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The Laplace Equation and Wave Equation

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Book cover Partial Differential Equations I

Part of the book series: Applied Mathematical Sciences ((AMS,volume 115))

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Abstract

In this chapter we introduce the central linear partial differential equations of the second order, the Laplace equation

$$\Delta u = f$$
(0.1)

and the wave equation

$$\left (\frac{{\partial }^{2}} {\partial {t}^{2}} -\Delta\right )u = f.$$
(0.2)

For flat Euclidean space \(\mathbb{R}^{n}\), the Laplace operator is defined by

$$\Delta u =\frac{{\partial }^{2}u} {\partial {x}_{1}^{2}} +\cdots +\frac{{\partial }^{2}u} {\partial {x}_{n}^{2}}.$$
(0.3)

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA

    Michael E. Taylor

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  1. Michael E. Taylor
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Correspondence to Michael E. Taylor .

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Taylor, M.E. (2011). The Laplace Equation and Wave Equation. In: Partial Differential Equations I. Applied Mathematical Sciences, vol 115. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7055-8_2

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