Using a quaternionic reformulation of the electrical impedance equation, we consider a two-dimensional separable-variables conductivity function and, posing two different techniques, we obtain a special class of Vekua equation, whose general solution can be approach by virtue of Taylor series in formal powers, for which is possible to introduce an explicit Bers generating sequence

Gutierrez S, Andrei

Ramirez, Marco P.

Rodriguez, Octavio

Sanchez N., V.D.

Digitale Bibliothek Thüringen

18th International Conference on the Application of ComputerScience and Mathematics in Architecture and Civil EngineeringK. Gu¨rlebeck and C. Ko¨nke (eds.)Weimar, Germany, 07–09 July 2009ON THE SOLUTIONS OF ELECTRICAL IMPEDANCE EQUATION: APSEUDOANALYTIC APPROACH FOR SEPARABLE-VARIABLESCONDUCTIVITY FUNCTIONA. Gutierrez S., M.P. Ramirez T., O. Rodriguez T., V.D. Sanchez N.Universidad La SalleBenjamı´n Franklin 47, Col. Hipo´dromo Condesa, Me´xico D.F. 06140E-mail: audreiev@gmail.com, mramirez@lci.ulsa.mxKeywords: Electrical impedance, quaternions, pseudoanalyticAbstract. Using a quaternionic reformulation of the electrical impedance equation, we con-sider a two-dimensional separable-variables conductivity function and, posing two differenttechniques, we obtain a special class of Vekua equation, whose general solution can be ap-proach by virtue of Taylor series in formal powers, for which is possible to introduce an explicitBers generating sequence.11 INTRODUCTIONThe study of the electrical impedance equationdiv (σgradu) = 0, (1)where σ is the conductivity function and u denotes the electric potential, is the base for well un-derstanding the electrical impedance tomography problem, also known as the two-dimensionalCalderon’s problem [3]. It is remarkable that in many classical works fully dedicated to thistopic, the authors could even think that it was impossible to express the general solution of (1)in analytic form (see e.g. [20, pag. 99]). But in 2006, K. Astala and L. Pa¨iva¨rinta [1] noticedfor the first time that the two-dimensional case of (1) was closely related with a Vekua equa-tion [19], and indeed it was through this relation that they solved the Calderon’s problem in theplane. As a consequence, in 2007 V. Kravchenko and H. Oviedo [11] used the theory devel-oped by L. Bers [2] to express the general solution of a two-dimensional electrical impedanceequation with a certain class of conductivity functions σ. In this work we review two differenttechniques that based onto a quaternionic differential equation completely equivalent to (1), al-low us to obtain a Vekua equation when considering the two-dimensional case, and using recentdiscoveries on the field of pseudoanalytic function theory [7], we show how to build the generalsolution for this equation in terms of Taylor series in formal powers, when the conductivityfunction σ is at least once differentiable in the plane, and it is separable-variables. This classof conductivity functions constitute an useful and quite common approach for the electricalimpedance tomography problem [5].2 PRELIMINARIES2.1 Elements of quaternionic analysisLet us denote the algebra of real quaternions by H (R) [6][10]. The elements belonging toH (R) have the form q =∑3k=0 qkek, where qk, k = 0, 3 are real-valued functions, e0 = 1 andek, k = 1, 3 are the standard quaternionic units possessing the multiplication propertiese1e2e3 = −1;e2k = −1, k = 1, 3.We will use the notation q = q0 + −→q , where q0 is the scalar part of q and −→q =∑3k=1 qkek isthe vectorial part of q. Notice every purely-vectorial quaternionic-valued function q = −→q canbe identified with a three-dimensional vectorial-valued function −→q ∈ R3, and that their relationis one-to-one. In virtue of this isomorphism, we can write the multiplication q = q0 + −→q andp = p0 +−→p as followsq · p = q0p0 + q0−→p + p0−→q − 〈−→q ,−→p 〉+ [−→q ×−→p ] .It is evident that, in general, q · p 6= p · q, thus we will use the notationMpq = p · qfor indicating the multiplication by the right-hand side of q by p. In the set of at least once-2differentiable quaternionic-valued functions, it is defined the Moisil-Theodoresco differentialoperator D [6]:D = e1∂∂x1+ e2∂∂x2+ e3∂∂x3.It acts onto a quaternionic-valued function q = q0 +−→q as followsDq = gradq0 − div−→q + rot−→q .2.2 Elements of pseudoanalytic functionsAccording to [2], let us consider a pair of complex valued functions (F,G) such thatIm(FG)> 0, (2)where F is the complex conjugation of F : F = ReF − iImF, and i is the standard imaginaryunit i2 = −1. Therefore, any complex-valued function W can be represented asW = φF + ψG,where φ and ψ are real-valued functions. A pair of functions satisfying (2) is called a Bersgenerating pair. The derivative in the sense of Bers or (F,G)-derivative of W is introduced asd(F,G)Wdz= (∂zφ)F + (∂zψ)G, (3)where ∂z = ∂∂x − i ∂∂y . This derivative will exist if and only if(∂zφ)F + (∂zψ)G = 0, (4)where ∂z = ∂∂x + i∂∂y. Let us introduce the the characteristic coefficients of the pair (F,G):A(F,G) = −F∂zG−G∂zFFG− FG , B(F,G) =F∂zG−G∂zFFG− FG , (5)a(F,G) = −F∂zG−G∂zFFG− FG , b(F,G) =F∂zG−G∂zFFG− FG .Using these notations, the (F,G)-derivative of W (3) can be written asd(F,G)Wdz= ∂zW − A(F,G)W −B(F,G)W, (6)and (4) becomes∂zW − a(F,G)W − b(F,G)W = 0. (7)This equation is called the Vekua equation [19]. The functions W fulfilling (7) are known as(F,G)-pseudoanalytic functions.The following statements were proposed in [2].Remark 1 The functions F and G of the generating pair (F,G) are (F,G)-pseudoanalytic,and their (F,G)-derivatives fulfilld(F,G)Fdz=d(F,G)Gdz= 0.3Definition 2 Let (F,G) and (F1, G1) be two generating pairs. If their characteristic coeffi-cients satisfy the relationsa(F,G) = a(F1,G1), B(F,G) = −b(F1,G1). (8)(F1, G1) will be called the successor pair of (F,G), as well (F,G) will be the predecessor pairof (F1, G1) .Theorem 3 Let W be a (F,G)-pseudoanalytic function, and let (F1, G1) be a successor pairof (F,G). Therefore the (F,G)-derivative of W will be a (F1, G1)-pseudoanalytic function.Definition 4 Let (F,G) be a generating pair. Its adjoint pair (F ∗, G∗) is defined by the formulasF ∗ = − 2FFG− FG, G∗ =2GFG− FG.It is also possible to introduced an (F,G)-integral of a complex-valued function W as∫ z1z0Wd(F,G)z = F (z1)Re∫ z1z0G∗Wdz +G (z1)Re∫ z1z0F ∗Wdz,such that if W = φF + ψG is (F,G)-pseudoanalytic, we will have that∫ zz0d(F,G)Wdzd(F,G)z = W (z)− φ (z0)F (z)− ψ (z0)G (z) .The last integral expression represents the antiderivative in the sense of Bers of the complex-valued functiond(F,G)Wdzsince the (F,G)-derivatives of F and G are zero.The complex-valued function w will be (F,G)-integrable iffRe∮G∗wdz + iRe∮F ∗wdz = 0.Theorem 5 The (F,G)-derivative of a (F,G)-pseudoanalytic function W is (F,G)-integrable.Theorem 6 Let the pair (F,G) be a predecessor of (F1, G1). A complex-valued function willbe (F1, G1)-pseudoanalytic iff it is (F,G)-integrable.Definition 7 Let {(Fm, Gm)} , m = 0,±1,±2,±3, ... be a sequence of generating pairs, andlet (Fm+1, Gm+1) be a successor of (Fm, Gm) . Thus we call {(Fm, Gm)} a generating sequence.If (F0, G0) = (F,G) we will say (F,G) to be embedded in {(Fm, Gm)} .Definition 8 The formal power Z(0)m (a, z0; z) with center at z0, coefficient a and exponent 0 isdefined as the linear combination of the elements of the generating pair Fm and Gm with realconstant coefficients