Solution to the mixed boundary-value problem for Laplace equation in multidimentional infinite layer

Authors: Algazin O.D., Kopaev A.V. Published: 08.02.2015
Published in issue: #1(58)/2015  
DOI: 10.18698/1812-3368-2015-1-3-13

Category: Mathematics and Mechanics  
Keywords: harmonic functions, Fourier transform, tempered distributions, filtration theory

The paper considers the solution to the mixed boundary-value problem of finding a harmonic function of n variables for the domain confined by two parallel hyperplanes. This function was determined by its values on a hyper-plane and its normal derivative on another hyper-plane. The obtained solution is presented as a sum of two integrals which kernels are expressed only in terms of elementary functions in the case of the even-dimension space. In contrast to the odd-dimension space they are also expressed through the Bessel functions. If the given boundary values are tempered distributions, then the solution is written as a convolution of the kernels with these functions. The opportunity of practical application of the obtained formulas is illustrated by the example of forming up a filtration flow under the spot dam with a aquiclude.


[1] Komech A.I. Linear partial differential equations with constant coefficients. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat.: Fund. Napr. [Results of science and technology. Ser. "Modern Problems of Mathematics: Fundamental Directions"], 1988, vol. 31, pp. 127-261 (in Russ.). Available at: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intf&paperid=113&option_lang=eng (accessed 01.11.2014).

[2] Mikhlin S.G. Lineynye uravneniya v chastnykh proizvodnykh [Linear partial differential equations]. Moscow, Vysshaya Shkola Publ., 1977. 430 p.

[3] Bitsadze A.V. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow, Fizmatgiz Publ., 1976. 296 p. (Engl. ed.: Bitsadze A.V. Equations of mathematical physics. Moscow, Mir Publishers, 1980. 318 p.). Available at: http://www.amazon.com/Equations-Mathematical-Physics-A-V-Bitsadze/dp/0714715433 (accessed 01.11.2014).

[4] Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki [Equations of mathematical physics]. Moscow, MGU Publ., 1999. 798 p. (Engl. ed.: Tikhonov A.N., Samarskii A.A. Equations of mathematical physics. Dover Publications; Reprint edition, 2011. 800 p.). Available at: http://www.amazon.com/Equations-Mathematical-Physics-Dover-Books/dp/0486664228 (accessed 01.11.2014).

[5] Polyanin A.D. Spravochnik po lineynym uravneniyam matematicheskoy fiziki [Handbook of linear equations of mathematical physics]. Moscow, Fizmatlit Publ., 2001. 576 p. (Engl. ed.: Polyanin A.D. Handbook of linear partial differential equations for engineers and scientists. USA, Chapman and Hall/CRC, 2001. 800 p.).

[6] Kas’yanov E.Yu., Kopaev A.V. On the solution of the Dirichlet problem for some multidimensional domains by the method of reproducing kernels. Izv. Vyssh. Uchebn. Zaved., Mat. [Russ. Math.], 1991, no. 6, pp. 17-20 (in Russ.).

[7] Vladimirov V.S. Obobshchennye funktsii v matematicheskoy fizike [Tempered distributions in mathematical physics]. Moscow, Nauka Publ., 1979. 320 p.

[8] Bochner S. Vorlesungen bber Fouriersche Integrale. Leipzig, Akademische Verlagsgesellschaft m.b. H. VIII, 1932. 229 p. (Engl. ed.: Bochner S. Lectures on Fourier integrals. Trans. from the Ger. Princeton Uni. Press, 1959. 333 p. Russ. ed.: Bokhner S. Lektsii ob integralakh Fur’e [Lectures on Fourier integrals]. Moscow, Fizmatgiz Publ., 1962. 360 p.).

[9] Ditkin V.A., Prudnikov A.P. Integral’nye preobrazovaniya i operatsionnoe ischislenie [Integral transforms and operator calculus]. Moscow, Fizmatgiz Publ., 1961. 524 p.

[10] Radygin V.M., Golubeva O.V. Primenenie funktsiy kompleksnogo peremennogo v zadachakh fiziki i tekhniki [Application of function of complex variable in physics and engineering problems]. Мoscow, Vysshaya Shkola Publ., 1983. 160 p.

[11] Golubeva O.V. Kurs mekhaniki sploshnykh sred [The course of continuum mechanics]. Мoscow, Vysshaya Shkola Publ., 1972. 367 p.