ[r], to be employed in (35), according to the formula
(38){θ[r]=2πrR}r=0R-1
for the iterative expressions (36) that can be performed at each angle θ[r]. When the full procedure is complete, we will possess a set of N+1 formal powers Z(n)(1,0;z), approached for R radii, and with P points per radius. Thus, by collecting the real parts of Z(n)(1,0;z) valued at the points z[P-1], which are precisely those located at the boundary Γ, we will obtain a numerical approximation of the set
(39){ReZ(n)(1,0;z)|Γ}n=0N.
An identical procedure can be performed for obtaining the elements of
(40){ReZ(n)(i,0;z)|Γ}n=0N.
Since the numerical approaching of the set (34) is complete, a standard Gram-Schmidt method will reach the orthonormal system (31) for approaching the boundary condition u|Γ.The effectiveness of this numerical approach has been analyzed in several works (see [8, 10]). Here we will only analyze two particular examples of separable-variables conductivity functions σ, before studying those that are not separable variables.


## 3. The Cases When the Conductivityσ Possesses a Separable-Variables Form

### 3.1. Example Whenσ Is an Exponential Function

Proposition 12.
Letσ=ex+y. Then the function u=e-x-y will be a particular solution of (1).Figure1 plots an illustration of the exponential conductivity.Figure 1
Example of a separable-variables exponential conductivity functionσ=ex+y.The numerical procedure described in Section2.4 will be employed for approaching the forward Dirichlet boundary value problem when σ possesses the form of the proposition above, imposing as the boundary condition the exact solution
(41)u|Γ=e-x-y.The errorℰ will be defined according to the Lebesgue measure
(42)ℰ=(∫Γ‍(u|Γ-∑n=02N+1‍αnun)2dl)1/2,
where the addition within the integral expression corresponds to the approached solution (32).Table1 contains the relation between the error ℰ and the number of formal powers N. The parameters R and P have been fixed at the value 100, since for this particular case, as displayed in the table, they do not significantly influence the diminution of the error when increasing.Table 1
Relation between the errorℰ and the number of formal powers, for the case when σ possesses an exponential form.
Number of radiiR
Points per radiusP
Number of formal powersN
ℰ
500
500
45
1.4370
×
1
0
-
15
100
100
45
1.9996
×
1
0
-
15
100
100
35
2.0001
×
1
0
-
15
100
100
25
2.0094
×
1
0
-
15
100
100
15
2.0524
×
1
0
-
15
100
100
5
3.1333
×
1
0
-
14


### 3.2. The Case Whenσ Has a Lorentzian Form

Proposition 13.
Let(43)σ=(1x2+0.1)(1y2+0.1).
Then the function
(44)u=x3+y33+0.1(x+y)
will be a particular solution of (1).An illustration of the conductivity function (43) is displayed in Figure 2.Figure 2
Example of a separable-variables Lorentzian conductivity functionσ=(x2+0.1)-1·(y2+0.1)-1.Once more, the exact solution (44) will be imposed as the boundary condition. The numerical results are shown in Table 2. This, indeed, is a more interesting case, since the number N of formal powers strongly influences the accuracy of the approach. The increment of the number of points per radius P and the number of radii R, as in the previous case, do not significantly increase the accuracy.Table 2
Relation between the errorℰ and the number of formal powers, for the case when σ has a Lorentzian form.
Number of radiiR
Points per radiusP
Number of formal powersN
ℰ
500
500
45
2.2968
×
1
0
-
8
100
100
45
2.3753
×
1
0
-
8
100
100
35
5.8069
×
1
0
-
7
100
100
25
1.9710
×
1
0
-
5
100
100
15
7.2801
×
1
0
-
4
100
100
5
4.8875
×
1
0
-
2


