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PAGE 1

Laplace's Equation

Heat eq:

\[ u_t = k u_{xx} \quad (1\text{-D}) \]

A simple sketch of a horizontal rod with endpoints labeled x=0 and x=L.

\[ u_t = k(u_{xx} + u_{yy}) \quad (2\text{-D}) \]

A sketch of a rectangular region in the xy-plane with boundaries labeled x=0, x=a, y=0, and y=b.

Wave eq:

\[ u_{tt} = a^2 u_{xx} \quad (1\text{-D}) \]

\[ u_{tt} = a^2(u_{xx} + u_{yy}) \quad (2\text{-D}) \]

the right side of them is : the second partial derivs with respect to space variables \( (x, y, z, \text{etc}) \)

they can all be represented by \( \nabla^2 u \)

\( \uparrow \) Laplacian operator

\[ \nabla^2 = \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} + \dots \right) \]

if \( u = u(x, y) \)

then \[ \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \]

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the value of \( \nabla^2 u \) at a point gives us the shape information

\[ u_{xx} > 0 \rightarrow \nabla^2 u > 0 \rightarrow \]

A small hand-drawn concave up parabola segment.

\( \leftarrow \) value of \( u \) here is lower than the average nearby

let's look at the 2D heat eq: \[ u_t = k \nabla^2 u \]

\( \rightarrow \) steady-state solution \( (u_t = 0) \)

we get \( \nabla^2 u = 0 \rightarrow \)

\[ u_{xx} + u_{yy} = 0 \]

Laplace's Eq.

set up:

\[ 0 < x < a \quad 0 < y < b \]

4 BCs: one for each edge

\( u(x, 0) = f_1(x) \)
lower edge
\( u(x, b) = f_2(x) \)
top edge
\( u(0, y) = g_1(y) \)
left edge
\( u(a, y) = g_2(y) \)
right edge
A rectangular region on x and y axes. The x-axis intercept is labeled 'a' and the y-axis intercept is labeled 'b'. An arrow points to the bottom edge with the label u(x,0) = f1(x).

goal: find \( u(x, y) \)

satisfying \( u_{xx} + u_{yy} = 0 \)

and ALL BCs

PAGE 3

The problem can be simplified by using the principle of superposition because Laplace's eq. is linear.

So, general solution

A rectangular domain in the xy-plane with boundary conditions g1 on the left, f2 on the top, g2 on the right, and f1 on the bottom.
=
Rectangular domain with non-zero boundary condition f1 on the bottom and zero on all other sides.
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Rectangular domain with non-zero boundary condition g2 on the right and zero on all other sides.
+
Rectangular domain with non-zero boundary condition f2 on the top and zero on all other sides.
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Rectangular domain with non-zero boundary condition g1 on the left and zero on all other sides.

→ make 3 BCs homogeneous (0), rotate which is nonhomogeneous

As an example, let's solve the 3rd case above:

\[ u_{xx} + u_{yy} = 0 \quad 0 < x < a, \quad 0 < y < b \]

\[ u(x, 0) = 0 \quad \text{(bottom)} \]

\[ u(0, y) = 0 \quad \text{(left)} \]

\[ u(a, y) = 0 \quad \text{(right)} \]

\[ u(x, b) = f(x) \quad \text{(top)} \]

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We will use the separation of variables again

\[ u(x, y) = X(x) Y(y) \]

\[ u_{xx} = X'' Y \quad u_{yy} = X Y'' \]

\[ u_{xx} + u_{yy} = 0 \]

\[ X'' Y + X Y'' = 0 \]

\[ \frac{X''}{X} = -\frac{Y''}{Y} = \text{constant} = -\lambda \quad \text{(just like in heat/wave eqs)} \]

ODEs that results:

\[ X'' + \lambda X = 0 \]

\[ Y'' - \lambda Y = 0 \]

homogeneous

BCs:

\[ u(x, 0) = 0 \rightarrow Y(0) = 0 \]

\[ u(0, y) = 0 \rightarrow X(0) = 0 \]

\[ u(a, y) = 0 \rightarrow X(a) = 0 \]

Solve for \( X \) or \( Y \) first whichever has complete BCs

(NOT always \( X \) first as in heat/wave eqs)

PAGE 5

Solving Differential Equations for Heat and Wave Equations

solve

\( X'' + \lambda X = 0 \)
\( X(0) = X(a) = 0 \)

in heat/wave eqs w/ \( a = L \)

\[ \lambda_n = \frac{n^2 \pi^2}{a^2} \]
\[ X_n = \sin\left( \frac{n\pi}{a} x \right) \]

\( n = 1, 2, 3, \dots \)

now

\( Y'' - \lambda Y = 0 \)
\( Y(0) = 0 \)
\[ Y'' - \frac{n^2 \pi^2}{a^2} Y = 0 \]
\[ Y(y) = A e^{\frac{n\pi y}{a}} + B e^{-\frac{n\pi y}{a}} \]

or

\[ Y(y) = C_1 \cosh\left( \frac{n\pi y}{a} \right) + C_2 \sinh\left( \frac{n\pi y}{a} \right) \]
choose either whichever is more convenient w/ BC

here, the hyperbolic ones are better

\( Y(0) = 0 \rightarrow C_1 = 0 \)
\[ Y_n = \sinh\left( \frac{n\pi y}{a} \right) \]
PAGE 6

General Solution and Boundary Conditions

for each \( n \), \( u_n = X_n Y_n \)

\[ u(x, y) = \sum_{n=1}^{\infty} A_n \sinh\left( \frac{n\pi y}{a} \right) \sin\left( \frac{n\pi x}{a} \right) \]

one last BC:

\( u(x, b) = f(x) \) (top)

\[ f(x) = \sum_{n=1}^{\infty} \left[ A_n \sinh\left( \frac{n\pi b}{a} \right) \right] \sin\left( \frac{n\pi x}{a} \right) \]

Sine series

\[ A_n \sinh\left( \frac{n\pi b}{a} \right) = \frac{2}{a} \int_{0}^{a} f(x) \sin\left( \frac{n\pi x}{a} \right) dx \]
Note: "L" for \( x \) because left side is \( f(x) \)

find \( A_n \) from that :

\[ A_n = \frac{2}{a \sinh\left( \frac{n\pi b}{a} \right)} \int_{0}^{a} f(x) \sin\left( \frac{n\pi x}{a} \right) dx \]
PAGE 7

Surface Plot of \(u(x, y)\) (Laplace Equation Solution)

Parameters:

  • \(a = 1\), \(b = 2\)
  • top edge = 3
  • (rest at 0)
A 3D surface plot showing the solution to the Laplace equation on a rectangular domain. The surface is flat at zero for most of the domain and rises sharply to a value of 3.0 along the edge where y equals 2.0. The color gradient transitions from dark purple at zero to bright yellow at the peak.
PAGE 8

Isotherms (Contour Plot) of \(u(x, y)\)

A 2D contour plot showing isotherms of the function u(x,y). The contours are concentrated near the top edge (y=2.0), where the value is highest, and curve downwards towards the center of the domain. The color scale on the right indicates values ranging from 0.0 to 3.2.

The contour lines represent paths of constant temperature (isotherms) for the steady-state heat distribution defined by the Laplace equation.