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PAGE 1

Higher-Dimension PDE

1-D \( u_t = k u_{xx} \)
A diagram of a 1D rod with boundaries at x=0 and x=L, labeled with the temperature function u(x,t).
2-D \( u_t = k (u_{xx} + u_{yy}) \)
A 2D rectangular domain in perspective with boundaries x=0, x=a, y=0, and y=b, labeled with u(x,y,t).

today: edges are kept at \( u = 0 \)

\( u_t = k (u_{xx} + u_{yy}) \) \( 0 < x < a \) \( 0 < y < b \)

BCs:

  • \( u(x, 0, t) = 0 \) (bottom)
  • \( u(a, y, t) = 0 \) (right)
  • \( u(x, b, t) = 0 \) (top)
  • \( u(0, y, t) = 0 \) (left)

IC:

\( u(x, y, 0) = f(x, y) \)
A rectangle plotted on an x-y coordinate plane with vertices at (0,0), (a,0), (a,b), and (0,b).
PAGE 2

Same method: separation of variables

\( u(x, y, t) = X(x) Y(y) T(t) \)

rewrite heat eq: \( u_t = k (u_{xx} + u_{yy}) \)

\( X Y T' = k (X'' Y T + X Y'' T) \)

\[ \underbrace{\frac{X''}{X}}_{\text{depends on } x \text{ alone}} = \underbrace{-\frac{Y''}{Y} + \frac{T'}{kT}}_{\text{depends on } y \text{ and } t} = \text{constant} = -\lambda \]

same separation constant

first ODE:

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

\( -\frac{Y''}{Y} + \frac{T'}{kT} = \text{constant} = -\lambda \)

rewrite:

\( \frac{Y''}{Y} = \frac{T'}{kT} + \lambda = \text{constant} = -\mu \)

(another constant)
PAGE 3

Separation of Variables: Additional ODEs and Boundary Conditions

From that we get two more ODEs:

\[ Y'' + \mu Y = 0 \]
\[ T' + k(\mu + \lambda)T = 0 \]

looks just like \( X \) equation

Boundary Conditions (BCs)

  • \( u(x, 0, t) = 0 \quad \rightarrow \quad Y(0) = 0 \)
  • \( u(x, b, t) = 0 \quad \rightarrow \quad Y(b) = 0 \)
  • \( u(0, y, t) = 0 \quad \rightarrow \quad X(0) = 0 \)
  • \( u(a, y, t) = 0 \quad \rightarrow \quad X(a) = 0 \)

Solving for \( X \)

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

\[ \lambda_n = \frac{n^2 \pi^2}{a^2} \]

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

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

Solving for \( Y \)

\( Y'' + \mu Y = 0, \quad Y(0) = Y(b) = 0 \)

\[ \mu_m = \frac{m^2 \pi^2}{b^2} \]
\[ Y_m = \sin\left(\frac{m \pi}{b} y\right) \]

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

PAGE 4

Solving for \( T \) and General Solution

\[ T' + k\left(\frac{m^2 \pi^2}{b^2} + \frac{n^2 \pi^2}{a^2}\right)T = 0 \]
\[ T_{mn} = e^{-k\left(\frac{m^2 \pi^2}{b^2} + \frac{n^2 \pi^2}{a^2}\right)t} \]

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

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

for each \( (m, n) \) pair, there is one solution

\[ u_{mn} = T_{mn} X_n Y_m \]

General Solution

General solution: sum over \( n \) and \( m \)

\[ u(x, y, t) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn} e^{-k\left(\frac{m^2 \pi^2}{b^2} + \frac{n^2 \pi^2}{a^2}\right)t} \sin\left(\frac{n \pi x}{a}\right) \sin\left(\frac{m \pi y}{b}\right) \]

Initial Condition (IC)

IC:

\( u(x, y, 0) = f(x, y) \)

initial heat distribution

PAGE 5
\[ f(x,y) = \sum_{m=1}^{\infty} \underbrace{\left[ \sum_{n=1}^{\infty} A_{mn} \sin\left(\frac{n\pi x}{a}\right) \right]}_{\text{"constant" if } x \text{ is fixed } \rightarrow \text{ call it } C} \sin\left(\frac{m\pi y}{b}\right) \]
double sine series

if \( x \) is fixed, it looks like

\[ f(x,y) = \sum_{m=1}^{\infty} C \sin\left(\frac{m\pi y}{b}\right) \]