### 3.3. Construction of a Piecewise Separable-Variables Conductivity Function

Consider a bounded domainΩ(ℝ2) (in this case the unit circle) and divide it into a finite number of subsections S, taking care that the point to be considered the center of the formal powers z0 (see Definition 7) does not reside onto the boundary of two subsections. For simplicity, let us make the division by employing a finite set of parallel lines to the x-axis, equidistant to one another, and let us locate z0=0.Supposing that the values of the electrical conductivity are defined for every point within the domainΩ(ℝ2), let us locate a straight line crossing every subsection, that does not intersect the bounding parallel lines of its corresponding subsection. Indeed, such lines can be simply parallel to the bounding ones.The following step is to collect finite sets of conductivity values, corresponding to a numberC of points located at the crossing lines. For every line, the quantity of collected values C must be big enough to warrant that an interpolating process, in this case performed with straight lines, will adequately approach all the remaining conductivity values defined over the line points.Since we have already assumed that every crossing line will be parallel to the subsection-bounding lines, and in consequence to thex-axis, all collected points corresponding to the same crossing line will possess the same y-coordinate. Let us propose that the conductivity within every subsection can be represented according to the expression
(45)σ=y+KΥ+K·f(x),
where Υ denotes the y-coordinate that is common to all points along the crossing line, f(x) is an interpolating function that approaches the collected conductivity values, and K is a real positive constant such that y+K≠0, for all y∈Ω.Applying this idea in every subsection, the conductivityσ within the bounded domain Ω can be approached by means of the piecewise-defined function:
(46)σ(x,y)={y+KΥ1+K·f1(x):y∈[y(1),y(2));y+KΥ2+K·f2(x):y∈[y(2),y(3));⋮y+KΥS+K·fS(x):y∈[y(S),y(S+1)].
Here y(1) represents the first y-coordinate found within the domain Ω when broaching the y-axis from ∞ up to -∞, whereas y(S) represents the last one. The pairs of y-axis parallel lines (y(j),y(j+1)), where j=1,…,S+1, are given by the common y-coordinates belonging to every pair of lines delimiting the subsections. Notice the piecewise function (46) is separable variables.Thus, according to Section2.2, it immediately follows that
(47)F0={(y+KΥ1+K·f1-1(x))1/2:y∈[y(1),y(2)),(y+KΥ2+K·f2-1(x))1/2:y∈[y(2),y(3)),⋮(y+KΥS+K·fS-1(x))1/2:y∈[y(S),y(S+1)],
whereas
(48)G0={i(Υ1+Ky+K·f1(x))1/2:y∈[y(1),y(2)),i(Υ2+Ky+K·f2(x))1/2:y∈[y(2),y(3)),⋮i(ΥS+Ky+K·fS(x))1/2:y∈[y(S),y(S+1)].
For the generating pair (F1,G1) we will have
(49)F1={(y+KΥ1+K·f1(x))1/2:y∈[y(1),y(2)),(y+KΥ2+K·f2(x))1/2:y∈[y(2),y(3)),⋮(y+KΥS+K·fS(x))1/2:y∈[y(S),y(S+1)],G1={i(Υ1+Ky+K·f1-1(x))1/2:y∈[y(1),y(2)),i(Υ2+Ky+K·f2-1(x))1/2:y∈[y(2),y(3)),⋮i(ΥS+Ky+K·fS-1(x))1/2:y∈[y(S),y(S+1)].We will use these piecewise-defined functions to perform the numerical procedure described in Section2.4. The first case to analyze will be the exponential conductivity σ=ex+y studied in Section 3.1, imposing as the boundary condition u|Γ=e-x-y the exact solution presented in Proposition 12.The second case will be the Lorentzian conductivity treated in Section3.2, whose boundary condition will be the exact solution u|Γ shown in Proposition 13. The results of these cases are exposed in Tables 3 and 4. For both, we can observe that the accuracy is strongly related with the number of sections S and the number of collected values per section C, as well as with the number N of formal powers. Once more, the increment of the number of points per radius P and the number of radii R do not significantly improve the accuracy.Table 3
Relation between the errorℰ and the number of sections S and samples per section C, when σ=ex+y.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
4.4935
×
1
0
-
5
100
100
45
10001
10001
6.9358
×
1
0
-
5
100
100
45
5001
5001
6.8149
×
1
0
-
4
100
100
45
2501
2501
5.7113
×
1
0
-
4
100
100
45
1001
1001
3.3381
×
1
0
-
3
100
100
45
501
501
1.5105
×
1
0
-
3
100
100
45
101
101
5.3048
×
1
0
-
3Table 4
Relation between the errorℰ and the number of formal powers N, when σ=ex+y.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
6.9358
×
1
0
-
5
100
100
35
10001
10001
4.0388
×
1
0
-
4
100
100
25
10001
10001
5.3651
×
1
0
-
4
100
100
15
10001
10001
3.4733
×
1
0
-
3
100
100
5
10001
10001
4.5714
×
1
0
-
2Beside, since the tables presented in the previous subsection have shown that the accuracy of the method is not considerably improved when neither the number of points per radiusP nor the number of radii R increase, we will fix both values P=R=100 hereafter.Tables3 and 4 show that the piecewise separable-variables conductivity function (46) can be positively employed for numerically approaching solutions of this boundary value problem, because even the magnitudes of the errors ℰ are considerable bigger than those obtained when employing the original σ, their magnitudes are still acceptable when compared with other classical numerical methods.More precisely, employing a standard finite element method technique for solving forward Dirichlet boundary value problems of elliptic partial differential equations in the plane, when considering the exponential conductivityσ=ex+y and imposing the boundary condition u|Γ=e-x-y, the error was ℰ=6.1638×10-4, utilizing 8257 nodes in the mesh, corresponding to 16256 triangular elements. For the Lorentzian conductivity σ=(x2+0.1)-1(y2+0.1)-1, with the boundary condition u|Γ=(1/3)(x3+y3)+0.1(x+y), employing identical number of nodes in the mesh, the resulting error was ℰ=2.6424×10-4.A more detailed description of the behavior of the method is provided in Tables5, 6, and 7.Table 5
Relation between the errorℰ and the number of sections S and samples per section C, when σ=(x2+0.1)-1(y2+0.1)-1.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
6.1214
×
1
0
-
6
100
100
45
10001
10001
1.1241
×
1
0
-
5
100
100
45
5001
5001
1.3062
×
1
0
-
5
100
100
45
2501
2501
7.2332
×
1
0
-
5
100
100
45
1001
1001
1.7079
×
1
0
-
4
100
100
45
501
501
3.5471
×
1
0
-
4
100
100
45
101
101
1.0553
×
1
0
-
3Table 6
Relation between the errorℰ and the number of formal powers N, when σ=(x2+0.1)-1(y2+0.1)-1.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
1.1241
×
1
0
-
5
100
100
35
10001
10001
9.1964
×
1
0
-
5
100
100
25
10001
10001
5.0855
×
1
0
-
4
100
100
15
10001
10001
4.2633
×
1
0
-
3
100
100
5
10001
10001
2.9065
×
1
0
-
2Table 7
Relation between the errorℰ and the number of sections S and samples per section C, when σ=exy.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
3.8759
×
1
0
-
6
100
100
45
10001
10001
5.4177
×
1
0
-
6
100
100
45
5001
5001
2.1470
×
1
0
-
5
100
100
45
2501
2501
1.5406
×
1
0
-
5
100
100
45
1001
1001
6.6917
×
1
0
-
5
100
100
45
501
501
1.5059
×
1
0
-
4
100
100
45
101
101
8.7576
×
1
0
-
4Notice that the comparison with the finite element method is given as a basic reference. Indeed, the number of nodes in the mesh was taken at8257, because it is the closest value to the number of points located in Ω obtained when we multiply P·R=10000.