λ and µ, satisfying the conditionλFm (z0) + µGm (z0) = a.The higher exponents n = 1, 2, ...are defined byZ(n)m (a, z0; z) = n∫ zz0Z(n−1)m+1 (a, z0; ς) d(Fm,Gm)ς.4The formal powers possesses the following properties.1. Z(n)m (a, z0; z) is (Fm, Gm)-pseudoanalytic.2. For a1 and a2 being real constants, we haveZ(n)m (a1 + ia2, z0; z) = a1Z(n)m (1, z0; z) + a2Z(n)m (i, z0; z) .3. The asymptotic formulas hold:limz→z0Z(n)m (a, z0; z) = a (z − z0)n .Remark 9 As shown in [2], any complex-valued function W, satisfying (7), accepts the expan-sionW =∞∑n=0Z(n) (an, z0; z) , (9)where the missing subindex m means that all formal powers belong to the generating pair(F,G). This is: expression (9) is an analytic representation of the general solution of (7).Definition 10 [14] A function Φ = φ+iψ of a complex variable z = x+iy is called p -analyticif∂∂xφ =1p∂∂yψ and∂∂yφ = −1p∂∂xψ. (10)Theorem 11 [8] The complex-valued functionW = φ · √p+ ψ · i√pwill be a solution of the Vekua equation∂zW −∂z√p√pW = 0 (11)if and only if the real-valued functions φ and ψ satisfy (10).Theorem 12 [7] Let (F,G) be a generating pair of the formF =√p = U(x)V (y) ,G =i√p=iU(x)V (y).Thus this generating pair is embedded in the generating sequence {(Fm, Gm)} , m = 0,±1,±2,±3, ...defined asFm = 2mU(x)V (y) ,Gm = i2mU(x)V (y);for even m, andFm =2mU(x1)V (x2) ,Gm = i2mV (x2)U(x1)for odd m.53 FROM THE QUATERNIONIC ELECTRICAL IMPEDANCE EQUATION TO THEVEKUA EQUATIONAs it was shown in [13],[15],[17] and [18], when we introduce the notations−→E = √σgradu, (12)−→σ = grad√σ√σ,the equation (1) turns into (D +M−→σ)−→E = 0. (13)Essentially, we will study two different paths to obtain the Vekua equation from (13). The firstone [16] is to consider at once−→E = E1e1 + E2e2,σ = σ (x1, x2)then, introducing the notationσ1 =1√σ∂∂x1√σ, σ2 =1√σ∂∂x2√σ, (14)the function −→σ in (12) will take the form−→σ = σ1e1 + σ2e2,thus from (13) we will haveD (E1e1 + E2e2) + (E1e1 + E2e2) (σ1e1 + σ2e2) = 0,from which the following system is obtained∂∂x1E1 + ∂∂x2E2 = −E1σ1 − E2σ2,∂∂x1E2 − ∂∂x2E1 = E3σ1 − E1σ2,∂∂x3E1 = ∂∂x3E2 = 0.Multiplying the second equation by −i and adding to the first, it follows(∂∂x1+ i∂∂x2)(E1 − iE2) + (σ1 − iσ2) (E1 − iE2) = 0,but according to (14)σ1 − iσ2 = 1√σ(∂∂x1− i ∂∂x2)√σ.6Considering this, and introducing the notationE = E1 − iE2, (15)∂z1 =∂∂x1− i ∂∂x2∂z1 =∂∂x1+ i∂∂x2we have∂z1E +∂z1√σ√σE = 0. (16)This is a Vekua equation for which we are able to build the general solution in terms of Taylorseries in formal powers, as we will show further, for the case when σ is a separable-variablesfunction. We must mention that a similar Vekua equation, but considering a bicomplex case,had been already deduced in [4] from a quaternionic Dirac equation corresponding to a massiveparticle under the influence of a special class of potentials. Let us study now a secund methodfor obtaining a Vekua equation from (1).In [12] is given the proof of the following statement.Theorem 13 Let the purely-vectorial quaternionic-valued functions−→E 1,−→E 2 and −→E 3 be linearindependent, and be all solutions of (13). Let the real scalar functions ϕ1, ϕ2 and ϕ3 be allsolutions of the equation3∑k=1(Dϕk)−→E k = 0. (17)Hence, the purely-vectorial quaternionic-valued function−→E =3∑k=1ϕk−→E k (18)is the general solution of (13).It is easy to check [15] that the following quaternionic-valued functions constitute a set ofthree linear independent solutions for the equation (13).