(\( x \) fixed)

"regular" sine series
\[ C = \underbrace{\frac{2}{b} \int_{0}^{b} f(x,y) \sin\left(\frac{m\pi y}{b}\right) dy}_{\text{function of } x} = \sum_{n=1}^{\infty} \underset{\uparrow \\ \text{constant}}{A_{mn} \sin\left(\frac{n\pi x}{a}\right)} \]
"regular" sine series
\[ A_{mn} = \frac{2}{a} \int_{0}^{a} \left[ \frac{2}{b} \int_{0}^{b} f(x,y) \sin\left(\frac{m\pi y}{b}\right) dy \right] \sin\left(\frac{n\pi x}{a}\right) dx \]
PAGE 6
\[ A_{mn} = \frac{4}{ab} \int_{0}^{a} \int_{0}^{b} f(x,y) \sin\left(\frac{m\pi y}{b}\right) \sin\left(\frac{n\pi x}{a}\right) dy dx \]

example

  • \( a = 1, b = 2, f(x,y) = 3 \)
  • \( k = 1 \)
A 2D Cartesian coordinate system showing a rectangular region defined by x from 0 to 1 and y from 0 to 2. An annotation indicates that within this region, the initial value u is 3 uniformly.
\[ u(x,y,t) = \sum_{m \text{ odd}}^{\infty} \sum_{n \text{ odd}}^{\infty} \frac{48}{mn\pi^2} \sin(n\pi x) \sin\left(\frac{m\pi y}{2}\right) e^{-\left(n^2\pi^2 + \frac{m^2\pi^2}{4}\right)t} \]
\( \downarrow \)
\( x \) decay rate (faster)
\( \swarrow \)
\( y \) decay rate
PAGE 7

Surface Plot at \( t = 0.002 \)

The following visualization represents the temperature distribution \( u \) across a 2D domain at a specific time step \( t = 0.002 \). The vertical axis represents the temperature \( u \), while the horizontal axes represent spatial coordinates \( x \) and \( y \).

A 3D surface plot showing temperature distribution over a rectangular domain. The surface is mostly flat and yellow at a high temperature of approximately 3.0, with steep gradients dropping to zero at the boundaries, indicated by blue and purple colors. The x-axis ranges from 0.0 to 1.0, the y-axis from 0.00 to 2.00, and the vertical u-axis from 0.0 to 3.0.

Note: The sharp drop-off at the edges indicates the boundary conditions where the temperature is held at zero.

PAGE 8

Heatmap at \( t = 0.002 \)

This top-down heatmap view provides an alternative perspective of the temperature distribution \( u(x, y) \) at time \( t = 0.002 \). The color intensity corresponds to the temperature value.

A 2D heatmap showing temperature distribution. A large central rectangular area is bright yellow, indicating a constant high temperature of 3.0. This area is surrounded by a narrow border of orange, red, and purple, showing a rapid decrease to 0.0 at the boundaries. The x-axis ranges from 0.0 to 1.0 and the y-axis from 0.00 to 2.00.

Observation:

The heatmap confirms that the interior of the domain remains at the initial high temperature, while the thermal diffusion effect is concentrated near the boundaries at this very early time step.

PAGE 9

Surface Plot at \( t = 0.05 \)

The following visualization represents the spatial distribution of temperature \( u \) across a 2D domain at a specific time step \( t = 0.05 \).

A 3D surface plot showing temperature distribution over a rectangular domain. The x-axis ranges from 0.0 to 1.0, the y-axis from 0.00 to 2.00, and the vertical u-axis (temperature) from 0.0 to 3.0. The surface forms a smooth, rounded peak centered in the domain, with colors transitioning from dark purple at the base (0.0) to green and yellow at the peak (approximately 2.0).

Observation:

At \( t = 0.05 \), the temperature profile shows a clear diffusion pattern, with the highest values concentrated in the center of the domain and tapering off towards the boundaries.