## 3.1. Example Whenσ Is an Exponential Function

Proposition 12.
Letσ=ex+y. Then the function u=e-x-y will be a particular solution of (1).Figure1 plots an illustration of the exponential conductivity.Figure 1
Example of a separable-variables exponential conductivity functionσ=ex+y.The numerical procedure described in Section2.4 will be employed for approaching the forward Dirichlet boundary value problem when σ possesses the form of the proposition above, imposing as the boundary condition the exact solution
(41)u|Γ=e-x-y.The errorℰ will be defined according to the Lebesgue measure
(42)ℰ=(∫Γ‍(u|Γ-∑n=02N+1‍αnun)2dl)1/2,
where the addition within the integral expression corresponds to the approached solution (32).Table1 contains the relation between the error ℰ and the number of formal powers N. The parameters R and P have been fixed at the value 100, since for this particular case, as displayed in the table, they do not significantly influence the diminution of the error when increasing.Table 1
Relation between the errorℰ and the number of formal powers, for the case when σ possesses an exponential form.
Number of radiiR
Points per radiusP
Number of formal powersN
ℰ
500
500
45
1.4370
×
1
0
-
15
100
100
45
1.9996
×
1
0
-
15
100
100
35
2.0001
×
1
0
-
15
100
100
25
2.0094
×
1
0
-
15
100
100
15
2.0524
×
1
0
-
15
100
100
5
3.1333
×
1
0
-
14


## 3.2. The Case Whenσ Has a Lorentzian Form

Proposition 13.
Let(43)σ=(1x2+0.1)(1y2+0.1).
Then the function
(44)u=x3+y33+0.1(x+y)
will be a particular solution of (1).An illustration of the conductivity function (43) is displayed in Figure 2.Figure 2
Example of a separable-variables Lorentzian conductivity functionσ=(x2+0.1)-1·(y2+0.1)-1.Once more, the exact solution (44) will be imposed as the boundary condition. The numerical results are shown in Table 2. This, indeed, is a more interesting case, since the number N of formal powers strongly influences the accuracy of the approach. The increment of the number of points per radius P and the number of radii R, as in the previous case, do not significantly increase the accuracy.Table 2
Relation between the errorℰ and the number of formal powers, for the case when σ has a Lorentzian form.
Number of radiiR
Points per radiusP
Number of formal powersN
ℰ
500
500
45
2.2968
×
1
0
-
8
100
100
45
2.3753
×
1
0
-
8
100
100
35
5.8069
×
1
0
-
7
100
100
25
1.9710
×
1
0
-
5
100
100
15
7.2801
×
1
0
-
4
100
100
5
4.8875
×
1
0
-
2


