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Higher-Dimension PDEs

1-D Heat eq: \( u_t = k u_{xx} \) \( 0 < x < L \)

A diagram of a one-dimensional rod with boundaries at x equals 0 and x equals L, with a point labeled u of x and t.

2-D Heat eq: \( u_t = k (u_{xx} + u_{yy}) \)

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

A 2D rectangular plate shown in perspective with boundaries x equals 0, x equals a, y equals 0, and y equals b, with an interior point labeled u of x, y, and t.

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

all four edges at \( u=0 \) (homogeneous)

BCs (four edges)

  • \( u(x, 0, t) = 0 \) (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 coordinate graph showing a rectangle in the first quadrant of the xy-plane with width a and height b.
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separation of variables: \( u(x, y, t) = X(x) Y(y) T(t) \)

\( u_t = k (u_{xx} + u_{yy}) \) becomes

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

ODEs:

\[ X'' + \lambda X = 0 \]
\[ -\frac{Y''}{Y} + \frac{T'}{k T} = -\lambda \]

rewrite:

\[ \underbrace{\frac{Y''}{Y}}_{\text{depends on } y} = \underbrace{\frac{T'}{k T} + \lambda}_{\text{depends on } t} = \text{constant} = -\mu \quad \text{(another separation constant)} \]
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Separation of Variables for 2D Heat Equation

more ODEs:

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

(looks just like \( X \) eq.)

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

Boundary Conditions (BCs)

  • \( u(x, 0, t) = 0 \rightarrow X(x)Y(0)T(t) = 0 \rightarrow Y(0) = 0 \)
  • \( u(a, y, t) = 0 \rightarrow \dots \rightarrow X(a) = 0 \)
  • \( u(x, b, t) = 0 \rightarrow \dots \rightarrow Y(b) = 0 \)
  • \( u(0, y, t) = 0 \rightarrow \dots \rightarrow X(0) = 0 \)
\[ X'' + \lambda X = 0 \quad X(0) = X(a) = 0 \]
\[ Y'' + \mu Y = 0 \quad Y(0) = Y(b) = 0 \]
\[ \lambda_n = \frac{n^2 \pi^2}{a^2} \]
\[ \mu_m = \frac{m^2 \pi^2}{b^2} \]
\[ X_n = \sin\left(\frac{n\pi}{a}x\right) \]

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

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

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

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

for each pair of \( (n, m) \)   \( u_{mn} = T_{mn} X_n Y_m \)

general solution: sum over \( m, n \)

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

Initial Condition (IC)

\( u(x, y, 0) = f(x, y) \) initial heat distribution

at \( t = 0 \)

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

looks like:

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

(\( x \) fixed)

"regular" sine series

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\[ \underbrace{C = \frac{2}{b} \int_{0}^{b} f(x,y) \sin\left(\frac{m\pi y}{b}\right) dy}_{\text{Sine series again}} = \sum_{n=1}^{\infty} A_{mn} \sin\left(\frac{n\pi x}{a}\right) \]

\[ A_{mn} = \frac{2}{a} \int_{0}^{a} \left[ \frac{2}{b} \int_{0}^{b} f(x,y) \sin\left(\frac{m\pi y}{b}\right) dy \right] \sin\left(\frac{n\pi x}{a}\right) dx \]

\[ A_{mn} = \frac{4}{ab} \int_{0}^{a} \int_{0}^{b} f(x,y) \sin\left(\frac{m\pi y}{b}\right) \sin\left(\frac{n\pi x}{a}\right) dy \, dx \]

Example

\( a=1, \quad b=2, \quad f(x,y)=3 \)

\( \kappa = 1 \)

A 2D plot on x and y axes showing a rectangular region bounded by x from 0 to 1 and y from 0 to 2, labeled as being initially heated to a uniform temperature of 3.
\[ u(x,y,t) = \sum_{m \text{ odd}}^{\infty} \sum_{n \text{ odd}}^{\infty} \frac{48}{mn\pi^2} \sin(n\pi x) \sin\left(\frac{m\pi y}{2}\right) e^{-\left(n^2\pi^2 + \frac{m^2\pi^2}{4}\right)t} \]

\( \leftarrow \) x decay rate (faster)

\( \leftarrow \) y decay rate

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Surface Plot at \( t = 0.002 \)

A 3D surface plot of temperature u at time t=0.002. The plot shows a plateau at u=3 across the interior of the domain defined by x in [0, 1] and y in [0, 2], with steep gradients near the edges where the temperature falls to zero. A vertical color bar on the right maps colors from dark blue (0.0) to yellow (3.0).
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Heatmap Visualization of Temperature Distribution

The following heatmap illustrates the temperature distribution \( u \) across a rectangular domain at a very early time step, \( t = 0.002 \).

A 2D heatmap showing temperature distribution on a rectangular grid from x=0 to 1 and y=0 to 2. The interior is a uniform light yellow (high temperature near 3.0), while the edges show a sharp gradient to dark purple (low temperature near 0.0).

