On explicitly solvable Vekua equations and explicit solution of the stationary Schrödinger equation and of the equation<b>div(σ∇u)=0</b>

Kravchenko, Vladislav V.; Oviedo, Hector

doi:10.1080/17476930601140343

Cited by 13 publications

(30 citation statements)

References 10 publications

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“…In the case of a real valued function f the relation between solutions of (18) and equations (20), (21) was observed in [20], and between solutions of (18) and equations (22), (23) in [18].…”

Section: Preliminariesmentioning

confidence: 99%

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On a factorization of second-order elliptic operators and applications

Kravchenko¹

2006

J. Phys. A: Math. Gen.

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We show that given a nonvanishing particular solution of the equation (div p grad +q)u = 0,the corresponding differential operator can be factorized into a product of two first order operators. The factorization allows us to reduce the equation (1) to a first order equation which in a two-dimensional case is the Vekua equation of a special form. Under quite general conditions on the coefficients p and q we obtain an algorithm which allows us to construct in explicit form the positive formal powers (solutions of the Vekua equation generalizing the usual powers (z − z0) n , n = 0, 1, . . .). This result means that under quite general conditions one can construct an infinite system of exact solutions of (1) explicitly, and moreover, at least when p and q are real valued this system will be complete in ker(div p grad +q) in the sense that any solution of (1) in a simply connected domain Ω can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of Ω. Finally we give a similar factorization of the operator (div p grad +q) in a multidimensional case and obtain a natural generalization of the Vekua equation which is related to second order operators in a similar way as its two-dimensional prototype does.

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Section: Preliminariesmentioning

confidence: 99%

“…Hence from the just proven equation (20) we obtain that W 1 is a solution of (22). In order to obtain equation (23) for W 2 it should be noted that…”

Section: Preliminariesmentioning

confidence: 99%

On a factorization of second-order elliptic operators and applications

Kravchenko¹

2006

J. Phys. A: Math. Gen.

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“…p-analytic functions found numerous applications in elasticity theory (see, e.g., [1,11,13]), in problems of fluid dynamics (see, e.g., [9,10,23,24,30]). The system describing p-analytic functions is closely related to the stationary Schrödinger equation and the conductivity equation [5,14,15,17,19,20]. One of the most important and intensely studied classes of p-analytic functions are x kanalytic functions, where k is any integer number (see, e.g., [1,6,9,12,21,22,25,29,30]).…”

Section: Introductionmentioning

confidence: 99%

On a transplant operator and explicit construction of Cauchy-type integral representations for p-analytic functions

Kravchenko

2008

Journal of Mathematical Analysis and Applications

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“…TWO-DIMENSIONAL CASE: PSEUDOANALYTIC APPROACH As it was explained in [17], recent discoveries in the theory of pseudoanalytic functions [11] has shown that equation (1), for the two-dimensional case, is completely equivalent to a special Vekua equation of the form…”

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confidence: 99%

“…The correspondence between W 1 and W 2 is one to one. Proposition 3: [11] Let W = W 1 + iW 2 be a solution of (9). Then the function…”

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confidence: 99%

On the advances of two and three-dimensional electrical impedance equation

Tachiquin

Nava

Jaso

et al. 2008

2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control

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We review the state of the art of solutions for the electrical impedance equation div( grad ') = 0;(1)where the function = + i!" is the admitivity, denotes the conductivity, ! is the frequency, " is the permittivity, i is the imaginary unit, and ' denotes the electric potential. Considering the three-dimensional case, we show that this equation is equivalent to a Schrödinger equation, and applying the algebra of quaternions, we study one method for factorizing (1) into two rst-order differential operators. We also discuss a method for rewriting equation (1) directly into a quaternionic rst-order linear differential equation and we cite some cases when it is possible to solve it analytically. For the two-dimensional case, we show that (1) is equivalent to a Vekua equation, and applying recently discovered properties of pseudoanalytic functions, we explore how to obtain solutions of (1) starting with solutions of a two-dimensional Schrödinger equation. Once we have discussed some elements of Pseudoanalytic Function Theory, we expose how to express the general solution of the Electrical Impedance Equation through Taylor series in formal powers, remarking the impact of this tools in such important questions as the Calderon problem in the plane.

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On explicitly solvable Vekua equations and explicit solution of the stationary Schrödinger equation and of the equationdiv(σ∇u)=0

Cited by 13 publications

References 10 publications

On a factorization of second-order elliptic operators and applications

On a factorization of second-order elliptic operators and applications

On a transplant operator and explicit construction of Cauchy-type integral representations for p-analytic functions

On the advances of two and three-dimensional electrical impedance equation

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