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Mathematics LibreTexts

9.7: The Laplace Equation

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The diffusion equation in two spatial dimensions is ut=D(uxx+uyy).

The steady-state solution, approached asymptotically in time, has ut=0 so that the steady-state solution u=u(x,y) satisfies the two-dimensional Laplace equation uxx+uyy=0.

We will consider the mathematical problem of solving the two dimensional Laplace equation inside a rectangular or a circular boundary. The value of u(x,y) will be specified on the boundaries, defining this problem to be of Dirichlet type.

Dirichlet Problem for a Rectangle

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Figure 9.7.1: Dirichlet problem for the Laplace equation in a rectangle.

We consider the Laplace equation (???) for the interior of a rectangle 0<x<a, 0<y<b, (see Fig. 9.7.1), with boundary conditions u(x,0)=0,u(x,b)=0,0<x<a;u(0,y)=0,u(a,y)=f(y),0yb.

More general boundary conditions can be solved by linear superposition of solutions.

We take our usual ansatz u(x,y)=X(x)Y(y), and find after substitution into (???), X with \lambda the separation constant. We thus obtain the two ordinary differential equations X''-\lambda X=0,\quad Y''+\lambda Y=0.\nonumber

The homogeneous boundary conditions are X(0) = 0,\: Y(0) = 0 and Y(b) = 0. We have already solved the equation for Y(y) in §9.5, and the solution yields the eigenvalues \lambda_n=\left(\frac{n\pi}{b}\right)^2,\quad n=1,2,3,\ldots ,\nonumber with corresponding eigenfunctions Y_n(y)=\sin\frac{n\pi y}{b}.\nonumber

The remaining X equation and homogeneous boundary condition is therefore X''-\frac{n^2\pi^2}{b^2}X=0,\quad X(0)=0,\nonumber and the solution is the hyperbolic sine function X_n(x)=\sinh\frac{n\pi x}{b},\nonumber times a constant. Writing u_n = X_nY_n, multiplying by a constant and summing over n, yields the general solution u(x,y)=\sum\limits_{n=0}^\infty c_n\sinh\frac{n\pi x}{b}\sinh\frac{n\pi y}{b}.\nonumber

The remaining inhomogeneous boundary condition u(a, y) = f(y) results in f(y)=\sum\limits_{n=0}^\infty c_n\sinh\frac{n\pi a}{b}\sin\frac{n\pi y}{b},\nonumber which we recognize as a Fourier sine series for an odd function with period 2b, and coefficient c_n \sinh (n\pi a/b). The solution for the coefficient is given by c_n=\frac{2}{b\sinh\frac{n\pi a}{b}}\int_0^b f(y)\sin\frac{n\pi y}{b}dy.\nonumber

Dirichlet Problem for a Circle

The Laplace equation is commonly written symbolically as \label{eq:2}\nabla ^2u=0, where \nabla^2 is called the Laplacian, sometimes denoted as \Delta. The Laplacian can be written in various coordinate systems, and the choice of coordinate systems usually depends on the geometry of the boundaries. Indeed, the Laplace equation is known to be separable in 13 different coordinate systems! We have solved the Laplace equation in two dimensions, with boundary conditions specified on a rectangle, using \nabla ^2=\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}.\nonumber

Here we consider boundary conditions specified on a circle, and write the Laplacian in polar coordinates by changing variables from cartesian coordinates. Polar coordinates is defined by the transformation (r,\theta )\to (x, y):

\label{eq:3}x=r\cos\theta,\quad y=r\sin\theta ; and the chain rule gives for the partial derivatives \label{eq:4}\frac{\partial u}{\partial r}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial r},\quad \frac{\partial u}{\partial\theta}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial\theta}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial\theta}.

After taking the partial derivatives of x and y using \eqref{eq:3}, we can write the transformation \eqref{eq:4} in matrix form as \label{eq:5}\left(\begin{array}{c}\partial u/\partial r \\ \partial u/\partial\theta\end{array}\right)=\left(\begin{array}{rr}\cos\theta &\sin\theta \\ -r\sin\theta & r\cos\theta\end{array}\right)\left(\begin{array}{c}\partial u/\partial x \\ \partial u/\partial y\end{array}\right).

