Week 8 Laplace transformation The basics The Shifting Theorems Applications of the LT to ODEs

1. The basics ۞ Consider f(t) defined for t ≥ 0. Then, if the integral exists, it is called the Laplace transform of f(t). Alternative notation: Example 2:

(integrate by parts n times) Laplace transformation is a linear – hence,

Example 3: Theorem 1: Existence of Laplace transforms Let f(t) be a piece-wise continuous function for t ≥ 0 and where M and a are positve constants. Then L[f(t)] exists for s ≥ a.

If we are given a Laplace transform F(s), we can find the original function f(t) through the inverse LT, Comment: Since the forward LT doesn’t ‘know’ how f(t) behaves at t < 0, the inverse LT can’t restore this part of f(t). There are two ways to find inverse transforms: by using the table of transforms and by evaluating a certain complex integral. Example 4: Find

The inverse LT is linear, i.e. Example 5: Find ۞ The Heaviside step function, u(t – a), is hence...

Oliver Heaviside (1850 –1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for solving differential equations (later found to be equivalent to the Laplace transformation), and predicted the existence of the “Heaviside layer” in the ionosphere. He didn’t go to university and his only paid employment was a telegraph operator job (while he was 18-24 years old). In later years his behaviour became quite eccentric: he would sign letters with the initials “W.O.R.M.” after his name, used granite blocks for furniture, and reportedly started painting his fingernails pink. For most of his life, Heaviside was at odds with the scientific establishment. Most of his recognition was gained posthumously.

Comment: Observe that i.e. the inverse transforms of two different functions coincide... how come? 2. The Shifting Theorems Theorem 2: 1st Shifting Theorem If L-1[F(s)] = f(t), then Proof: By definition of the LT...

hence, hence, Apply L-1 to this equality to obtain the desired result. █ Example 6: Find

Solution: F(s) = (s – 2)–4 is not in the table, but, using the 1st Shifting Theorem with a = –2, we can reduce the problem to F(s) = s–4: Example 7: Find The answer:

Theorem 3: 2nd Shifting Theorem If L-1[F(s)] = f(t), then Alternative formulation: Example 8: Find Solution: Use the 2nd Shifting Theorem with a = 6 and F(s) = s–4...