## 3.3. Construction of a Piecewise Separable-Variables Conductivity Function

Consider a bounded domainΩ(ℝ2) (in this case the unit circle) and divide it into a finite number of subsections S, taking care that the point to be considered the center of the formal powers z0 (see Definition 7) does not reside onto the boundary of two subsections. For simplicity, let us make the division by employing a finite set of parallel lines to the x-axis, equidistant to one another, and let us locate z0=0.Supposing that the values of the electrical conductivity are defined for every point within the domainΩ(ℝ2), let us locate a straight line crossing every subsection, that does not intersect the bounding parallel lines of its corresponding subsection. Indeed, such lines can be simply parallel to the bounding ones.The following step is to collect finite sets of conductivity values, corresponding to a numberC of points located at the crossing lines. For every line, the quantity of collected values C must be big enough to warrant that an interpolating process, in this case performed with straight lines, will adequately approach all the remaining conductivity values defined over the line points.Since we have already assumed that every crossing line will be parallel to the subsection-bounding lines, and in consequence to thex-axis, all collected points corresponding to the same crossing line will possess the same y-coordinate. Let us propose that the conductivity within every subsection can be represented according to the expression
(45)σ=y+KΥ+K·f(x),
where Υ denotes the y-coordinate that is common to all points along the crossing line, f(x) is an interpolating function that approaches the collected conductivity values, and K is a real positive constant such that y+K≠0, for all y∈Ω.Applying this idea in every subsection, the conductivityσ within the bounded domain Ω can be approached by means of the piecewise-defined function:
(46)σ(x,y)={y+KΥ1+K·f1(x):y∈[y(1),y(2));y+KΥ2+K·f2(x):y∈[y(2),y(3));⋮y+KΥS+K·fS(x):y∈[y(S),y(S+1)].
Here y(1) represents the first y-coordinate found within the domain Ω when broaching the y-axis from ∞ up to -∞, whereas y(S) represents the last one. The pairs of y-axis parallel lines (y(j),y(j+1)), where j=1,…,S+1, are given by the common y-coordinates belonging to every pair of lines delimiting the subsections. Notice the piecewise function (46) is separable variables.Thus, according to Section2.2, it immediately follows that
(47)F0={(y+KΥ1+K·f1-1(x))1/2:y∈[y(1),y(2)),(y+KΥ2+K·f2-1(x))1/2:y∈[y(2),y(3)),⋮(y+KΥS+K·fS-1(x))1/2:y∈[y(S),y(S+1)],
whereas
(48)G0={i(Υ1+Ky+K·f1(x))1/2:y∈[y(1),y(2)),i(Υ2+Ky+K·f2(x))1/2:y∈[y(2),y(3)),⋮i(ΥS+Ky+K·fS(x))1/2:y∈[y(S),y(S+1)].
For the generating pair (F1,G1) we will have
(49)F1={(y+KΥ1+K·f1(x))1/2:y∈[y(1),y(2)),(y+KΥ2+K·f2(x))1/2:y∈[y(2),y(3)),⋮(y+KΥS+K·fS(x))1/2:y∈[y(S),y(S+1)],G1={i(Υ1+Ky+K·f1-1(x))1/2:y∈[y(1),y(2)),i(Υ2+Ky+K·f2-1(x))1/2:y∈[y(2),y(3)),⋮i(ΥS+Ky+K·fS-1(x))1/2:y∈[y(S),y(S+1)].We will use these piecewise-defined functions to perform the numerical procedure described in Section2.4. The first case to analyze will be the exponential conductivity σ=ex+y studied in Section 3.1, imposing as the boundary condition u|Γ=e-x-y the exact solution presented in Proposition 12.The second case will be the Lorentzian conductivity treated in Section3.2, whose boundary condition will be the exact solution u|Γ shown in Proposition 13. The results of these cases are exposed in Tables 3 and 4. For both, we can observe that the accuracy is strongly related with the number of sections S and the number of collected values per section C, as well as with the number N of formal powers. Once more, the increment of the number of points per radius P and the number of radii R do not significantly improve the accuracy.Table 3
Relation between the errorℰ and the number of sections S and samples per section C, when σ=ex+y.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
4.4935
×
1
0
-
5
100
100
45
10001
10001
6.9358
×
1
0
-
5
100
100
45
5001
5001
6.8149
×
1
0
-
4
100
100
45
2501
2501
5.7113
×
1
0
-
4
100
100
45
1001
1001
3.3381
×
1
0
-
3
100
100
45
501
501
1.5105
×
1
0
-
3
100
100
45
101
101
5.3048
×
1
0
-
3Table 4
Relation between the errorℰ and the number of formal powers N, when σ=ex+y.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
6.9358
×
1
0
-
5
100
100
35
10001
10001
4.0388
×
1
0
-
4
100
100
25
10001
10001
5.3651
×
1
0
-
4
100
100
15
10001
10001
3.4733
×
1
0
-
3
100
100
5
10001
10001
4.5714
×
1
0
-
2Beside, since the tables presented in the previous subsection have shown that the accuracy of the method is not considerably improved when neither the number of points per radiusP nor the number of radii R increase, we will fix both values P=R=100 hereafter.Tables3 and 4 show that the piecewise separable-variables conductivity function (46) can be positively employed for numerically approaching solutions of this boundary value problem, because even the magnitudes of the errors ℰ are considerable bigger than those obtained when employing the original σ, their magnitudes are still acceptable when compared with other classical numerical methods.More precisely, employing a standard finite element method technique for solving forward Dirichlet boundary value problems of elliptic partial differential equations in the plane, when considering the exponential conductivityσ=ex+y and imposing the boundary condition u|Γ=e-x-y, the error was ℰ=6.1638×10-4, utilizing 8257 nodes in the mesh, corresponding to 16256 triangular elements. For the Lorentzian conductivity σ=(x2+0.1)-1(y2+0.1)-1, with the boundary condition u|Γ=(1/3)(x3+y3)+0.1(x+y), employing identical number of nodes in the mesh, the resulting error was ℰ=2.6424×10-4.A more detailed description of the behavior of the method is provided in Tables5, 6, and 7.Table 5
Relation between the errorℰ and the number of sections S and samples per section C, when σ=(x2+0.1)-1(y2+0.1)-1.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
6.1214
×
1
0
-
6
100
100
45
10001
10001
1.1241
×
1
0
-
5
100
100
45
5001
5001
1.3062
×
1
0
-
5
100
100
45
2501
2501
7.2332
×
1
0
-
5
100
100
45
1001
1001
1.7079
×
1
0
-
4
100
100
45
501
501
3.5471
×
1
0
-
4
100
100
45
101
101
1.0553
×
1
0
-
3Table 6
Relation between the errorℰ and the number of formal powers N, when σ=(x2+0.1)-1(y2+0.1)-1.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
1.1241
×
1
0
-
5
100
100
35
10001
10001
9.1964
×
1
0
-
5
100
100
25
10001
10001
5.0855
×
1
0
-
4
100
100
15
10001
10001
4.2633
×
1
0
-
3
100
100
5
10001
10001
2.9065
×
1
0
-
2Table 7
Relation between the errorℰ and the number of sections S and samples per section C, when σ=exy.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
3.8759
×
1
0
-
6
100
100
45
10001
10001
5.4177
×
1
0
-
6
100
100
45
5001
5001
2.1470
×
1
0
-
5
100
100
45
2501
2501
1.5406
×
1
0
-
5
100
100
45
1001
1001
6.6917
×
1
0
-
5
100
100
45
501
501
1.5059
×
1
0
-
4
100
100
45
101
101
8.7576
×
1
0
-
4Notice that the comparison with the finite element method is given as a basic reference. Indeed, the number of nodes in the mesh was taken at8257, because it is the closest value to the number of points located in Ω obtained when we multiply P·R=10000.