Inversion of \eqref{eq:5} can be determined from the following result, commonly proved in a linear algebra class. If A=\left(\begin{array}{cc}a&b \\ c&d\end{array}\right),\quad\det A\neq 0,\nonumber then A^{-1}=\frac{1}{\det A}\left(\begin{array}{cc}d&-b \\ -c&a\end{array}\right).\nonumber

Therefore, since the determinant of the 2\times 2 matrix in \eqref{eq:5} is r, we have \label{eq:6}\left(\begin{array}{c}\partial u/\partial x \\ \partial u/\partial y\end{array}\right)=\left(\begin{array}{rr}\cos\theta &-\sin\theta /r \\ \sin\theta &\cos\theta /r\end{array}\right)\left(\begin{array}{c}\partial u/\partial r \\ \partial u/\partial\theta\end{array}\right).

Rewriting \eqref{eq:6} in operator form, we have \label{eq:7}\frac{\partial}{\partial x}=\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta},\quad\frac{\partial}{\partial y}=\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial\theta}.

To find the Laplacian in polar coordinates with minimum algebra, we combine \eqref{eq:7} using complex variables as \label{eq:8}\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}=e^{i\theta}\left(\frac{\partial}{\partial r}+\frac{i}{r}\frac{\partial}{\partial\theta}\right), so that the Laplacian may be found by multiplying both sides of \eqref{eq:8} by its complex conjugate, taking care with the computation of the derivatives on the righthand-side:

\begin{aligned}\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}&=e^{i\theta}\left(\frac{\partial}{\partial r}+\frac{i}{r}\frac{\partial}{\partial\theta}\right)e^{-i\theta}\left(\frac{\partial}{\partial r}-\frac{1}{r}\frac{\partial}{\partial\theta}\right) \\ &=\frac{\partial ^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial ^2}{\partial\theta ^2}.\end{aligned}

We have therefore determined that the Laplacian in polar coordinates is given by \label{eq:9}\nabla^2=\frac{\partial ^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial ^2}{\partial\theta^2}, which is sometimes written as \nabla^2 =\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}\right)+\frac{1}{r^2}\frac{\partial ^2}{\partial\theta ^2}.\nonumber

We now consider the solution of the Laplace equation in a circle with radius r < a subject to the boundary condition \label{eq:10}u(a,\theta )=f(\theta),\quad 0\leq\theta\leq 2\pi.

An additional boundary condition due to the use of polar coordinates is that u(r, \theta ) is periodic in \theta with period 2\pi. Furthermore, we will also assume that u(r, \theta ) is finite within the circle.

The method of separation of variables takes as our ansatz u(r,\theta )=R(r)\Theta (\theta ),\nonumber and substitution into the Laplace equation \eqref{eq:2} using \eqref{eq:9} yields R''\Theta +\frac{1}{r}R'\Theta +\frac{1}{r^2}R\Theta ''=0,\nonumber or r^2\frac{R''}{R}+r\frac{R'}{R}=-\frac{\Theta ''}{\Theta }=\lambda,\nonumber where \lambda is the separation constant. We thus obtain the two ordinary differential equations r^2R''+rR'-\lambda R=0,\quad \Theta ''+\lambda\Theta =0.\nonumber

The \Theta equation is solved assuming periodic boundary conditions with period 2\pi. If \lambda < 0, then no periodic solution exists. If \lambda = 0, then \Theta can be constant. If \lambda = \mu^2 > 0, then \Theta (\theta )=A\cos\mu\theta +B\sin\mu\theta ,\nonumber and the requirement that \Theta is periodic with period 2\pi forces \mu to be an integer. Therefore, \lambda_n=n^2,\quad n=0,1,2,\ldots ,\nonumber with corresponding eigenfunctions \Theta_n(\theta )=A_n\cos n\theta +B_n\sin n\theta.\nonumber

The R equation for each eigenvalue \lambda_n then becomes \label{eq:11}r^2R''+rR'-n^2R=0, which is an Euler equation. With the ansatz R = r^s, \eqref{eq:11} reduces to the algebraic equation s(s − 1) + s − n^2 = 0, or s^2 = n^2. Therefore, s = ±n, and there are two real solutions when n > 0 and degenerate solutions when n = 0. When n > 0, the solution for R(r) is R_n(r)=Ar^n+Br^{-n}.\nonumber

