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The Euler's Method applet helps
us understand numerical methods
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for approximating solutions
to differential equations.
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I can choose the
differential equation
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using this pull
down menu, and I've
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selected the equation y
prime equals y squared
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minus x, the same equation that
we used in the isocline applet.
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The graphing window shows a
slope field, the slope field
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of this equation.
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And the value of the slope
field can be read off
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by rolling over the window.
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It's read off on the
right-hand side here. f
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of x, y is various values
depending on where I'm located.
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I've also chosen an initial
condition, initial value,
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of x equals-- x_0 is
zero, and y_0 is minus 1.
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I can see the actual solution
with that initial condition
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by pressing Actual from this
set of boxes and checking Start.
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Now a curve is drawn
on the graphing plane.
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This is the solution with
that initial condition.
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And a table of values shows
up in this left table.
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We can see that this is
one of the solutions which
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is sucked into the funnel.
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So we understand
the values of y of x
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quite well when x is large.
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But what if I want to
know the value of y of 1?
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According to the
table over here,
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the value is
approximately minus 0.83.
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But how do we know that?
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Euler's method is the
simplest numerical method.
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It uses the tangent
line approximation.
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If I set the step size to be
1, I can then click Start,
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and this will draw a tangent
line segment, with delta x
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equal to 1, starting at
my initial condition,
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and with slope given by the
slope field at that point.
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So the tangent line
approximation to y of 1
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is the value zero.
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Well, that's not very good.
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But I can improve things by
using a smaller step size.
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So let's go down to a step
size of 1/4, start again.
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Now I've drawn a
tangent line segment,
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but the horizontal
distance is only 1/4.
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Let's see if we can
see this more clearly
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by pressing the zoom key.
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This will zoom in on the same
picture that we had before.
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I can measure the slope
field at the end point
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of this green line segment.
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It seems to be about 0.32.
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And by pressing Next Step,
I can draw a line segment
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moving off with that slope.
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So this now, it produces a
polygon, the Euler polygon,
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which will stay closer
to the actual curve
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than the simple tangent
line approximation did.
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I can continue this process by
continuing to say Next Step.
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The table of values
appears on the left,
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and we discover that
the Euler approximation
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to y of 1 with step
size 1/4, is minus 0.75.
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Much better than the
earlier value we had.
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And I can improve
things still further
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by choosing a smaller step size.
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In fact, you get as close as
you want to the actual solution
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by selecting sufficiently
small step sizes.
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Let's do one more example
with step size of 1/8.
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Now I will click 8 times
to produce an Euler polygon
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with 8 segments.
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And I have an
estimate of minus 0.8.
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All of these estimates
are too large.
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All of these curves,
these polygons,
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lie above the actual
solution curve.
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Let's see if we can
see why this is.
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You'll notice that the slope
field is given by the formula
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y squared minus x.
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So as x increases, the slope
field decreases in value.
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So as we're moving out along
one of these Euler struts,
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the slope field is
decreasing under it.
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And that causes
the actual solution
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to fall below the Euler polygon.
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And that process will continue
as I iterate the Euler process.
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So the general rule is, if the
direction field is decreasing
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in the x-direction, you should
expect the actual solution
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to be less than
the Euler estimate.
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There are lots of
things that can go wrong
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in this kind of numerical work.
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To see one of
them, let's unzoom.
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Zoom back, clear the screen,
redraw the actual solution,
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and choose step size 1.
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Now instead of wanting
to compute y of 1,
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suppose that I wanted to compute
the value of the solution
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at x equals 6.
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Well if I try doing this
using step size of 1,
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let's see what happens.
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So I begin.
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I have the same
strut I had before.
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It's too large, but
now the slope field
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has a negative value so
that comes back down.
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Things are looking better.
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In the next step, I've overshot.
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And if I take another
step, then I've
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overshot again in the other
direction, more dramatically.
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And now the slope field
is even more negative.
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So when I take the
next step, I've
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overshot yet again,
more dramatically.
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And if I take the next step, now
my estimate for the solution,
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which is down here, at x
equals 6 is the value 7,
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this is in the range where
the slope field continues
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to increase forever.
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And so my estimated solution
will zoom off towards infinity,
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while the actual
curve is down here.
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I call this
catastrophic overshoot.
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It's just one of a number
of different things
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that can go wrong when you try
to use these numerical methods.