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PAGE 1

Laplace's Equation

Heat eq:

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

A simple horizontal rod representing a 1D domain from x=0 to x=L.

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

A parallelogram representing a 2D rectangular region in the xy-plane bounded by 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: \( u_{xx} + u_{yy} \), \( u_{xx} \), etc. (2nd partials w/ space variables)

all can be represented by the Laplacian operator \( \nabla^2 \)

  • if \( u(x,y) \), then \( \nabla^2 u = u_{xx} + u_{yy} \)
  • \( u(x,y,z) \), then \( \nabla^2 u = u_{xx} + u_{yy} + u_{zz} \)
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\( \nabla^2 u \) tells us the shape of \( u \) and its relationship to the average nearby

\( \nabla^2 u \) if \( u = u(x) \) is \( u_{xx} \)

\[ \curvearrowleft \quad u_{xx} < 0 \]

\( u \) here is higher than avg \( u \) nearby

let's look at 2-D heat eq: \( u_t = k \nabla^2 u = k(u_{xx} + u_{yy}) \)

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

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

Laplace's Eq.

set up:

\( 0 < x < a \) , \( 0 < y < b \)

4 BCs: for each edge

  • \( u(x, 0) = f_1(x) \) (bottom)
  • \( u(x, b) = f_2(x) \) (top)
  • \( u(0, y) = g_1(y) \) (left)
  • \( u(a, y) = g_2(y) \) (right)
A coordinate graph showing a rectangle in the first quadrant. The horizontal axis is x, vertical is y. The rectangle has width a and height b. The four sides are labeled with boundary condition functions: f1(x) at the bottom, f2(x) at the top, g1(y) at the left, and g2(y) at the right.

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

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

AND all BCs

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the equation is linear so superposition applies

idea:

A series of coordinate graphs illustrating the principle of superposition for a rectangular domain. The main graph shows a rectangle with boundary conditions f1, f2, g1, and g2. It is equated to the sum of four graphs, each showing the same rectangle but with only one non-zero boundary condition (f1, g2, f2, or g1 respectively) and the other three sides set to zero.

→ make 3 BCs homogeneous (zero), rotate, solve, combine.

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

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

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

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

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

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

can be the insulated version w/ partial = 0

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we will use separation of variables again

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

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

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

\[ u_{yy} = XY'' \]

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

\[ X''Y = -XY'' \to \frac{X''}{X} = -\frac{Y''}{Y} = \text{constant} = -\lambda \]

(same as in heat/wave)

two ODEs result:

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

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

BCs: \[ u(x, 0) = 0 \to Y(0) = 0 \]

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

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

Solve for \( X \) or \( Y \) first whichever has complete BCs (NOT always \( X \) first)

here, \( X \) first

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\[ X'' + \lambda X = 0 \]
\[ X(0) = X(a) = 0 \]
\[ \lambda_n = \frac{n^2 \pi^2}{a^2} \]

a is L here

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

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

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

or

\[ Y(y) = C_1 \cosh\left( \frac{n \pi}{a} y \right) + C_2 \sinh\left( \frac{n \pi}{a} y \right) \]
choose the form that is convenient for the BC

\[ Y(0) = C_1 \rightarrow \]

\[ Y_n = \sinh\left( \frac{n \pi}{a} y \right) \]
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for each \( n, \quad u_n = X_n Y_n \)

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

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

\( \uparrow \)
"L" for x

\( A_n \) comes from that:
\[ A_n = \frac{2}{a \sinh\left( \frac{n \pi b}{a} \right)} \int_{0}^{a} f(x) \sin\left( \frac{n \pi x}{a} \right) dx \]
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Surface Plot of \(u(x, y)\) (Laplace Equation Solution)

\(a = 1, \quad b = 2\)

\(f(x) = 3\) (top edge)

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 constant value of 3.0 along the top edge at y equals 2.0. The color gradient transitions from dark purple at zero to bright yellow at 3.0.
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Isotherms (Contour Plot) of \(u(x, y)\)

A 2D contour plot representing isotherms of the Laplace equation solution. The x-axis ranges from 0.0 to 1.0 and the y-axis from 0.0 to 2.0. Concentric semi-elliptical contour lines are clustered near the top edge at y equals 2.0, where the value is highest, and spread out as they approach the bottom of the domain.