1. 1 Week 10 5. Applications of the LT to PDEs (continued) Example 1: This problem describes propagation of a signal generated at the end of a semi-infinite string. (2) (3) (1) Solve the following initial-boundary-value problem:

2. 2 Solution: Take the LT of Eq. (1) and use IC (3): hence, (5) (4) hence, (4) yields Take the LT of BC (2): Where do we get another condition to determine A(s) and B(s) ?...

3. 3 It can be seen from (4), that U(x, s) may grow as x → +∞, so we should make sure that it doesn’t! Evidently, the behaviour of (4) as x → +∞ depends on the sign of Re s... so, what should we assume it to be? after which (4)-(5) yield Given that the path of integration in the inverse LT can be moved arbitrarily to the right, we can safely assume that Re s > 0. Hence, (4) is bounded as x → +∞ only if Take the inverse LT...

4. 4 hence, using the 2 nd Shifting Theorem with a = x/c, hence, Example 2: Solve the following initial-boundary-value problem:

5. 5 This problem describes spreading of heat in a half-space from a source at the boundary. Solution: The usual routine yields hence, where (6)

6. 6 Rearranging (6) using the convolution theorem, we obtain Comment: where f(t) is a given function (the BC) and We shall use a formula from Q4c of TS6: (8) (7)

7. 7 Summarising (7)-(8), we obtain (9)

8. 8 Comment: where Re-write (9) in the form (observe the highlighted parts) or, equivalently, G(x, t) is called the Green’s function of this problem.

9. 9 Comment: Generally, a Green’s function G provides means to represent the solution of a problem by a convolution integral of G with the function describing the boundary or initial condition. In the latter case, the solution has the form where (a, b) is the domain where the problem is to be solved.