Abstract
In this chapter, the discussion is made over topics about modelling, analysis and interpretation related to Poisson’s equation, with Laplace’s equation as its special case. First, a number of physical phenomena that can be described by Poisson’s equation are examined, and their physical analogies are summarised then. To solve PDE problems for Poisson’s equation, the concepts of variational principle and Green’s function are introduced in turn. In the end of this chapter, the issue of solution uniqueness is looked into in the context of Poisson’s equation, and the corresponding (strong) maximum principle is mentioned.
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Notes
- 1.
Since \(\mathbf {r} = \rho (\sin \theta \cos \varphi , \sin \theta \sin \varphi , \cos \theta )^{\mathrm {T}}\), \(\mathbf {r}^* = \rho ^* (\sin \theta ^*\cos \varphi ^*, \sin \theta ^*\sin \varphi ^*, \cos \theta ^*)^{\mathrm {T}}\) the angle between them given by \(\gamma \) satisfies
$$ \cos \gamma = \cos \theta \cos \theta ^* + \sin \theta \sin \theta ^* \cos (\varphi - \varphi ^*). $$.
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Zhu, Y. (2021). Poisson’s Equation. In: Equations and Analytical Tools in Mathematical Physics. Springer, Singapore. https://doi.org/10.1007/978-981-16-5441-1_3
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DOI: https://doi.org/10.1007/978-981-16-5441-1_3
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Publisher Name: Springer, Singapore
Print ISBN: 978-981-16-5440-4
Online ISBN: 978-981-16-5441-1
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