# Article

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Keywords:
quadrature formula method; surface phenomenon; skin effect; existence; uniqueness
Summary:
The paper is a mathematical background of the paper of D. Mayer, B. Ulrych where the mathematical model of the skin effect is established and discussed. It is assumed that the currents passing through parallel conductors are under effect of a variable magnetic field. The phasors of the density of the current are solutions of $f(x)-jq \sum{^k_{i=1}}b_i \int_{Si} f(y) V(|y-x|)dy-c_p=h(x)$ for $x\in S_p, p=1,\dots,k$, $\int_{Si}f(x)dx=I_i, i=1,\dots, k$, where $j$ is the imaginary unit, $b_i,q,I_i$ are given constants, #h(x)$is a given function and$f(x)$is an unknown function and$c_i\$ are unknown constants. The first and the second section of this paper are devoted to the problem of existence and unicity of a solution. The third section is devoted to a numerical method.
References:
[1] D. Mayer B. Ulrych: Integrální model povrchového jevu a jeho numerické řešení. Elektrotechnický časopis 35, 1984, č. 10.
[2] L. Liusternik V. Sobolev: Elements of Functional Analysis. New York: Frederick Ungar Publishing Company Inc. 1961. MR 0141967
[3] M. В. Канторович В. И. Крылов: Приближенные методы высшего анализа. Гос. Изд. Тех.-Теор. Лит., Москва 1952. Zbl 1225.62076
[4] L. Collatz: Numerische Behandlung von Differentialgleichungen. Springer Verlag, Berlin- Göttingen-Heidelberg 1951. MR 0043563 | Zbl 0054.05101
[5] D. A. H. Jacobs ed.: The State of the Art in Numerical Analysis. Acad. Press, London-New York-San Francisco 1977. MR 0440858

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