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8.1: Laplace Equations

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    35368
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    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 \(\theta.\)


    This page titled 8.1: Laplace Equations is shared under a not declared license and was authored, remixed, and/or curated by Pamini Thangarajah.

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