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PAGE 1

Heat Eq. (2)

last time:

\[ u_t = k u_{xx} \] \( 0 < x < L \) \( t > 0 \)
\( u(0, t) = 0 \)
left temp = 0
\( u(L, t) = 0 \)
right temp = 0
\( u(x, 0) = f(x) \)
initial temp. profile
A horizontal rod of length L with endpoints at x=0 and x=L. A wavy curve labeled f(x) represents the initial temperature distribution along the rod.
\[ u(x, t) = X(x) T(t) \]
\[ X(x) = \sin\left(\frac{n\pi x}{L}\right) \] spatial solution
\[ T(t) = e^{-kn^2\pi^2 t / L^2} \] temporal solution
\[ u(x, t) = \sum_{n=1}^{\infty} c_n e^{-kn^2\pi^2 t / L^2} \sin\left(\frac{n\pi x}{L}\right) \]
\[ c_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx \]
PAGE 2

example

Copper plate/slab thickness 4 cm

\( k = 1.15 \text{ cm}^2/s \)

A 3D rectangular copper slab with a thickness of 4 cm. A small cylindrical core sample is highlighted being extracted from the center of the slab.

“core sample”

heat the entire interior uniformly to \( 100^\circ\text{C} \) at \( t = 0 \)

keep right and left faces at \( 0^\circ\text{C} \) for all \( t \)

A horizontal cylindrical core sample of length 4, with endpoints at x=0 and x=4.

Entire plate is uniformly heated and made of same material (same \( k \))

  • \( \rightarrow \) no lateral heat flow
  • \( \rightarrow \) only toward the faces (ends)
\[ u(x, t) = \sum_{n=1}^{\infty} c_n e^{-1.15 n^2 \pi^2 t / 16} \sin\left(\frac{n\pi x}{4}\right) \]

initial: \( f(x) = 100 \)

\( \vdots \) \[ c_n = \frac{2}{4} \int_{0}^{4} 100 \sin\left(\frac{n\pi x}{4}\right) dx = \frac{200 [1 - (-1)^n]}{n\pi} \]
PAGE 3

Heat Equation Solution Analysis

as \( t \to \infty \), the time solution \( \to 0 \) \( \to \) heat flows out of rod through ends

space solution \( \to \) periodic \( \to \) same temps on ends

Temperature Calculation

what is the temp at the mid point \( (x=2) \) 3 seconds later \( (t=3) \)

\[ u(2, 3) = \sum_{n=1}^{\infty} \frac{200(1 - (-1)^n)}{n\pi} e^{-1.15n^2\pi^2 \cdot 3/16} \sin\left(\frac{n\pi \cdot 2}{4}\right) \]
\( \leftarrow \) infinite sum

in practice, the negative exponential makes this a fast converging series \( \to \) only few terms needed to get "good" approx.

1-term approx \( (n=1 \text{ only}) \to 15.16^{\circ}\text{C} \)

PAGE 4

Surface Plot of Heat Equation Solution \( u(x, t) \)

A 3D surface plot showing temperature u as a function of position x and time t. The x-axis ranges from 0 to 4, the t-axis from 0 to 5, and the u-axis from 0 to 100. The surface shows a high initial temperature distribution that decays rapidly over time, with the highest temperatures at the center of the rod and zero temperature at the boundaries x=0 and x=4.
PAGE 5

Temperature Profile \(u(x, t)\) at various times \(t\)

A line graph showing the temperature profile u(x, t) across position x from 0 to 4.0 at different time intervals. At t=0, the profile is a rectangular step function at u=100. As time increases (t=0.1, 0.5, 1.0, 2.0, 5.0), the curves smooth out and decrease in peak height, converging toward u=0.

The graph illustrates the spatial distribution of temperature \(u\) along a one-dimensional domain \(x \in [0, 4]\) for several discrete time steps.

  • At \(t = 0\), the initial condition is a uniform temperature of 100 across the domain, with boundary conditions fixed at 0 at \(x=0\) and \(x=4\).
  • As time progresses, the heat dissipates through the boundaries, causing the temperature profile to evolve into a series of bell-shaped curves that flatten over time.
  • By \(t = 5.0\), the temperature across the entire domain has significantly decayed toward the steady-state value of 0.
PAGE 6

Temperature \(u(x, t)\) over time at various positions \(x\)

A line graph showing temperature u(x, t) decaying over time t from 0 to 5. Three curves represent different spatial positions: x=0.5, x=1.0, and x=2.0. All curves start at u=100 and decay exponentially toward 0, with the position closest to the boundary (x=0.5) decaying the fastest.

This graph depicts the temporal evolution of temperature at specific fixed points in space.

  • The curve for \(x = 0.5\) shows the most rapid cooling, as it is closest to the zero-temperature boundary.
  • The curve for \(x = 2.0\) (the center of the domain) maintains its temperature longer than the other points, though it also eventually decays to zero.
  • All points exhibit an exponential-like decay characteristic of the heat equation solution.
PAGE 7

Heat Conduction in Composite Slabs

Two such slabs put together (\(4\text{ cm}\) thick each)

Two outer faces at \(0^\circ\text{C}\)

Left slab heated to \(50^\circ\text{C}\) at \(t=0\)

Right slab heated to \(100^\circ\text{C}\) at \(t=0\)

A 3D sketch of two rectangular slabs placed side-by-side. A cylindrical rod is shown passing through the center of both. Arrows indicate the outer faces are held at 0 degrees Celsius.
A 1D horizontal line representing the rod, with endpoints labeled x=0 and x=8.