## 4. Analysis of Conductivity Functions That Are Not Separable Variables

This section is fully dedicated to study conductivity functions that do not possess a separable-variables form, but for which we are able to obtain exact solutions of (1), in order to impose them as boundary conditions.


### 4.1. The Nonseparable-Variables Exponential Case

Proposition 14.
Let the conductivity function(50)σ=exy.
Then, a particular solution of (1) will be
(51)u=e-xy.Figure3 illustrates the conductivity funcion of (50).Figure 3
Example of nonseparable-variables conductivity functionσ=exy.Tables7 and 8 contain the information about the behavior of the error ℰ when changing the values of S, C, and N. In this case the diminution of the number of sections S and diminution of the samples per section C do influence the accuracy, but their influence is not so significant as the diminution of the formal powers N.Table 8
Relation between the errorℰ and the number of formal powers N, when σ=exy.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
5.4177
×
1
0
-
6
100
100
35
10001
10001
2.6425
×
1
0
-
5
100
100
25
10001
10001
3.2504
×
1
0
-
5
100
100
15
10001
10001
3.4497
×
1
0
-
5
100
100
5
10001
10001
1.0798
×
1
0
-
2The error obtained when employing the finite element method approach wasℰ=2.3966×10-4, employing 8257 points in the mesh.


### 4.2. The Nonseparable-Variables Lorentzian Case

Proposition 15.
Let the conductivity function have the form(52)σ=1(x+y)2+1.
An exact solution for (1) is
(53)u=(x+y)33+x+y.The conductivity function with the form (52) is illustrated in Figure 4.Figure 4
Example of nonseparable-variables Lorentzian conductivity functionσ=((x+y)2+1)-1.Table9 shows an abnormal behavior of the error ℰ when decreasing the number of sections S and samples C. That is, it does not follow a clear pattern. But more important is to remark that the finite element method reported an error ℰ=5.4311×10-4, which implies that is better situated for analyzing this class of conductivity functions. Table 10 states that the diminution of the number of formal powers N decreases the accuracy.Table 9
Relation between the errorℰ and the number of sections S and samples per section C, when σ=((x+y)2+1)-1.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
8.8364
×
1
0
-
3
100
100
45
10001
10001
8.9246
×
1
0
-
3
100
100
45
5001
5001
9.4573
×
1
0
-
3
100
100
45
2501
2501
9.9164
×
1
0
-
3
100
100
45
1001
1001
1.0330
×
1
0
-
2
100
100
45
501
501
8.6507
×
1
0
-
3
100
100
45
101
101
9.5800
×
1
0
-
3Table 10
Relation between the errorℰ and the number of formal powers N, when σ=((x+y)2+1)-1.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values  per sectionC
ℰ
100
100
45
10001
10001
8.9246
×
1
0
-
3
100
100
35
10001
10001
4.3227
×
1
0
-
2
100
100
25
10001
10001
5.9625
×
1
0
-
2
100
100
15
10001
10001
8.2125
×
1
0
-
2
100
100
5
10001
10001
1.4064
×
1
0
-
1


### 4.3. The Nonseparable-Variables Polynomial Case

Proposition 16.
Let one assume that the conductivity function has the form(54)σ=x+y+10,
thus the function
(55)u=ln(x+y+10)
will be a solution of (1).In Figure5 it is illustrated the polynomial conductivity function of (54).Figure 5
Examples of nonseparable-variables polynomial conductivity functionσ=x+y+10.Tables11 and 12 show a regular behavior of the error ℰ, because it decreases when the values S and C do. This case possesses a singular characteristic when compared to the cases above: The diminution of the number of formal powers N does not provoke a considerable loss of the accuracy. This exception will be remarked since it has not been detected in other class of conductivity functions.Table 11
Relation between the errorℰ and the number of sections S and samples per section C, when σ=x+y+10.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
3.0640
×
1
0
-
6
100
100
45
10001
10001
2.3420
×
1
0
-
6
100
100
45
5001
5001
8.1962
×
1
0
-
6
100
100
45
2501
2501
1.9286
×
1
0
-
5
100
100
45
1001
1001
6.4916
×
1
0
-
5
100
100
45
501
501
1.5166
×
1
0
-
4
100
100
45
101
101
4.5938
×
1
0
-
4Table 12
Relation between the errorℰ and the number of formal powers N, when σ=x+y+10.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
2.3420
×
1
0
-
6
100
100
35
10001
10001
1.0616
×
1
0
-
5
100
100
25
10001
10001
1.6275
×
1
0
-
5
100
100
15
10001
10001
1.8396
×
1
0
-
5
100
100
5
10001
10001
1.9909
×
1
0
-
5On the other hand, the finite element method proved to provide a very similar accuracy for analyzing this case, because the obtained error wasℰ=2.8958×10-6.


