Experiments With the Laplace Transform
Part 3. A partial differential equation problem
In this part we will use the Laplace transform to investigate another problem involving the one-dimensional heat equation. Other modules dealing with this equation include Introduction to the One-Dimensional Heat Equation, The One-Dimensional Heat Equation, and Fourier Transform I.
This time we will be looking at a semi-infinite rod. We will impose a boundary condition at the left end, but none at the right which we assume stretches off to infinity. For the second boundary condition, we assume that the desired solution is bounded. The initial temperature distribution will be constant at 1. We will keep the left end constant at 1 as well until time t = 5, when we drop it to 0. Here is the symbolic description of the problem:
The graph of the left boundary condition is shown below
Suppose that u(x,t) is the solution of this problem. Let U(x,s) be the Laplace transform of u(x,t) with respect to t. Since the transform variable is t, we may assume that the transform of uxx(x,t) is just Uxx(x,s).
- Show that U(x,s) must satisfy the following conditions. (Here your justification will be a combination of text and computer algebra computations.)
In addition it can be shown that
(You are not required to show this.)
- For each fixed s, the transformed partial differential equation is a second-order linear differential equation in x. Explain why a solution U(x,s) of the differential equation must have the form
Here the coefficients a and b may vary with s but not x.
- Explain why a(s) must be identically 0.
- Use the remaining condition to show that U(x,s) must be
- Now we want to take the inverse Laplace transform of U(x,s). The second term presents no problem. We think of the first term as a product of two functions of s. The first factor is
and the second factor is
Find functions h1 and h2 so that the first factor is the Laplace transform of h1 and the second the transform of h2. (Here t is the variable and x will appear in h2 as a parameter.)
- We may use the convolution property for the Laplace transform discussed in Part 1 to write the solution u(x,t) as a convolution integral plus a second term. Evaluate this convolution integral, and define u(x,t).
- Examine the graph of u(x,t) as a function of t for a range of different values of x. Then look at the surface z = u(x,t). Explain why the function u(x,t) is behaving as you would expect for this problem. (If the function u(x,t) is not behaving as you would expect, decide whether the problem lies with your function or your expectations, and adjust accordingly.)