Initial temp:

\[ f(x) = \begin{cases} 50 & 0 < x < 4 \\ 100 & 4 < x < 8 \end{cases} \]

\(\vdots\)

\[ u(x,t) = \sum_{n=1}^{\infty} \frac{100}{n\pi} \left[ 1 + \cos\left(\frac{n\pi}{2}\right) - 2(-1)^n \right] e^{-1.15n^2\pi^2t/64} \sin\left(\frac{n\pi x}{8}\right) \]
PAGE 8

Surface Plot \(u(x, t)\) (\(L = 8, k = 1.15\), non-uniform initial)

A 3D surface plot of temperature u(x,t). The x-axis represents position from 0 to 8, the t-axis represents time from 0 to 10, and the vertical axis represents temperature from 0 to 100. A color bar on the right indicates temperature values from 20 to 80.

The plot visualizes the heat diffusion over time in the composite slab system. The initial temperature discontinuity at \(x=4\) quickly smooths out as heat flows toward the cooler boundaries at \(x=0\) and \(x=8\).

PAGE 9

Temperature Profile \( u(x, t) \)

The following graph illustrates the spatial temperature distribution \( u(x, t) \) across a rod of length \( L = 8 \) with a thermal diffusivity constant \( k = 1.15 \). The initial condition at \( t = 0 \) is a step function, which smooths out over time as heat diffuses through the medium.

A line graph showing temperature u(x, t) versus position x from 0 to 8. At t=0, the profile is a step function with values of 50 for x between 0 and 4, and 100 for x between 4 and 8. As time t increases from 0.1 to 10.0, the sharp corners of the step function smooth out and the overall temperature decreases toward zero across the entire length.

Parameters and Legend

  • Rod Length: \( L = 8 \)
  • Thermal Diffusivity: \( k = 1.15 \)
  • Time intervals plotted: \( t = 0, 0.1, 0.5, 1.0, 2.0, 5.0, 10.0 \)
PAGE 10

Temperature Evolution Over Time

This graph depicts the temporal evolution of temperature \( u(x, t) \) at specific fixed positions \( x \) along the rod. It demonstrates how the temperature at different points converges as time progresses.

A line graph showing temperature u versus time t from 0 to 10 for three positions: x=2.0, x=4.0, and x=6.0. The curve for x=6.0 starts highest at 100 and decays rapidly. The curve for x=4.0 starts at 75 and decays. The curve for x=2.0 starts at 50 and decays more slowly. All curves converge toward lower temperatures as time increases.

Observation Points

  • \( x = 2.0 \) (Blue line)
  • \( x = 4.0 \) (Yellow line)
  • \( x = 6.0 \) (Green line)
PAGE 11

Nonhomogeneous Boundary Conditions

now let's relax the ends at \( 0^{\circ}C \) constraint

A simple diagram of a horizontal rod with its left end at x equals 0 and its right end at x equals L.

\[ u_t = k u_{xx} \quad 0 < x < L \quad t > 0 \]

\[ u(0, t) = T_1 \]

\[ u(L, t) = T_2 \]

if \( T_1, T_2 \) are not zero, the BC's are nonhomogeneous

steady-state temp solution

  • \( \rightarrow t \rightarrow \infty \) or \( u_t = 0 \) (time is not a factor any more)
  • in \( u_t = k u_{xx} \) if \( u_t = 0 \)

\[ u_{xx} = 0 \rightarrow u = C_1 + C_2 x \text{ using } u(0) = T_1, u(L) = T_2 \]

we get

\[ u = \frac{T_2 - T_1}{L} x + T_1 = v(x) \quad \text{steady-state temp.} \]

PAGE 12
A coordinate graph with horizontal axis x and vertical axis v. A straight line segment connects the point (0, T1) to the point (L, T2).

but what about the transient solution (when time still matters)

\[ u_t = k u_{xx} \]

\[ u(0, t) = T_1 \]

\[ u(L, t) = T_2 \]

\[ u(x, 0) = f(x) \]

\[ v(x) = \frac{T_2 - T_1}{L} x + T_1 \]

define \( w(x, t) = u(x, t) - v(x) \)

\[ w_t = u_t \quad w_{xx} = u_{xx} \rightarrow w_t = k w_{xx} \]

same heat eq.

\[ w(0, t) = 0 \]

\[ w(L, t) = 0 \]

\( \leftarrow \) BC's are homogeneous in \( w \)

solution is known

\[ w(x, t) = \sum_{n=1}^{\infty} B_n e^{-k n^2 \pi^2 t / L^2} \sin\left(\frac{n \pi x}{L}\right) \]

PAGE 13

General Solution and Initial Conditions

\[ u(x,t) = v(x) + \sum_{n=1}^{\infty} B_n e^{-k n^2 \pi^2 t / L^2} \sin\left(\frac{n \pi x}{L}\right) \]

initial condition: \( u(x,0) = f(x) \)

\[ f(x) = v(x) + \sum_{n=1}^{\infty} B_n \sin\left(\frac{n \pi x}{L}\right) \]
\[ f(x) - v(x) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n \pi x}{L}\right) \]
Sine series
\[ B_n = \frac{2}{L} \int_{0}^{L} [f(x) - v(x)] \sin\left(\frac{n \pi x}{L}\right) dx \]