### 4.4. The Nonseparable-Variables Sinusoidal Case

Proposition 17.
Let one consider the sinusoidal conductivity(56)σ=2+sin(x+y).
One can verify by direct substitution that the function
(57)u=23arctan(2tan((x+y)/2)+13)
is a solution of (1).The Figure6(a) illustrates the conductivity (56), whereas the Figure 6(b) illustrates the case (58).Examples of sinusoidal conductivity functions.
(a)
σ
=
2
+
sin
(
x
+
y
)
(b)
σ
=
2
+
sin
5
(
x
+
y
)Tables13 and 14 show that the behavior of ℰ is not fully regular, because a diminution of the number S and C does not necessarily imply a reduction of the accuracy, whereas the diminution of the formal powers N provokes significant changes of the error ℰ. For this case, the finite element method reported an error ℰ=0.0192.Table 13
Relation between the errorℰ and the number of sections S and samples per section C, when σ=2+sin(x+y).
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
7.5333
×
1
0
-
2
100
100
45
10001
10001
7.6129
×
1
0
-
2
100
100
45
5001
5001
7.4343
×
1
0
-
2
100
100
45
2501
2501
6.9999
×
1
0
-
2
100
100
45
1001
1001
8.4363
×
1
0
-
2
100
100
45
501
501
6.4758
×
1
0
-
2
100
100
45
101
101
9.0737
×
1
0
-
2Table 14
Relation between the errorℰ and the number of formal powers N, when σ=2+sin(x+y).
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
7.6129
×
1
0
-
2
100
100
35
10001
10001
1.3551
×
1
0
-
1
100
100
25
10001
10001
2.0886
×
1
0
-
1
100
100
15
10001
10001
3.3854
×
1
0
-
1
100
100
5
10001
10001
5.3026
×
1
0
-
1Notice that the exact solution (57) that was imposed as the boundary condition u|Γ cannot be employed when a constant factor multiplies the argument of the trigonometric function, because the tangent would not be defined at certain points. Nevertheless, one last experiment is worth of research.Let us suppose that the conductivity function possesses the form(58)σ=2+sin5(x+y),
and let us approach the solution to the forward problem keeping as the boundary condition the expression (57). Tables 15 and 16 display the biggest error ℰ found in the present work, even the behavior of ℰ could be considered regular, because it decreases while S, C, and N do. The reader will appreciate that the accuracy is, in general, unsatisfactory for applications. Indeed, the finite element method obtained an error ℰ=0.0043, hence it is quite better situated for analyzing this kind of conductivity functions.Table 15
Relation between the errorℰ and the number of sections S and samples per section C, when σ=2+sin5(x+y).
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
7.5333
×
1
0
-
2
100
100
45
10001
10001
7.6129
×
1
0
-
2
100
100
45
5001
5001
7.4343
×
1
0
-
2
100
100
45
2501
2501
6.9999
×
1
0
-
2
100
100
45
1001
1001
8.4363
×
1
0
-
2
100
100
45
501
501
6.4758
×
1
0
-
2
100
100
45
101
101
9.0737
×
1
0
-
2Table 16
Relation between the errorℰ and the number of formal powers N, when σ=2+sin5(x+y).
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
7.6129
×
1
0
-
2
100
100
35
10001
10001
1.3551
×
1
0
-
1
100
100
25
10001
10001
2.0886
×
1
0
-
1
100
100
15
10001
10001
3.3854
×
1
0
-
1
100
100
5
10001
10001
5.3026
×
1
0
-
1


## 4.1. The Nonseparable-Variables Exponential Case

Proposition 14.
Let the conductivity function(50)σ=exy.
Then, a particular solution of (1) will be
(51)u=e-xy.Figure3 illustrates the conductivity funcion of (50).Figure 3
Example of nonseparable-variables conductivity functionσ=exy.Tables7 and 8 contain the information about the behavior of the error ℰ when changing the values of S, C, and N. In this case the diminution of the number of sections S and diminution of the samples per section C do influence the accuracy, but their influence is not so significant as the diminution of the formal powers N.Table 8
Relation between the errorℰ and the number of formal powers N, when σ=exy.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
5.4177
×
1
0
-
6
100
100
35
10001
10001
2.6425
×
1
0
-
5
100
100
25
10001
10001
3.2504
×
1
0
-
5
100
100
15
10001
10001
3.4497
×
1
0
-
5
100
100
5
10001
10001
1.0798
×
1
0
-
2The error obtained when employing the finite element method approach wasℰ=2.3966×10-4, employing 8257 points in the mesh.


