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Return to Part VI of the course APMA0330
First, we remind the definition of a piecewise continuous function. For our applications, we don't need the general definition of such function made previously. Instead, we restrict ourself with the following simplified version.
A function f(t), defined on semi-infinite interval [0, ∞), is called piecewise continuous or intermittent if this interval can be broken into finite number of subintervals
so that the function f(t) is continuous on each of them and has finite limit values at the endpoints. A piecewise continuous function can be defined as
where each function fk(t), k = 1, 2, … , m, is continuous on the interval (𝑎k, bk) and has finite limit values at 𝑎k and
bk. Since we are going to apply the Laplace transformation to these intermittent functions, we require that the function fm(t) grows no faster than exponential function at infinity in order to define its Laplace transform:
In order for integral \eqref{EqInput.2} to converge, the function f(t) should grow at infinity no faster than exponential. There is known a very wide class of functions for which we know for sure that their Laplace transformations converge. It is referred to as functions-original, and they were discussed in the first section.
We usually do not specify the values of the piecewise continuous functions at the points of discontinuity (if any) because they do not effect the value of Laplace's integral \eqref{EqInput.2}. However, the inverse Laplace transformation always defines the value of the function at the point of discontinuity to be the mean value of its left and right limit values.
The key to handle the Laplace transformation of intermittent functions lies in a notational one. We need a way to write a piecewise continuous function as simple formula so that it may be handled in convenient manner. This involves the unit step or Heaviside function:
\begin{equation} \label{EqInput.3}
H(t) = \begin{cases}
1 , & \ \mbox{ for }\ t > 0,
\\
1/2, & \ \mbox{ for }\ t=0,
\\
0, & \ \mbox{ for }\ t < 0.
\end{cases}
\end{equation}
For the Laplace transformation \eqref{EqInput.2}, it does not matter what is the value of the Heaviside function at t = 0. You can define the unit step function as
\[
u(t) = \begin{cases}
1 , & \ \mbox{ for }\ t > 0,
\\
\mbox{whatever you want}, & \ \mbox{ for }\ t=0,
\\
0, & \ \mbox{ for }\ t < 0.
\end{cases}
\]
independently on the value of these functions at t = 0.
However, the inverse Laplace transform restores only the Heaviside function, but not u(t). This is the reason why we utilize the definition of the Heaviside function, but not any other unit step function---we need a one-to-one correspondence.
The shifted Heaviside function H(t−c) can be thought of as an “on”/“off” switch with a trigger value c. If we look to the left of c, the function evaluates to zero (the “off” state), and if we look to the right of c, the function evaluates to one (the “on” state).
The importance of the Heaviside function lies in the fact that it can be combined with itself and other functions to generalize the notion of turn functions “on” or “off” over certain regions of t. In particular, if
we a given two positive numbers 0 < 𝑎 < b, we can define the window function
\begin{equation} \label{EqInput.4}
W(t; a, b) = H (t-a) - H(t-b) = \begin{cases}
0 , & \ \mbox{ for }\ t < a,
\\
1, & \ \mbox{ for }\ a < t < b,
\\
0, & \ \mbox{ for }\ t > b.
\end{cases}
\end{equation}
In other words, we are only in the “on” state in the region 𝑎 ≤ t ≤ b; otherwise, we are “off”. So this form allows us to define intervals that are “on”. Of course, what we are interested in turning “off” and “on” is not simply the value one. Rather, if we are giving some function g(t) on the interval [𝑎, b], then we are manipulating this function as
\[
g(t)\, W(t; a, b) = g(t) \left[ H(t-a) - H(t-b) \right] =
\begin{cases}
0 , & \ \mbox{ for }\ t < a,
\\
g(t), & \ \mbox{ for }\ a < t < b,
\\
0, & \ \mbox{ for }\ t > b.
\end{cases}
\]
With window function, we can represent a piecewise continuous function as the sum:
The problem of finding the Laplace transformation of the piecewise continuous function \eqref{EqInput.5} is reduced to determination of the Laplace transform of every term in this series. Since the Laplace transform of the window function is known
The Laplace transformation exists for many functions of a positive real variable (usually associated with time) including discontinuous functions.
Of course, the Laplace transform does not exist for arbitrary functions, but only for those that belong to special classes. Previously, we identified that the Laplace transform exists for functions with finite jumps and that grow no faster than an exponential function at infinity. The algorithm finding a Laplace transform of an intermittent function consists
of two steps:
Rewrite the given piecewise continuous function through shifted Heaviside functions.
Use the shift rule \( {\cal L} \left[ H(t-a)\, f(t-a) \right] = e^{a\lambda}\, {\cal L} \left[ f(t) \right] . \)
We demonstrate this approach in numerous examples.
Example 1:
Consider the following continuous function
which is commonly called a “tent” function. This could model, for instance, an external signal that begins to climb at t = 1, then begins to fall at
t = 2, and eventually reaches zero (i.e. no signal) at t = 3.
 
We plot the tent function:
Plot[Piecewise[{{0, 0 < t < 1}, {t - 1, 1 < t < 2}, {3 - t,
2 < t < 3}}], {t, 0, 4.5}, PlotStyle -> {Thickness[0.02], Blue}]
Tent function.
Mathematica code
To find the Laplace transform of function (1.1), we represent it as the sume of functions involving the window function:
because the Laplace transform of the exponential function is
\( {\cal L} \left[ e^{-at} \right] = \left( \lambda + a \right)^{-1} . \)
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