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 "7-5-S-Laplace Transform Issues and Opportunities in ",
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Cell["\<\
The Laplace Transform is a mathematical construct that has proven very useful \
in both solving and understanding differential equations. We introduce it and \
show its power here.\
\>", "Subsubsection",
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Cell["\<\
Definition of Laplace Transform of a function f(t) if the integral converges:\
\>", "Subsubsection",
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Cell["\<\
In order to evaluate such an integral we might be using integration by parts \
for f(t) of a different \"sort\" than exponential functions.  However, we \
turn to Mathematica to do the integrations for us and use its own powerful \
computation, look-up, and pattern matching algorithms to do Laplace \
Transforms.\
\>", "Text",
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This \"strange\" integral transforms most functions (if the integral \
converges) from a t domain to an s domain where s is (in our case) a real \
positive number.\
\>", "Subsubsection"],

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Notice the restriction that Re[s]>0 in order for us to have convergence to \
the quantity, s/(9+s^2). Thus s/(9+s^2) IS the Laplace Transform of the \
function cos(3t). \
\>", "Text",
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 "Tables of these exist and we could visit them, however since ",
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of Laplace Transforms in ",
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 "."
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 "Often the s domain is called the ",
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 ", for if the following integral - the Laplace Transform - is to make sense \
with respect to units then the units of s must be 1/Time, for in this way s*t \
has no units and so Exp[-s*t] has no units. Thus the Laplace Transform \
integral is obtained by just integrating a function of t over an interval of \
time,  [0, \[Infinity]) - resulting in a function of s, we call it \
\[ScriptCapitalL] (f(t))(s) = F(s)."
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 " can compute these Laplace Transforms and we see how to do this.  Here are \
the definitions brought up by query from ",
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 ".  If one needs more, such as an example, then go to Help Menu and type \
\"LaplaceTransform\" and you will have a window offering up a definition with \
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We have a tougher trick in undoing the Laplace Transform and this is done \
using the InverseLaplaceTransform command.\
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So here we go . . . taking Laplace Transforms of functions . . . . We Build \
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Cell[TextData[{
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functions."
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Cell[" Trig functions", "Subsubsection"],

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Continuing in this manner of discovery we could fill in a typical Laplace \
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If we are given the Laplace Transform Y(s) how might we obtain the original, \
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Let us give LaplaceTransform[y[t],t,s] a name, say Y[s]. Then things look \
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\>", "Subsubsection"],

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Can we take the Laplace Transform of a whole  differential equations? Let's \
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Let us try to replace LaplaceTransform[y[t],t,s] with Y[s] here too.\
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We can solve for the Laplace of the function in the above differential \
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our original differential equation a y\[CloseCurlyQuote]\[CloseCurlyQuote](t) \
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system - not the driver sin(t). This characterizes the system response and \
the roots of the characteristic equation, a*",
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Engineers get to \"know\" systems and the systems' behaviors by their \
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eigenvalues \"poles.\"\n\nIndeed, if we let y(0) = 0 and y'(0) = 0 then \
YSol[s] = L[s]*Q[s], and so Q[s] = Ysol[s]/L[s], where YSol[s] is the \
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effectively shows that the Laplace Transform of the Solution is \
\[OpenCurlyDoubleQuote]simply\[CloseCurlyDoubleQuote] a multiplication of \
Laplace Transform of the input or driver function, i.e. L[s], and the \
transfer function Q[s] = Q(s) = 1/ (a*",
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\[CloseCurlyDoubleQuote] of the differential equation model terms   a*y\
\[CloseCurlyQuote]\[CloseCurlyQuote](t) + b*y\[CloseCurlyQuote](t)   + c y(t) \
while the former, L[s], contains the \[OpenCurlyDoubleQuote]action\
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So when we actually use reasonable initial conditions (y(0) = 0, no initial \
displacement, and y\[CloseCurlyQuote](0) = 0, no initial velocity - depending \
upon the driver to get some activity going) on the inverse Laplace Transform, \
YSol[s], of our differential equation \
a*y\[CloseCurlyQuote]\[CloseCurlyQuote](t) + b*y\[CloseCurlyQuote](t)   + \
c*y(t)  = sin(t)\
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First we use DSolve and some formatting to grab the solution for comparison.\
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We consider  this same differential equation and apply Laplace Transforms to \
all terms, i.e. both sides of the differential equation.\
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Now we solve for Y[s], the Laplace Transform of y[t] in order to get ready \
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Now we apply the inverse transform to get the solution for y[t].\
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How does that compare to the DSolve solution? Exactly the same as we see \
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So what is the big deal?  Who would someone choose Laplace Transforms to \
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Here is the point (well one of the many points of advantage) of Laplace \
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and then we return to calculus land. More formally, the Laplace Transform \
transforms the problem from the time domain (t) to the frequency domain in \
(s). \
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We consider two additional \"Real Good\" reasons for Laplace Transforms\
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Unit Step or Heaviside Function - what does this function do.\
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Cell["\<\
We examine  the Dirac Delta function some more.  This function represents an \
instantaneous change.  E.g., we change the salt concentration by \"dumping\" \
in more salt ALL AT ONCE!  OR we \"shock\" a spring mass giving it an \
instantaneous force ALL AT ONCE!\
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The integral of an interval containing x = 0 for the DiracDelta[x] function \
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DiracDelta[x] imparts force of size 1 unit to our system, BUT all at once \
like the quick bang of a hammer, rather than a gradual application of force.\
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Transforms of UnitStep and Dirac Function - where does this fit into our \
Table of Laplace transforms?\
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UnitStep function represents a \"step\" or finite immediate change in \
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\>", "Subsubsection"],

