We could solve these equations and analyse the solution using the following steps.

1. Find the steady-state solution (2) by setting u, = 0. The steady-state equation will be an

inhomogeneous ODE. Solve the homogeneous version to find the homogeneous solution (2)

and verify that u, gives the inhomogeneous part. The steady-state solution is the sum of

these, and thus u₂(x) = u₁(x) + U₁-

2. Subtract the steady-state solution to find a PDE and conditions for v(x, t) = u(x, t) - u₂(x).

3. Use separation of variables to find v(r, t). This should include details of how to find all

coefficients in the solution.

4. Combine your results to find u(r, t). Describe the behaviour of your solution, including the

effects of k and h.

5. It can be harder to find the coefficients in the solution, so we could examine an example just

by picking one or two non-zero coefficients in your solution, which creates an initial condition

o(r). From this, you can look at the behaviour over time. For a preform, we often heat the

left end to about 700°C while the right end is at room temperature. For glass, you could

take k 0.8 and h1 but feel free to explore other values.