We will now cover some more complicated Lapalce Transformations:
The Convolution Theorem states:
\( \mathcal {L} \{ f \ast g \} = F(s) G(s) \)
Where \(f \ast g\) is the convolution operator:
\( f \ast g = \displaystyle \int_0^t f(\tau) g(t-\tau) d\tau\)
The inverse of the theorem is simply:
\( \mathcal {L}^{-1} \{ F(s) G(s) \} = f \ast g \)
Evaluate \( \mathcal {L} \{ e^{-2t} \ast t^4 \} \).
Evaluate \( \mathcal {L}^{-1} \{ \frac{s}{s^4-9} \} \).
A special case is when \(g(t) = 0\) which gives:
\( f \ast 1 = \int_0^t {f(\tau) d\tau}\)
The Lapalce Transformation is:
\( \mathcal{L} \{ \int_0^t {f(\tau) d\tau}\} = \cfrac{F(s)}{s} \)
The inverse Laplace is:
\( \mathcal {L}^{-1} \{ \cfrac{F(s)}{s} \} = \int_0^t {f(\tau) d\tau} \)
This allows us to solve integro-differential equations.
Solve \( f(t) = 2t -4 \int_0^t \sin (\tau) f(t-\tau) d\tau \).
Consider a piecewise continuous functions with period \(T\) such as:
\( f(t) = \begin{cases} 0, & 0 \le t \lt \frac{T}{2} \\ 1, & \frac{T}{2} \lt t \le T \end{cases} \)
The Lapalce Transformation is:
\( \mathcal{L} \{f\} = \cfrac{1}{1-e^{-sT}} \displaystyle \int_0^T {e^{-st} f(t)} dt\)
Evaluate \( \mathcal{L} \{f\} \) for the function above for a period of 4.
The Dirac function is a special unit impluse where the "on" time is squeezed to only a moment in time:
\( \delta (t-t_0) = \lim\limits_{a \to \infty} \delta_a (t-t_0) \)
Where the unit impluse is defined as:
\( \delta_a (t-t_0) = \begin{cases} 0, & 0 \le t \lt t_0 - a \\ \frac{1}{2a}, & t_0 - a \le t \lt t_0 + a \\ 0, & t_0 + a \ge t \end{cases} \)
That means the Dirac Function is infinitiy when \(t = t_0\). It is found in applications with temporary external forces or voltage, for example. The Lapalce Transformation is:
\( \mathcal{L} \{\delta (t-t_0) \} = e^{-st_0}\)
Solve \(y'' + 2y' +2y = \delta(t-\pi)\) subject to \( y(0) = 1, y'(0)=0 \).