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Keywords:
exact solutions; mixed problems; half-space; harmonic function; Integral representations
exact solutions; mixed problems; half-space; harmonic function; Integral representations
Summary:
A new and elegant procedure is proposed for the solution of mixed potential problems in a half-space with a circular line of division of boundary conditions. The approach is based on a new type of integral operators with special properties. Two general external problems are solved; i) An arbitrary potential is specified at the boundary outside a circle, and its normal derivative is zero inside; ii) An arbitrary normal derivative is given outside the circle, and be potential is zero inside. Several illustrative examples are considered. Certain methods of application of the proposed technique to the solution of a few complex problems are also discussed.
References:
[1] I. N. Sneddon: Mixed boundary value problems in potential theory. North-Holland Publishing Company, Amsterdam, 1966. MR 0216018 | Zbl 0139.28801
[2] T. S. Sankar V. I. Fabrikant: Investigations of a two-dimensional integral equation in the theory of elasticity and electrostatics. Journal de Mécanique Théorique et Appliquée, Vol. 2, No. 2, 1983, pp. 285-299. MR 0702239
[3] I. S. Gradshtein I. M. Ryzhik: Table of Integrals, Series and Products. AP, New York, 1965.
[4] H. Bateman A. Erdelyi: Higher transcendental functions. Vol. 1, McGraw-Hill, 1953. MR 0058756
[5] E. W. Hobson: On Green's function for a circular disk, with application to electrostatic problems. Transactions of Cambridge Philosophical Society, Vol. 18, 1900, pp. 277-291.
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