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Numerical treatment of a skew-derivative problem for the Laplace equation in the exterior of an open arc. (English) Zbl 1126.78018
The paper presents a numerical solution for the two-dimensional Laplace equation in the exterior of an open are with skew-derivative boundary condition. This direct problem requires solving for the electric field potential $$u$$ belonging to a certain class of functions and satisfying the Laplace equation $$\nabla^2u= 0$$ in the exterior of an open arc $$\Gamma\in{\mathcal C}^{2,\lambda},\lambda\in(0, 1]$$ parametrized by the arc length $$s:\Gamma=\{x=x(s)=(x_1(s),x_2(s)),s\in[a,b] \}$$, subject to the skew-derivative boundary condition
$\frac{\partial u} {\partial n}+\beta\frac{\partial u}{\partial\tau}=f\quad\text{on }\Gamma,$ where $$\beta$$ is a real given constant, $$\underline n=(\sin(\alpha(s)),-\cos (\alpha(s)))=(x_2'(s),-x_1'(s))$$ is the normal vector to $$\Gamma$$ at $$x(s)$$, $$\tau=(\cos(\alpha(s)),\sin(\alpha(s)))=(x_1'(s),x_2'(s))$$ is the tangent vector to $$\Gamma$$ at $$x(s)$$, and $$f\in C^{0,\lambda} ([a,b])$$ is a given function, and to the infinity conditions
$|u(x)|\leq\text{Const}., \quad |\nabla u(x)|\leq o(|x|^{-1}),\quad\text{as}\quad |x|=\sqrt {x^2_1+x^2_2}\to\infty.$ The numerical method is based on an analytic evaluation of the singular part in the single-layer potential and the use of the angular potential instead of the double-layer potential. Numerical results are presented and discussed.

##### MSC:
 78M15 Boundary element methods applied to problems in optics and electromagnetic theory 82D37 Statistical mechanical studies of semiconductors
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##### References:
  Lifanov IK (1996) Singular integral equations and discrete vortices. Zeist. VSP, International Sceience Publishers, The Netherlands  Lifanov IK, Poltavskii LN, Vainikko GM (2004) Hypersingular integral equations and their applications. CRC Press, Boca Raton · Zbl 1061.45001  Hall EH (1879) On a new action of the magnet on electric currents. Am J Maths 2:287 · JFM 11.0767.01 · doi:10.2307/2369245  Putley EH (1960) The Hall effect and related phenomena. Butterworth, London  Seeger K (1973) Semiconductor physics. Springer, NY  Sze SM (1981) Physics of semiconductor devices. Wiley, NY  Krutitskii PA, Krutitskaya NCh, Malysheva GYu (1999) A problem related to the Hall effect in a semiconductor with an electrode of an arbitrary shape. Math Prob Eng 5:83–95 · Zbl 0929.35154 · doi:10.1155/S1024123X99001003  Gabov SA (1977) An angular potential and its applications. Math USSR Sbornik 32:423–436 · Zbl 0396.31002 · doi:10.1070/SM1977v032n04ABEH002396  Krutitskii PA (1994) Dirichlet problem for the Helmholtz equation outside cuts in a plane. Comp Maths Math Phys 34:1073–1090 · Zbl 0835.35033  Muskhelishvili NI (1968) Singular Integral Equations. Nauka, Moscow and Noordhoff, Groningen (1972) · Zbl 0174.16202  Krutitskii PA (1994) Neumann problem for the Helmholtz equation outside cuts in a plane. Comp Maths Math Phys 34:1421–1431 · Zbl 0835.35034  Belotserkovskii SM, Lifanov IK (1993) Method of discrete vortices. CRC Press, Boca Raton  Vainikko G (2001) Fast solvers of generalized airfoil equation of index 1. Operator theory: advances and applications 121:498–516 · Zbl 1004.65145  Saranen J, Vainikko G (2002) Periodic integral and pseudodifferential equations with numerical approximation. Springer, Berlin · Zbl 0991.65125
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