8.1: Laplace Equations

Introduction

The following partial differential equation is called the two-dimensional Laplace equation:

$$\begin{equation} \displaystyle \frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{\partial y^{2}}=0 \label{laplace}\end{equation}$$

where $w(x,\ y)$ is the unknown function with two variables $x$ and $y$. The problem is to find a solution to this equation, namely, find a function $w(x,\ y)$ which satisfies the Equation $\ref{laplace}$. This equation is used to model various physical quantities.

Example $\PageIndex{1}$

Let $k$ be a real number. Show that the functions $w=e^{kx} cos(ky)$ and $w=e^{kx} sin(ky)$ satisfy the lapalce Equation $\ref{laplace}$ at every point in $\mathbb {R^2}$.

Solution

Let $w=e^{kx} cos(ky)$. Then we have,
$\dfrac{\partial w}{\partial x}=ke^{kx}\cos(ky),\ \dfrac{\partial w}{\partial y}=-ke^{kx}\sin(ky),$
which implies $\displaystyle \frac{\partial^{2}w}{\partial x^{2}}=k^{2}e^{kx}\cos(ky),\ \displaystyle \frac{\partial^{2}w}{\partial y^{2}}=-k^{2}e^{kx}\cos(ky)$.

Consider $\displaystyle \frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{\partial y^{2}}=k^{2}e^{kx}\cos(ky)-k^{2}e^{kx}\cos(ky)=0$.

Therefore, that the function $w=e^{kx}\cos(ky)$ satisfies the Equation $\ref{laplace}$. Similarly, the function $w=e^{kx}\sin(ky)$ satisfies the Equation $\ref{laplace}$.

Definition

A function $w(x,y)$ of two variables having continuous second partial derivatives in a region of the plane is said to be harmonic if it satisfies the Laplace Equation $\ref{laplace}$.

Exercise $\PageIndex{1}$

Show that $\ln(y^2+x^2)$ is hamornic everywhere except at the origin.

Converting Laplace's equation to polar co-ordinates

Consider the transformation to polar coordinates, $x=r \cos(\theta), y=r \sin(\theta),$ imples that $r^2=x^2+y^2$ and $\tan(\theta)= y/x.$ We can use these equations to express $\dfrac{\partial ^2w}{\partial x^2}+\dfrac{\partial ^2w}{\partial y^2}$ in terms of partials of $w$ with respect to $r$ and $\theta.$