Abstract
This chapter is devoted to the theory of linear elliptic partial differential equations and the related problems of potential theory. The basic concept of the Green’s function and the source solution are introduced and explored. This is followed by a detailed analysis of the Helmholtz equation in one, two, and three dimensions with applications to the diffraction problem for monochromatic waves. The inhomogeneous media case sets the stage for the Helmholtz equation with a variable coefficient and an analysis of waves in waveguides. The latter can be reduced to the celebrated Sturm–Liouville problem, and we study properties of its eigenvalues and eigenfunctions.
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Notes
- 1.
As in Volume 1, ϕ stands, usually, for an infinitely differentiable test function with compact support contained in V (in this case).
- 2.
Recall that G 0 appears here as a function of the Euclidean distance, so it is also a function of two variables.
- 3.
See, e.g., the classic monograph Higher Transcendental Functions by H. Bateman and A. Erdélyi, Mc Graw-Hill, Inc., New York 1953.
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Saichev, A.I., Woyczyński, W.A. (2013). Potential Theory and Fundamental Solutions of Elliptic Equations. In: Distributions in the Physical and Engineering Sciences, Volume 2. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-4652-3_1
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DOI: https://doi.org/10.1007/978-0-8176-4652-3_1
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Publisher Name: Birkhäuser, New York, NY
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Online ISBN: 978-0-8176-4652-3
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