Mathematical Background

  • Balgaisha Mukanova
  • Igor Modin
Part of the Innovation and Discovery in Russian Science and Engineering book series (IDRSE)


In this chapter, we provide well-known mathematical facts that we rely upon in the following discussion. We give elements of the potential theory for Laplace’s equation and then formulate the most important theorems of solvability of integral equations. This is followed by consideration of integral transforms, which are the most useful in geophysical problems. More detailed exposition of the material in this chapter can be found in the cited literature on integral equations, equations of mathematical physics, and functional analysis.


Simple layer potential Laplace’s equation Integral equation Integral transforms Solvability of integral equations 


  1. B.K. Driver, Compact and Fredholm operators and the spectral theorem. In Analysis Tools with Applications, Chapter 35 (Springer, Berlin, 2003), pp. 579–600Google Scholar
  2. N.M. Gunther, La théorie Du Potentiel et Ses Applications Aux problèmes Fondamentaux de la Physique Mathématique (Gauthier-Villars, Paris, 1934)Google Scholar
  3. I.G. Petrovskii, Lekcii po teorii integral’nykh uravneniy (Nauka, Moscow, 1965)Google Scholar
  4. A.N. Tikhonov, A.A. Samarskii, Equations of Mathematical Physics. (Translation from Russian) (Pergamon, 1963)Google Scholar
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  6. V.S. Vladimirov, Equations of Mathematical Physics (Marcel Dekker, New York, 1971)Google Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  • Balgaisha Mukanova
    • 1
  • Igor Modin
    • 2
  1. 1.L.N. Gumilyov Eurasian, National UniversityAstanaKazakhstan
  2. 2.Lomonosov Moscow State UniversityMoscowRussia

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