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Cell["\<\
We examine an animation or a Manipulation of del[x,a] as a gets smaller an \
smaller. We say DiracDelta[x] is the limit of del[x,a] as a\[Rule]0. \
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Cell["\<\
Here are the plots in Manipulation to illustrate this point.  By clicking on \
the + box in this command one can see the values of a and by using the slider \
next to a one can see the effect of decreasing the value of a in the plot of \
the resulting function.\
\>", "Text",
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Cell["\<\
Suppose we have a spring mass system  and we want to see the effect of \
hitting it with  spike force of 4 units of force after two (t = 2) seconds of \
\"usual\" motion.\
\>", "Subsection",
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Now we solve for Y[s], the Laplace Transform of y[t] in order to get ready \
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\>", "Text"],

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Cell["\<\
From the plot we see the change in the action at t = 2 due to our applying a \
spike force at that time. Up until the spring mass was at rest due to it \
initial conditions of y(0) = 0 and y\[CloseCurlyQuote](0) = 0.\
\>", "Text",
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Cell["\<\
Let us consider a tank system in which we \"dump\" stuff (DiracDelta \
function) and we alter amounts suddenly (UnitStep function).  First the \
system without any drastic dumping. Can you draw a scenario for this system \
given the equations?\
\>", "Subsection",
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Let us consider a tank system in which we (1) \"dump\" stuff (DiracDelta \
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of these. What are we dumping into the system?  Where are we dumping it in?  \
When do we start dumping?  How often are we dumping it in?\
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We might then modify the model system with the new \"stuff\" function and \
Laplace away!\
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Now use LaplaceTransform syntax, solve, plot, and examine solution.\
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Cell["\<\
After solving using Laplace Transform techniques we plot our solution for \
x(t). We see that from the dumping due to stuff[t] = 40*DiracDelta[t-25] of \
40 additional units instantly at time t = 25 there is a jump in x(t) at time \
t = 25 and an attendant rapid change in the second tank, y(t).\
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Cell["\<\
Suppose we have a spring mass system  and we want to see the effect of \
turning on a force and turning off that force, over and over again, perhaps \
by placing the mass in an alternating magnetic field.   We model this  with \
small rectangular functions of forces, 0.5 units high and 1 unit long, spaced \
2 units apart, centered at 1, 3, 5, etc.\
\>", "Subsection",
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}, Open  ]],

Cell["\<\
Here is our solution plot. Notice the effects of turning on and turning off \
the driver force.\
\>", "Text",
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Cell["\<\
We examine the input function (square wave) and the output function (rounded \
wave).\
\>", "Text"],

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