ENGI 9420 Engineering Analysis
Heat PDE Example

Example 8.06.1

Heat is conducted along a thin homogeneous bar extending from x = 0 to x = 2. There is no heat loss from the sides of the bar. The two ends of the bar are maintained at temperatures 100° (at x = 0) and 200° (at x = 2). The initial temperature throughout the bar at the cross-section x is

Find the temperature at any point in the bar at any subsequent time.

As noted on page 8.37 of the lecture notes, the complete solution is

complete solution

The Maple session below produces this animation:

animation of the solution

> # G.H. George ENGI 9420 2007 07 17 updated 2012 11 20

# 2012 Fall Example 8.06.1.

# Animation of the temperature distribution satisfying the heat equation du/dt = 9 * d2u/dx2

# with the initial temperature distribution (0 < x < 2).

# The left end of the bar (at x = 0) is kept at 100 C.

# The right end of the bar (at x = 2) is kept at 200 C.

> v(n,x,t) := 2320/(n*Pi)^3 * ((-1)^n - 1) * sin(n*Pi*x/2)
* exp(-(3*n*Pi/2)^2 * t);

> u(x,t) := sum(v(n,x,t),n=1..10)
+ 50*x + 100;

> with(plots,animate):

> animate(u(x,t), x=0..2, t=0..0.2, frames=50, color=blue,
thickness=2);

> # click the right mouse button on the diagram above

> # select "Animation" and select "Play".

> f(x) := sum(u(x,t),t=0..0);

> plot(f(x), x=0..2, f=-50..200, color=blue, thickness=2,
title=`t = 0.000`);

> f(x) := eval(u(x,t),t=0.001);

> plot(f(x), x=0..2, f=-50..200, color=blue, thickness=2,
title=`t = 0.001`);

> f(x) := eval(u(x,t),t=0.010);

> plot(f(x), x=0..2, f=-50..200, color=blue, thickness=2,
title=`t = 0.010`);

> f(x) := eval(u(x,t),t=0.050);

> plot(f(x), x=0..2, f=-50..200, color=blue, thickness=2,
title=`t = 0.050`);

> f(x) := eval(u(x,t),t=0.100);

> plot(f(x), x=0..2, f=-50..200, color=blue, thickness=2,
title=`t = 0.100`);

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Dr. G.H. George