Laplace’s Equation

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


In Chapter 4 we have discussed the PDEs that control the heat flow in two and three dimensional spaces given by
$$ \begin{gathered} u_t = \bar k(u_{xx} + u_{yy} ), \hfill \\ u_t = \bar k(u_{xx} + u_{yy} + u_{zz} ), \hfill \\ \end{gathered} $$
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
$$ \begin{gathered} u_{xx} + u_{yy} = 0, \hfill \\ u_{xx} + u_{yy} + u_{zz} = 0. \hfill \\ \end{gathered} $$
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 [10] because u defines the potential function.


Neumann Problem Superposition Principle Order Differential Equation Homogeneous Boundary Condition Variational Iteration Method 
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Copyright information

© 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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