The requirement that u(r,\theta ) is finite in the circle forces B = 0 since r^{−n} becomes unbounded as r\to 0. When n = 0, the solution for R(r) is R_n(r)=A+B\ln r,\nonumber and again finite u in the circle forces B = 0. Therefore, the solution for n = 0,\: 1,\: 2,\ldots is R_n = r^n. Thus the general solution for u(r,\theta ) may be written as \label{eq:12} u(r,\theta )=\frac{A_0}{2}+\sum\limits_{n=1}^\infty r^n (A_n\cos n\theta +B_n\sin n\theta ), where we have separated out the n = 0 solution to write our solution in a form similar to the standard Fourier series given by (9.3.1). The remaining boundary condition \eqref{eq:10} specifies the values of u on the circle of radius a, and imposition of this boundary condition results in \label{eq:13} f(\theta )=\frac{A_0}{2}+\sum\limits_{n=1}^\infty a^n (A_n\cos n\theta +B_n\sin n\theta ).

Equation \eqref{eq:13} is a Fourier series for the periodic function f(\theta ) with period 2\pi, i.e., L = \pi in (9.3.1). The Fourier coefficients a^nA_n and a^nB_n are therefore given by (9.3.5) and (9.3.6) to be \begin{align} a^nA_n&=\frac{1}{\pi}\int_0^{2\pi} f(\phi )\cos n\phi d\phi, \quad n=0,1,2,\ldots ; \nonumber \\ a^nB_n&=\frac{1}{\pi}\int_0^{2\pi}f(\phi )\sin n\phi d\phi ,\quad n=1,2,3,\ldots ,\label{eq:14}\end{align} where we have used \phi for the dummy variable of integration.

A remarkable fact is that the infinite series solution for u(r, \theta ) can be summed explicitly. Substituting \eqref{eq:14} into \eqref{eq:12}, we obtain \begin{aligned}u(r,\theta )&=\frac{1}{2\pi}\int_0^{2\pi}d\phi f(\phi )\left[ 1+2\sum\limits_{n=1}^\infty \left(\frac{r}{a}\right)^n (\cos n\theta\cos n\phi +\sin n\theta\sin n\phi )\right] \\ &=\frac{1}{2\pi}\int_0^{2\pi} d\phi f(\phi)\left[1+2\sum_{n=1}^\infty \left(\frac{r}{a}\right)^n\cos n(\theta -\phi)\right].\end{aligned}

We can sum the infinite series by writing 2 cos n(\theta − \phi ) = e^{in(\theta−\phi)} + e^{−in(\theta −\phi)}, and using the sum of the geometric series \sum_{n=1}^\infty z^n = z/(1 − z) to obtain \begin{aligned} 1+2\sum\limits_{n=1}^\infty \left(\frac{r}{a}\right)^n\cos n(\theta -\phi)&=1+\sum\limits_{n=1}^\infty \left(\frac{re^{i(\theta -\phi)}}{a}\right)^n +\sum\limits_{n=1}^\infty \left(\frac{re^{-i(\theta -\phi )}}{a}\right)^n \\ &=1+\left(\frac{re^{i(\theta -\phi)}}{a-re^{i(\theta -\phi )}}+c.c.\right) \\ &=\frac{a^2-r^2}{a^2-2ar\cos (\theta -\phi )+r^2}.\end{aligned}

Therefore, u(r,\theta )=\frac{a^2-r^2}{2\pi}\int_0^{2\pi}\frac{f(\phi )}{a^2-2ar\cos (\theta -\phi )+r^2}d\phi,\nonumber an integral result for u(r, \theta ) known as Poisson’s formula. As a trivial example, consider the solution for u(r, \theta ) if f(\theta ) = F, a constant. Clearly, u(r, \theta ) = F satisfies both the Laplace equation and the boundary conditions so must be the solution. You can verify that u(r, \theta ) = F is indeed the solution by showing that \int_0^{2\pi}\frac{d\phi}{a^2-2ar\cos (\theta -\phi )+r^2}=\frac{2\pi }{a^2-r^2}.\nonumber


This page titled 9.7: The Laplace Equation is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Jeffrey R. Chasnov via source content that was edited to the style and standards of the LibreTexts platform.

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