PAGE 10

Heatmap at \( t = 0.05 \)

This top-down heatmap view provides an alternative perspective of the temperature distribution \( u \) at \( t = 0.05 \), highlighting the symmetry of the diffusion process.

A 2D heatmap plot of temperature distribution. The horizontal x-axis ranges from 0.0 to 1.0 and the vertical y-axis ranges from 0.00 to 2.00. A color scale on the right indicates temperature u from 0.0 (dark purple) to 3.0 (bright yellow). The heatmap shows a central elliptical region of high temperature (orange/yellow) surrounded by concentric gradients of lower temperature (purple/black) towards the edges.

Key Details:

  • Domain: \( x \in [0, 1] \), \( y \in [0, 2] \)
  • Peak Temperature: Approximately \( 2.0 \) at the center.
  • Boundary Conditions: Temperature appears to approach zero at the boundaries, consistent with Dirichlet conditions.
PAGE 11

Surface Plot at \( t = 0.1 \)

A 3D surface plot showing temperature distribution over a 2D rectangular domain. The surface forms a smooth, rounded peak in the center, with values decreasing towards the boundaries. A color bar on the right indicates temperature values ranging from 0.0 (dark purple) to 3.0 (yellow).

The visualization represents the temperature distribution \( u \) across a spatial domain defined by \( x \) and \( y \) coordinates at a specific time step \( t = 0.1 \).

  • Vertical Axis (\( u \)): Represents temperature, ranging from 0.0 to 3.0.
  • Horizontal Axes (\( x, y \)): Represent spatial dimensions, with \( x \) from 0.0 to 1.0 and \( y \) from 0.00 to 2.00.
  • Color Mapping: Higher temperatures are shown in yellow/green, while lower temperatures are in dark purple/blue.
PAGE 12

Heatmap at \( t = 0.1 \)

A 2D heatmap representing the same temperature data as the surface plot. The x-axis ranges from 0.0 to 1.0 and the y-axis from 0.00 to 2.00. A bright reddish-pink core in the center indicates the highest temperature, fading to dark purple and black at the edges. A vertical color bar on the right maps colors to temperature values from 0.0 to 3.0.

This 2D heatmap provides a top-down view of the temperature field \( u(x, y) \) at \( t = 0.1 \).

Spatial Domain:
\( x \in [0, 1] \) and \( y \in [0, 2] \).
Temperature Gradient:
The central region shows the highest thermal concentration, which dissipates as it approaches the boundaries where \( u \approx 0 \).
PAGE 13

Surface Plot at \( t = 0.15 \)

The following visualization represents the temperature distribution \( u \) across a 2D domain defined by coordinates \( x \) and \( y \) at a specific time step \( t = 0.15 \).

A 3D surface plot showing temperature u over a rectangular domain. The x-axis ranges from 0.0 to 1.0, and the y-axis ranges from 0.00 to 2.00. The surface is mostly flat and dark purple, indicating low temperature values near 0.0 across the entire domain at this time step.

Plot Characteristics

  • Vertical Axis (\( u \)): Represents temperature, scaled from 0.0 to 3.0.
  • Horizontal Axes (\( x, y \)): Define the spatial domain, with \( x \in [0, 1] \) and \( y \in [0, 2] \).
  • Color Scale: A vertical color bar on the right maps colors to temperature values, ranging from dark purple (0.0) to bright yellow (3.0).
PAGE 14

Heatmap at \( t = 0.15 \)

This heatmap provides a top-down view of the temperature distribution \( u(x, y) \) at time \( t = 0.15 \), corresponding to the surface plot on the previous page.

A 2D heatmap showing temperature distribution. The horizontal x-axis ranges from 0.0 to 1.0 and the vertical y-axis ranges from 0.00 to 2.00. The plot is predominantly dark purple, with a slightly lighter purple circular region in the center, indicating a very low and relatively uniform temperature distribution.

Heatmap Details

  • Spatial Coordinates: The x-axis is labeled from 0.0 to 1.0, and the y-axis is labeled from 0.00 to 2.00.
  • Temperature Intensity: The color bar on the right indicates that the dark purple regions correspond to a temperature \( u \approx 0.0 \).
  • Observation: At this time step, the heat has dissipated or has not yet significantly permeated the domain, resulting in a near-zero temperature profile.