−→E 1 = e1K1e−Rσ1dx1+Rσ2dx2 ,−→E 2 = e2K2eRσ1dx1−Rσ2dx2 ,−→E 3 = e3K3eRσ1dx1+Rσ2dx2 ,where K1, K2 and K3 are real constants, and σ1, σ2 have the form (14). Let us consider the casewhen ϕ3 = 0 in (17), which precisely corresponds, as we will see, to the two-dimensional caseof (14). We have thenDϕ1 · −→E 1 +Dϕ2 · −→E 2 = 0.Expanding the expression, we obtain the following system of differential equations∂ϕ1∂x2=1p1∂ϕ2∂x1,∂ϕ1∂x1= − 1p1∂ϕ2∂x2,∂ϕ1∂x3=∂ϕ2∂x3= 0;7where p1 = Ke−2Rσ1dx1+2Rσ2dx2 and K is a real constant. The first pair of equations constitutethe differential system of the so called p-analytic functions, introduced in Definition 10, and byTheorem 11, its equivalent Vekua equation will have the form∂z2W −∂z2√p1√p1W = 0, (19)where ∂z2 = ∂∂x2 + i∂∂x1andW = ϕ1p1 + iϕ2p1,that in fact we can also write asW =W1 + iW2,where W1 and W2 are real-valued functions.An small but interesting detail that was not pointed out in [15] is that∂z2√p1√p1= −i (σ1 + iσ2) .Substituting this into (19) we have∂z2W + i (σ1 + iσ2)W = 0,but in fact∂z2 = i(∂∂x1− i ∂∂x2),therefore (∂∂x1− i ∂∂x2)W + (σ1 + iσ2)W = 0.Considering the complex conjugation of this expression we have(∂∂x1+ i∂∂x2)W + (σ1 − iσ2)W = 0,and taking into account the notations (15), by simply identifying W1= E ′1 and W2= E ′2, the lastequation turns into∂z1E ′ +∂z1√σ√σE ′ = 0, (20)where E ′ = E ′1 − iE ′2, a completely equivalent equation to (16).3.1 Building the general solution for the two-dimensional electrical impedance equationwhen the conductivity function is separable-variablesFirst of all, we should notice when we select a generating pair [9] [11]F =√σ, G =i√σ; (21)8the corresponding characteristic coefficients (6) becomeA(F,G) = a(F,G) = 0,B(F,G) =∂z1√σ√σ,b(F,G) =∂z1√σ√σ.Therefore the Vekua equation will have the form∂z1W −∂z1√σ√σW = 0. (22)According to Definition 2, the successor pair (F1, G1) of (21) must have characteristic coeffi-cientsa(F1,G1) = 0,b(F1,G1) = −∂z1√σ√σ,and by virtue of Theorem 3, the derivative in the sense of Bers d(F,G)Wdzof a (F,G)-pseudoanalyticfunction W solution of (22) will be (F1, G1)-pseudoanalytic. This is, it will be a solution of theVekua equation∂z1(d(F,G)Wdz)−B(F1,G1)(d(F,G)Wdz)= 0,or more precisely∂z1(d(F,G)Wdz)+∂z1√σ√σ(d(F,G)Wdz)= 0.Considering (15), by simply denotingE = d(F,G)Wdz,the last Vekua equation becomes∂z1E +∂z1√σ√σE = 0, (23)which evidently coincides with the Vekua equations (16) and (20) corresponding to the two-dimensional electrical impedance equation (1).For the case when σ = U2 (x1)V 2 (x2) , a very important and general case in the fieldof electrical impedance tomography, according to Theorem 12, we can introduce an explicitgenerating sequence {(Fm, Gm)} , m = 0,±1,±2,±3, ... in which the generating pairF =√σ = U (x1)V (x2) ,G =i√σ=iU (x1)V (x2),9is embedded. Therefore, by Definitions, Theorems and Remarks 2-9, we are able to express thegeneral solution of (22) in term of Taylor series in formal powersW =∞∑n=0Z(n) (a, z0, z) ,where z = x1 + ix2, and by Theorem 3, the (F,G)-derivative of W will be the general solutionof (23). It follows, using (12), to obtain the electric potential u, which will be the generalsolution of (1).REFERENCES[1] K. Astala, L. Pa¨iva¨rinta, Calderon’s inverse conductivity problem in the plane, Annals ofMathematics, Vol. 163, pp. 265-299, 2006.[2] L. Bers Theory of pseudoanalytic functions, IMM, New York University, 1953.[3] A.P. Calderon, On an inverse boundary value problem, Seminar on Numerical Analysisand its Applications to Continuum Physics, Soc. Brasil. Mat., pp. 65-73, 1980.[4] A. Castan˜eda, V. Kravchenko, New applications of pseudoanalytic function theory toDirac equation, Journal of Physics A: Mathematical and General, Vol. 38, pp. 9207-9219,2005.[5] E. Demidenko, Separable Laplace equation, magic Toeplitz matrix, and generalizedOhm’s law, Applied Mathematics and Computation, pp. 1313-1327, Elsevier, 2007.