## 4.2. The Nonseparable-Variables Lorentzian Case

Proposition 15.
Let the conductivity function have the form(52)σ=1(x+y)2+1.
An exact solution for (1) is
(53)u=(x+y)33+x+y.The conductivity function with the form (52) is illustrated in Figure 4.Figure 4
Example of nonseparable-variables Lorentzian conductivity functionσ=((x+y)2+1)-1.Table9 shows an abnormal behavior of the error ℰ when decreasing the number of sections S and samples C. That is, it does not follow a clear pattern. But more important is to remark that the finite element method reported an error ℰ=5.4311×10-4, which implies that is better situated for analyzing this class of conductivity functions. Table 10 states that the diminution of the number of formal powers N decreases the accuracy.Table 9
Relation between the errorℰ and the number of sections S and samples per section C, when σ=((x+y)2+1)-1.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
8.8364
×
1
0
-
3
100
100
45
10001
10001
8.9246
×
1
0
-
3
100
100
45
5001
5001
9.4573
×
1
0
-
3
100
100
45
2501
2501
9.9164
×
1
0
-
3
100
100
45
1001
1001
1.0330
×
1
0
-
2
100
100
45
501
501
8.6507
×
1
0
-
3
100
100
45
101
101
9.5800
×
1
0
-
3Table 10
Relation between the errorℰ and the number of formal powers N, when σ=((x+y)2+1)-1.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values  per sectionC
ℰ
100
100
45
10001
10001
8.9246
×
1
0
-
3
100
100
35
10001
10001
4.3227
×
1
0
-
2
100
100
25
10001
10001
5.9625
×
1
0
-
2
100
100
15
10001
10001
8.2125
×
1
0
-
2
100
100
5
10001
10001
1.4064
×
1
0
-
1


## 4.3. The Nonseparable-Variables Polynomial Case

Proposition 16.
Let one assume that the conductivity function has the form(54)σ=x+y+10,
thus the function
(55)u=ln(x+y+10)
will be a solution of (1).In Figure5 it is illustrated the polynomial conductivity function of (54).Figure 5
Examples of nonseparable-variables polynomial conductivity functionσ=x+y+10.Tables11 and 12 show a regular behavior of the error ℰ, because it decreases when the values S and C do. This case possesses a singular characteristic when compared to the cases above: The diminution of the number of formal powers N does not provoke a considerable loss of the accuracy. This exception will be remarked since it has not been detected in other class of conductivity functions.Table 11
Relation between the errorℰ and the number of sections S and samples per section C, when σ=x+y+10.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
3.0640
×
1
0
-
6
100
100
45
10001
10001
2.3420
×
1
0
-
6
100
100
45
5001
5001
8.1962
×
1
0
-
6
100
100
45
2501
2501
1.9286
×
1
0
-
5
100
100
45
1001
1001
6.4916
×
1
0
-
5
100
100
45
501
501
1.5166
×
1
0
-
4
100
100
45
101
101
4.5938
×
1
0
-
4Table 12
Relation between the errorℰ and the number of formal powers N, when σ=x+y+10.
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
2.3420
×
1
0
-
6
100
100
35
10001
10001
1.0616
×
1
0
-
5
100
100
25
10001
10001
1.6275
×
1
0
-
5
100
100
15
10001
10001
1.8396
×
1
0
-
5
100
100
5
10001
10001
1.9909
×
1
0
-
5On the other hand, the finite element method proved to provide a very similar accuracy for analyzing this case, because the obtained error wasℰ=2.8958×10-6.


## 4.4. The Nonseparable-Variables Sinusoidal Case

Proposition 17.
Let one consider the sinusoidal conductivity(56)σ=2+sin(x+y).
One can verify by direct substitution that the function
(57)u=23arctan(2tan((x+y)/2)+13)
is a solution of (1).The Figure6(a) illustrates the conductivity (56), whereas the Figure 6(b) illustrates the case (58).Examples of sinusoidal conductivity functions.
(a)
σ
=
2
+
sin
(
x
+
y
)
(b)
σ
=
2
+
sin
5
(
x
+
y
)Tables13 and 14 show that the behavior of ℰ is not fully regular, because a diminution of the number S and C does not necessarily imply a reduction of the accuracy, whereas the diminution of the formal powers N provokes significant changes of the error ℰ. For this case, the finite element method reported an error ℰ=0.0192.Table 13
Relation between the errorℰ and the number of sections S and samples per section C, when σ=2+sin(x+y).
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
7.5333
×
1
0
-
2
100
100
45
10001
10001
7.6129
×
1
0
-
2
100
100
45
5001
5001
7.4343
×
1
0
-
2
100
100
45
2501
2501
6.9999
×
1
0
-
2
100
100
45
1001
1001
8.4363
×
1
0
-
2
100
100
45
501
501
6.4758
×
1
0
-
2
100
100
45
101
101
9.0737
×
1
0
-
2Table 14
Relation between the errorℰ and the number of formal powers N, when σ=2+sin(x+y).
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
7.6129
×
1
0
-
2
100
100
35
10001
10001
1.3551
×
1
0
-
1
100
100
25
10001
10001
2.0886
×
1
0
-
1
100
100
15
10001
10001
3.3854
×
1
0
-
1
100
100
5
10001
10001
5.3026
×
1
0
-
1Notice that the exact solution (57) that was imposed as the boundary condition u|Γ cannot be employed when a constant factor multiplies the argument of the trigonometric function, because the tangent would not be defined at certain points. Nevertheless, one last experiment is worth of research.Let us suppose that the conductivity function possesses the form(58)σ=2+sin5(x+y),
and let us approach the solution to the forward problem keeping as the boundary condition the expression (57). Tables 15 and 16 display the biggest error ℰ found in the present work, even the behavior of ℰ could be considered regular, because it decreases while S, C, and N do. The reader will appreciate that the accuracy is, in general, unsatisfactory for applications. Indeed, the finite element method obtained an error ℰ=0.0043, hence it is quite better situated for analyzing this kind of conductivity functions.Table 15
Relation between the errorℰ and the number of sections S and samples per section C, when σ=2+sin5(x+y).
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
20001
20001
7.5333
×
1
0
-
2
100
100
45
10001
10001
7.6129
×
1
0
-
2
100
100
45
5001
5001
7.4343
×
1
0
-
2
100
100
45
2501
2501
6.9999
×
1
0
-
2
100
100
45
1001
1001
8.4363
×
1
0
-
2
100
100
45
501
501
6.4758
×
1
0
-
2
100
100
45
101
101
9.0737
×
1
0
-
2Table 16
Relation between the errorℰ and the number of formal powers N, when σ=2+sin5(x+y).
Number of radiiR
Points per radiusP
Number of formal powersN
Number of sectionsS
Collected values per sectionC
ℰ
100
100
45
10001
10001
7.6129
×
1
0
-
2
100
100
35
10001
10001
1.3551
×
1
0
-
1
100
100
25
10001
10001
2.0886
×
1
0
-
1
100
100
15
10001
10001
3.3854
×
1
0
-
1
100
100
5
10001
10001
5.3026
×
1
0
-
1


