Question

Define the laplace equation in 2d and 3d also comment regarding the classification of Laplace equation and its application?

Answer 1

Laplace equation can be written as

$\nabla^2f = 0 \; \textrm{or} \; \Delta f=0$

In 2D in Cartesian coordinates this can be re-written as

$\displaystyle \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2}=0$

In 3D Laplace equation is

$\displaystyle \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2}+ \frac{\partial^2f}{\partial z^2}=0$

Classification. Laplace equation is second-order partial differential equation. By the classification of quasi-linear second order PDE, Laplace equation is elliptic.The solutions of Laplace's equation are the harmonic functions, which are important in many fields of science, including electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat equation.