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In this segment what we are going to do is, we are going to talk about the different mathematical procedures for which for which
numerical methods are used. So I want to introduce you to some of them.
So these are some of the major mathematical procedures for which you use numerical
methods; nonlinear equations, differentiation, simultaneous linear equations,
curve fitting which includes interpolation and regression, and then we have integration,
ordinary differential equations. Some undergraduate courses also do include
some talk about partial differential equations, optimization, and fast fourier
transforms. So let's go ahead and, in this segment I'm going to just talk about the
first four of them. So I am going to talk about nonlinear equations, differentiation, simultaneous
linear equations and interpolation.
So if you look at the first case, we are going to talk about nonlinear equations. A sinple example here is that
of a floating ball here. So we have a floating ball and we want to be able to figure
out how much is the distance or how much of the ball is submerged under
water. So if we want to be able to find that out, for example, let's suppose that somebody tells
us that the diameter of the ball is 0.11 meters and the specific gravity is 0.6
Then, what you are able to figure out is that you are able to get this
equation which is a nonlinear equation. This particular equation which you are seeing here is a cubic
equation which you have to solve. And once you find out the value of x you will be able to figure out how much of the
ball is submerged under water. Somebody might say hey, why is this important? This can be important that,
let's suppose you are filling in this tank with water and you are using this ball
as a control so that when the ball goes all the way to the top right here, that
the flow of the water stops. So we do need to be able to figure out how much of the ball is submerged
under water when we are starting this process here. This
particular cubic equation which you are getting here is basically obtained by
using Newton's Third Law of Motion and Newton's Second Law of Motion which is basically that
the upward force is the buoyancy force
and the downward is just the weight of the ball.
So if you take, if you use the Archimedes Principle, that the weight
of the ball is equal to the weight of the water displaced, eventually you are going to end up with
the cubic equation which is right here.
Now, if we are just going to plot this particular left hand side
of this equation, this is what you are able to obtain. This is what you get as the
cubic equation. Now, what you are finding out is that you have three roots. You have this as a
root of the equation, this as another root of this equation, and this as another root of this
equation because you have a cubic equation, so cubic equation you are going to get three roots. They may be three real
roots as the case for this particular equation, or you might have two
complex roots and one real root. That's another possibility but for this particular
nonlinear equation which you see here you are getting three roots
so someboby might say hey, since we are getting three roots it means that the ball can float up to three levels. That's not the
case because we know that the diameter of the ball which we had is
0.11 meters so that means that the value of x can going from
0 to 0.11 meters.
It cannot be anything less than zero, it cannot be anything more than 0.11.
So if you look at it this way then 0 is right here and
0.11 is almost right here. And you can very well see that there is only one root
which is between 0 and 0.11. And the reason why I bring this up is because we do
need to understand that even if we have some of the best computer algorithms available to
us or best of the computer programs available to us or even availability of computational
packages such as Matlab, Mathematica, Maple, etc etc, we
still have to use the physics of the problem to be able to solve an engineering or a science problem. And you can
see this is a typical case where you are getting three real roots of the cubic equation and you are using the
physics of the problem that x can be only between 0 and 0.11 and you are able to
hone on into the root of interest to you.
The second mathematical procedure we are talking about
is differentiation here. So let's suppose somebody gives you the velocity of this rocket which is going up
as a function of time and wants you to find acceleration. So if you are trying to find
acceleration it means that you are trying to figure out what is the derivative of the velocity expression which
is given to you. Now, surely for this particular problem you can use your differential calculus
knowledge to be able to differentiate this velocity expression to get the acceleration
and put in the value of time, t, equal to 7 seconds which you are asked to do but how would if
you are writing, doing this is in a computer program, where let's suppose this function is generated within
the program itself or you are just looking for the numerical values as opposed to trying to do
this symbolically. You will have to use some kind of a numerical technique to find out the
acceleration of the rocket by using some kind of a numerical
differentiation to do so. Many times
when we continue with the discussion on differentiation, is that many times what you will have is that
you may have to find the acceleration of a particular body when you don't know what the velocity
is as a function of time, as a continuous function, but it is only given to you at
discrete data points. How are you then going to find out what the acceleration is at
time, t, equals 7 seconds? So those are the kinds of things which you talk about in numerical methods.
You're only given the value of the velocity at only at three data points let's suppose
and you are asked to find out what the acceleration is at t equal to 7 seconds. How do we go about
doing that? Do we just simply draw a straight line from this point to this point
and try to look at the slope of that straight line? How do we know that even if we
take the slope of this line, how do we know that how acurate that is? Because one of the things which you
got to understand about numerical methods is that you have to be concerned about two things
from an engineer's point of view. One is be able to get the answer and also to be able to figure out
how good that answer is. So thats something which we will talk about when we talk about
numerical differentiation. The
next mathematical procedure which you will see where numerical methods are used are simultaneous
linear equations. So this is a typical example of being able to set up
simultaneous linear equations. Here you are given velocity vs time data at three
different times and what you want to do is, you want to be able to develop a velocity profile
from 5 to 12 so that you can able to figure out what the velocity at a particular time
is. So what you will do is, over here what we are doing is we are assuming a second order
polynomial which is going through those three data points which are given right here at 5, 8
and 12. And what we want to be able to do is set up the equations so that we can find out what
the constants of this polynomial model are, this
second order polynomial profile are. So we have to find a, b and c. So what we do is we
simply substitute the values of time, t, equal to 5, 8 and 12 and since we know the velocities
of those three times we will be able to set up three equations and three
unknowns. So the question rises how so I go about solving three equations, three unknowns. That might
seem to be simple and straight forward that you already know that from your high school alegbra or from some other
courses such as physics and statics but when you have a million equations, million unknowns, what do you do then?
So thats why we do need to know about numerical methods of solving simultaneous
linear equations. The next mathematical procedure which we are talking about is
interpolation. What we need to know about interpolation is that if somebody
gives us data as a function of, well let's suppose data as a function of one
variable, so here we know velocity as a function of time. So let's suppose somebody says hey,
can you tell me what the velocity is at 7 seconds? So mainly what that means, the data
is given to you as specific points and you want to find the data at some other point which is given to you.
So what you are requested is that you are given the value of the velocity at 5,
the value of the velocity at 7, and the value of the velocity at 12 is given to you and you want to
find out what the velocity at 7 is. Now somebody might say hey, I have already taken my thermodynamics class
and we used steam tables to be able to find, let's suppose, specific heat at a particular temperature which was not
given to us in the table and all I did was I did simply drew a straight line between
the two data points and I had this formula, this single formula and I was able to calculate what the value of
the specific heat is at a particular temperature. Well, as you can see here is that, what I have done is that
this is a straight line which you are getting here and this is the second order polynomial. The blue line which you are seeing is
a second order polynomial which has been drawn through those three data points. If you try to calculate what the value
of the velocity at time, t, equal to 7 second is you will get two different values.
So again, what you are able to figure out is not
just a question of getting an answer. I can get the anwser about finding the velocity at 7 seconds by simply by
drawing a straight line between two consecutive data points and being able to use
my straight line formula to calculate the interpolation value. But right here you are finding out that the blue line
which you are seeing here, the red line is a stright line that I just drew by visual
aid and then the blue line is a second order polynomial which I drew by using
Matlab. What you are finding out, we are getting two different values for the velocity.
Which one is correct? Or, are either of them correct? So as we said in numerical methods,
not only are you supposed to find the answer, but also how good that answer is or how accurate that answer
is. And that's why we have to study numerical methods about interpolation also.
And that's the end of this segment.
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