## 5. Conclusions

The comparison of the numerical method posed in this work and the finite element method is far to be complete. Thus the selected examples will be considered a basic reference from which many other trials must be adequately performed, since they could not be included in these pages because they constitute independent research topics. For instance, a detailed description of the computational complexity [12] of both methods would provide a point of view less dependent on the platform on which the methods are programmed; thus we could focus our attention into the basic properties.Still, we will remark an important characteristic of the pseudoanalytic functions-based method. It provides information about the solutions of the forward Dirichlet boundary value problem for (1), that is not clear how to obtain when utilizing the finite element method.More precisely, the numerical approach based upon the pseudoanalytic analysis reaches a full set of functions defined at the boundary, that might contain relevant data if we desire to examine the electrical impedance tomography problem.For instance, let us consider the base functions{un}n=02N+1 approached for the case when σ=exy, as examined in (50). The absolute values of the coefficients αn, employed for approaching the boundary condition u|Γ=e-xy, are displayed in the semilogarithmic graphic of Figure 7. This calculation was performed considering S=20001 sections, C=10001 samples per section, R=100 radii, P=101 points per radii, and N=45 formal powers; thus we will have 2N+1=91 base functions un, defined at the boundary Γ.Figure 7
Absolute values of the coefficientsαn employed for approaching the boundary condition u|Γ=e-xy, utilizing the base elements {un} corresponding to the conductivity function σ=exy.Notice that in Figure7 the values of the coefficients αn possess a particular distribution, even we will not point an evident pattern. But since the set {un}n=02N+1 is an orthonormal base, it can be used to approach any other boundary condition. In particular, if we use this base for approaching the condition
(59)u|Γ=13(x+y)3+0.1(x+y),
that indeed corresponds to the conductivity σ=((x+y)2+0.1)-1, we will observe that the values of the coefficients αn, plotted in Figure 8, behave in a very different way.Figure 8
Absolute values of the coefficientsαn employed for approaching the boundary condition u|Γ=(1/3)(x+y)3 + 0.1(x+y), utilizing the base elements {un} corresponding to the conductivity function σ=exy.As a complementary example, let us consider the base functions{un} corresponding to a Lorentzian conductivity σ=((x+y)2+0.1)-1 studied in Section 4.2. Figure 9 displays the absolute values of the coefficients αn when we impose the boundary condition u|Γ=(1/3)(x+y)3+0.1(x+y), as posed in (53).Figure 9
Absolute values of the coefficientsαn employed for approaching the boundary condition u|Γ=(1/3)(x+y)3 + 0.1(x+y), utilizing the base elements {un} corresponding to the conductivity function σ=((x+y)2+0.1)-1.But if we impose a boundary condition of the form(60)u|Γ=e-xy,
upcoming from (51), we can appreciate, as plotted in Figure 10, that the absolute values of the obtained coefficients αn are, once more, very different to those of Figure 9.Figure 10
Absolute values of the coefficientsαn employed for approaching the boundary condition u|Γ=e-xy, utilizing the base elements {un} corresponding to the conductivity function σ=((x+y)2+0.1)-1.This implies that the behavior of the values of the coefficientsαn could well serve as a criterion for understanding if some conductivity function σ can effectively provoke a certain electric potential distribution at the boundary Γ of same domain Ω(ℝ2), when physical measurements provide the boundary condition.Taking into account that most of the algorithms dedicated to approach solutions of the electrical impedance tomography problem recursively solve the forward problem [2], this novel information will be used to introduce changes in σ; thus the boundary condition is fulfilled with a better convergence, and with less computational time.Even it is not possible yet to estimate when this new method will be employed to approach solutions for the electrical impedance tomography problem, we will remark that the main contribution of this work is to provide a new technique for fully applying the elements of the modern Pseudoanalytic function theory into the analysis of conductivity functions that originated from physical experimental models, which will certainly provide additional information for better understanding the inverse Dirichlet boundary value problem of (1) in the plane.In other words, before this proposal, the mathematical advances were exclusively improving the study of separable-variables conductivity functions, rarely useful for representing physical cases. These pages have shown that, under certain limits, any conductivity function is susceptible to be analyzed employing pseudoanalytic functions, whether they possess an exact mathematical form or not. As a matter of fact, it is enough that the conductivity values are fully defined over the points within a bounded domainΩ, with a smooth enough boundary Γ. The research will continue to better understand the limiting cases and the certain conditions of smoothness of the boundary; thus the posed ideas can be adequately employed in the applied sciences.



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*Source: 493483-2013-06-24.xml*