Observation:

At \( t = 0.002 \), the initial temperature of approximately \( 3.0 \) is maintained throughout most of the interior, with steep gradients appearing only at the boundaries where the temperature is fixed at \( 0 \).

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Surface Plot of Temperature Decay

As time progresses to \( t = 0.05 \), the temperature distribution \( u(x, y, t) \) evolves into a smooth, curved surface as heat diffuses out of the domain boundaries.

A 3D surface plot showing the temperature u over the domain x in [0, 1] and y in [0, 2] at t = 0.05. The surface forms a smooth mound shape, peaking in the center and tapering to zero at all four boundaries.

Analysis:

The surface plot demonstrates the smoothing effect of the heat equation. The sharp transitions seen in the earlier heatmap have dissipated, resulting in a characteristic bell-like distribution that satisfies the zero-temperature boundary conditions.

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Heatmap Visualization of Temperature Distribution

The following heatmap illustrates the spatial distribution of temperature \( u \) across a rectangular domain at a specific time step \( t = 0.05 \). The domain is defined by the coordinates \( x \) ranging from 0.0 to 1.0 and \( y \) ranging from 0.00 to 2.00.

A 2D heatmap titled 'Heatmap at t = 0.05' showing temperature distribution. The x-axis ranges from 0.0 to 1.0 and the y-axis from 0.00 to 2.00. A color bar on the right indicates temperature 'u' from 0.0 (dark purple) to 3.0 (bright yellow). The center of the domain shows a high-temperature orange-red region, which fades to dark purple at the boundaries, indicating cooling at the edges.

Observation:

At \( t = 0.05 \), the heat is concentrated in the center of the domain, with the highest temperatures reaching approximately 2.5 to 2.8 units. The boundaries are maintained at a lower temperature, near 0.0, suggesting Dirichlet boundary conditions.

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Surface Plot of Temperature Evolution

This surface plot provides a three-dimensional perspective of the temperature field \( u(x, y) \) at a later time step, \( t = 0.1 \). It visualizes the magnitude of temperature as the height of the surface above the \( xy \)-plane.

A 3D surface plot titled 'Surface Plot at t = 0.1'. The vertical axis represents temperature 'u' from 0.0 to 3.0. The horizontal axes are x (0.0 to 1.0) and y (0.00 to 2.00). The surface is a smooth, rounded peak centered in the domain, colored with a gradient from dark purple at the base (u=0) to teal/blue at the peak (u ≈ 1.5). A color bar on the right confirms the temperature scale.

Analysis of Decay:

Comparing this to the previous state at \( t = 0.05 \), the peak temperature has significantly decreased from nearly 3.0 to approximately 1.5. This demonstrates the diffusive nature of the heat equation over time as energy spreads and leaves the system through the boundaries.

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Heatmap Visualization of Temperature Distribution

This page presents a 2D heatmap representing the temperature distribution across a rectangular domain at a specific time step.

Heatmap at \( t = 0.1 \)

A 2D heatmap showing temperature distribution over a rectangular domain with x-axis from 0.0 to 1.0 and y-axis from 0.00 to 2.00. The temperature, indicated by a color scale from 0.0 (dark purple) to 3.0 (yellow), is highest in the center (reddish-pink) and decreases towards the boundaries (dark purple).

The heatmap illustrates the spatial variation of temperature \( u \) at time \( t = 0.1 \). The central region shows higher thermal energy, which gradually dissipates towards the edges of the domain.

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3D Surface Plot of Temperature Distribution

This page provides a three-dimensional perspective of the temperature field, showing the magnitude of temperature as height.

Surface Plot at \( t = 0.15 \)

A 3D surface plot showing temperature u as a function of spatial coordinates x and y. 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. The surface forms a smooth, low-profile mound peaking in the center, with colors corresponding to a scale from dark purple (0.0) to yellow (3.0).

The surface plot at \( t = 0.15 \) visualizes the temperature \( u(x, y) \) as a height above the \( xy \)-plane. Compared to earlier time steps, the peak temperature has decreased as heat diffuses across the domain.

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Heatmap Visualization of Temperature Distribution

The following figure illustrates the spatial distribution of temperature \( u \) across a two-dimensional domain at a specific time step \( t = 0.15 \). The domain is defined by the coordinates \( x \) and \( y \).

A 2D heatmap showing temperature distribution at time t equals 0.15. The x-axis ranges from 0.0 to 1.0, and the y-axis ranges from 0.00 to 2.00. The temperature u is represented by a color scale from 0.0 (dark purple/black) to 3.0 (bright yellow). The center of the plot shows a slightly higher temperature region in dark purple, while the edges are black, indicating lower temperatures.

Observation

At \( t = 0.15 \), the temperature \( u \) remains relatively low throughout the domain, with values mostly below 0.5 as indicated by the dark purple and black regions. The color bar on the right provides a reference for the temperature scale \( u \).