[6] K. Gu¨rlebeck, W. Spro¨ssig, Quaternionic analysis and elliptic boundary value problems.Berlin: Akademie-Verlag, 1989.[7] V. V. Kravchenko, Applied Pseudoanalytic Function Theory, Series: Frontiers in Mathe-matics, ISBN: 978-3-0346-0003-3, 2009.[8] V. V. Kravchenko, Recent developments in applied pseudoanalytic function theory, Bei-jing: Science Press ”Some topics on value distribution and differentiability in complexand p-adic analysis”, eds. A. Escassut, W. Tutschke and C. C. Yang, 267-300, 2008.[9] V.V. Kravchenko, On the relation of pseudoanalytic function theory to the two-dimensional stationary Schro¨dinger equation and Taylor series in formal powers for itssolutions, Journal of Physics A: Mathematical and General, Vol. 38, No. 18, pp. 3947-3964, 2005.[10] V. Kravchenko, Applied Quaternionic Analysis, Researches and Exposition in Mathemat-ics, Vol. 28, Heldermann Verlag, 2003.[11] V. Kravchenko, H. Oviedo, On explicitly solvable Vekua equations and explicit solutionof the stationary Schro¨dinger equation and of the equation div (σ∇u) = 0, ComplexVariables and Elliptic Equations, Vol. 52, No. 5, pp. 353-366, 2007.10[12] V.V. Kravchenko, M. P. Ramirez, On Bers generating functions for first order differentialequations of Mathematical Physics (in process of preparation).[13] V. Kravchenko, M.P. Ramirez T., A quaternionic reformulation of Maxwell’s equationsin inhomogeneous media, First International Workshop on Mathematical Modeling onPhysical Processes in Inhomogeneous Media, Universidad de Guanajuato, IEEE, 2001.[14] G. N. Polozhy, Generalization of the theory of analytic functions of complex variables: p-analytic and (p, q)-analytic functions and some applications. Kiev University Publishers(in Russian), 1965.[15] M.P. Ramirez T., O. Rodriguez T., J. J. Gutierrez C., New Exact Solutions for the Three-Dimensional Electrical Impedance Equation with a Separable-Variables ConductivityFunction, 6th International Conference on Electrical Engineering, Computing Science andAutomatic Control (CCE), Mexico City, Mexico, 2009 (in process of acceptation).[16] M.P. Ramirez T., V.D. Sanchez Nava, O. Rodriguez Torres, A. Gutierrez S., On the generalsolution of the two-dimensional electrical impedance equation for a separable-variablesconductivity function, Proceedings of World Congress on Engineering, IAENG ISBN:978-988-17012-5-1, U.K., 2009 (to be published).[17] M.P. Ramirez Tachiquin, V.D. Sanchez Nava, A. Fleiz Jaso, O. Rodriguez Torres, On theadvances of two and three-dimensional Electrical Impedance Equation, 5th InternationalConference on Electrical Engineering, Computing Science and Automatic Control (CCE),Me´xico City, Mexico, (2008), IEEE Catalog Number: CFP08827-CDR, ISBN: 978-1-4244-2499-3, Library of Congress: 2008903800, 978-1-4244-2499-3/08,IEEE.[18] M.P. Ramirez T., V.D. Sanchez Nava, A. Fleiz Jaso, On the solutions of the ElectricalImpedance Equation, applying Quaternionic Analysis and Pseudoanalytic Function The-ory, 12-th International Conference on Mathematical Methods in Electromagnetic Theory,IEEE: CFP08761-PRT, ISBN: 978-1-4244-2284-5, pp. 190-192, Ukraine, 2008.[19] I.N. Vekua, Generalized Analytic Functions, International Series of Monographs on Pureand Applied Mathematics, Pergamon Press, 1962.[20] J.G. Webster, Electrical Impedance Tomography, Adam Hilger, Bristol and New York,1990.11

ON THE SOLUTIONS OF ELECTRICAL IMPEDANCE EQUATION: A PSEUDOANALYTIC APPROACH FOR SEPARABLE-VARIABLES CONDUCTIVITY FUNCTION

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ON THE SOLUTIONS OF ELECTRICAL IMPEDANCE EQUATION: A PSEUDOANALYTIC APPROACH FOR SEPARABLE-VARIABLES CONDUCTIVITY FUNCTION

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