# Laplace’s Equation

• Abdul-Majid Wazwaz
Part of the Nonlinear Physical Science book series (NPS)

## Abstract

In Chapter 4 we have discussed the PDEs that control the heat flow in two and three dimensional spaces given by
$$\begin{gathered} u_t = \bar k(u_{xx} + u_{yy} ), \hfill \\ u_t = \bar k(u_{xx} + u_{yy} + u_{zz} ), \hfill \\ \end{gathered}$$
(7.1)
respectively, where $$\bar k$$ is the thermal diffusivity. If the temperature u reaches a steady state, that is, when u does not depend on time t and depends only on the space variables, then the time derivative u t vanishes as t→∞. In view of this, we substitute u t =0 into (7.1), hence we obtain the Laplace’s equations in two and three dimensions given by
$$\begin{gathered} u_{xx} + u_{yy} = 0, \hfill \\ u_{xx} + u_{yy} + u_{zz} = 0. \hfill \\ \end{gathered}$$
(7.2)
The Laplace’s equation [2,3,5] is used to describe gravitational potential in absence of mass, to define electrostatic potential in absence of charges [4,7], and to describe temperature in a steady-state heat flow. The Laplace’s equation is often called the potential equation  because u defines the potential function.

## Keywords

Neumann Problem Superposition Principle Order Differential Equation Homogeneous Boundary Condition Variational Iteration Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2009

## Authors and Affiliations

• Abdul-Majid Wazwaz
• 1
1. 1.Department of MathematicsSaint Xavier UniversityChicagoUSA

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