5.2 Laplace Transformation

Purpose

We would like to know the original function (A or C) when only its derivative (A’) is known.

In other words, we want to solve differential equation!

Why differential equation?

Most of the real world problems can be modeled using differential equation, linear or nonlinear, coupled or uncoupled.

Assumption

Let A, B, C denote Function, and let T denote a Transformation.

Let A, B, T be known, and let C be unknown.

If {T(A) = B and T(C) = B} then at least ONE of C is A .

Definition of Laplace Transformation

\[ L(f(t)) = \int_{0}^{\infty} e^{-st}f(t)dt \]

Laplace Theorem

\[ L \left( \frac{df(t)}{dt} \right) = s \cdot L \left( f(t) \right) - f(0) \]

Linear Property of Laplace Transformation

If a, b are constants;

\[ L \left( a \cdot f(t) + b \cdot g(t) \right) = a \cdot L(f(t)) + b \cdot L(g(t)) \]

Use

1. If an equation of derivatives is given,

   \(\frac{df(t)}{dt}\) = given expression

2. Both side of equation can be Laplace transformed.

   \(L \left( \frac{d(f(t)}{dt} \right)\) = L(given expression)

   \(s \cdot L \left( f(t) \right) - f(0)\) = L(given expression)

   (Left side replaced by Laplace theorem, Right side by Linear property)

3. Rearrange into the following forms;

   L(f(t)) = rearranged expression not containing L(f(t))

4. Search right side expression in the Laplace Transformation Table.

5. If it is found, the left side expression of the table is the original function, i.e. f(t)!

Laplace transformation is very useful solving linear coupled differential equation.