EulerMethod Excel Implementation Elementary Differential Equations

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Math 240 Home, Textbook Contents, Online Homework Home

Geometric Interpretation - Numerical Methods

Initial Discussion

So far we have learned some algebraic techniques for solving first order differential equations of

various special forms. We will learn more techniques in the future, but we will still be restricted to

solving just selected equations of special form. Most first order equations can't be solved explicitly,even in integral form. That being the case, it is often most useful to have a numerical technique to

approximate the solution. We will consider the two simplest numerical methods for approximating

solutions, Euler's method and the improved Euler's method. These techniques build on the

geometric interpretation of differential equations that we developed in the last lab. We will consider

the example

Geometrically, this solution is the curve

that follows the arrows of the slope field

and passes through the point (0,1). We

can build an approximation to this curveas follows. Suppose we want to

approximate the value of at

Start at the point (0,1). At this point the

slope is So our arrow

points down with slope We follow

this arrow as pictured at the left to get

Since the

exact value of , this

is a reasonably good approximation.

You might observe that this is exactly

the same thing as using the tangent line

approximation from Calculus. And just

like the tangent line approximation, it works

reasonably well as long as your target (in this

case 0.1) is close enough to your starting (in

this case 0). It breaks down if you try to get too

far away from your initial value. For example, if

you try to approximate using this technique,

you get , but the exact value

is . As you can see from the

picture above, the slope field changes at each

point causing the yellow solution curve to bend up

instead of just continuing in the same downwarddirection of the initial arrow. To deal with the

problem of changing slopes, we need to make many small steps following each arrow for a short

distance and then computing the new slope (the new arrow), repeating the process until we reach

our target value. We call the

horizontal length of each short step

the step size in this process. It will

be helpful to introduce some

notation as we start repeating our

approximation process for multiple

steps. For the initial value problem

so the starting point is , we

mentary Differential Equations https://www.math.ksu.edu/math240/book/chap1/numerical.php

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call our step size and then define

For example, to approximate using a step size of 0.1 we compute

Implementing Euler's Method in Excel

This sort of repetitive calculation is exactly what computers were invented to do. While you can

carry out the assignment below by hand, you will probably want to program this algorithm into a

computer. A spreadsheet is well suited to this process, though if you want to cook up a program in

your favorite language to carry out the computations instead, be my guest.

Open Excel (or your favorite spreadsheet if you are working on your own computer). We'll start by

labeling the columns to keep track of what goes where. Enter x into cell A1, y' into cell B1, y into

cell C1, and h into cell H1. Next put the initial values and step-size into the appropriate starting

spots with 0 in cell A2, 1 in cell C2, and 0. 1 in cell H2. Then put in the formulas with =A2+$H$2 in

cell A3, =2*A2- C2 in cell B3, and =C2+$H$2*B3 in cell C3 (we'll discuss the meaning of the $ signs in

these formulas below). You now have the x column set to add the step size h as you move down

each row, the y' column to compute the slope based on the starting x and y values, and the y

column to add h times y' to the previous y. Now click and drag to select cells A3..C3 and then click

on the small square at the lower right of the box around those cells and drag it down for a dozen

rows or so. This will copy the formulas down and automates the repetitive calculation of each step.

Note that if you click now on cell A4 it shows the formula =A3+$H$2. Comparing this to the original

formula you copied from cell A3, =A2+$H$2, you see the spreadsheet recognizes that as you pull

down the formula, you want to update the reference from cell A2 to cell A3 as you pull down.

However, the reference to cell H2 hasn't changed (which is good because you still wanted toreference the same step-size). This is what the $ signs did - they told the spreadsheet the reference

was absolute, not relative, and shouldn't update when you pulled the formula down.

Naturally, you can use this technique to handle other equations. The only thing you need to edit is

the formula for y' in column B. Do be sure to copy the edited formula down the whole column.

Since our example equation is linear, we can find the exact answer to our initial value

problem is . This allows us to check how close Euler's method comes to

approximating the true solution. In cell D1 of the spreadsheet enter y exact and in cell E1 enter

rel error (which stands for relative error). Then in cell D2 enter =3*exp( - A2) +2*A2- 2 and in cell

E2 enter =( D2- C2) / D2. It will be convenient to right-click on cell E2 and Format Cells as Percentage

since relative error is a percent. Now click and drag to select cells D2 and E2 and drag those


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formulas down. This will show you how accurate the approximation produced by Euler's method is.

As you can see, Euler's method isn't particularly accurate. We can improve the approximation by

taking smaller steps. If you change cell H2 to 0. 05 to cut the step size in half, you find that the

error in the value of has been cut from 5.87% to 2.85%. Of course you had to take 10 steps

of size 0.05 to get to 0.5 while it only took 5 steps of size 0.1. So you do twice as much work but

you cut the error roughly in half. This is actually the standard pattern.

The Improved Euler Method (Heun's Method)

Just shrinking the step size isn't always practical as a way to improve accuracy in Euler's method.

Since you use the results from each step in the next round of calculations, roundoff error will build

up as you proceed. If you make the step size too small, you can build up so much roundoff error

that it swamps the "discretization error" built into the approximation. You also slow down your

calculations with small step sizes since it takes so many steps to get where you are going. Some

extra work getting a more accurate approximation than the tangent line approximation will pay off.

This leads to the Improved Euler Method, which is also sometimes called Heun's Method. So how

can we improve our approximation for

, ? Well what we

are looking for is not the tangent line to the

curve, we are looking for the "secant" line

between the points and

. Then the formula for the secant

line is where is the slope of the secant line. So how

can we approximate the slope of the secant

line? Euler's method uses the slope of the


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tangent line at the left endpoint

as the approximation to

the slope of the secant line. Geometrically,

there is no reason why the left endpoint

should give a better approximation than the slope at the right endpoint, . In fact,

the average of the slopes of the tangent lines at each endpoints

is better still. Unfortunately, we only know the left endpoint ; the right endpoint

is what we are trying to find (which is why we used the left endpoint in Euler's method). But once

we have carried out Euler's method, we have an approximation to the right endpoint, . Now

we can use this approximation to compute the approximate slope of the tangent line at the right

endpoint and then the approximate slope of the secant line

All those "approximates" in the last sentence may look scary, but the process is actually fairly

straightforward. For the initial value problem

we set and and compute and

as in Euler's method. We then improve our approximation for from to

We repeat this procedure as many times as needed to reach our desired point just as with the

original Euler's method. For example, to approximate using a step size of 0.1 we compute

the following (calculations here have only been reported to 4 decimal places):

Implementing The Improved Euler Method In Excel

Just as for Euler's method, the repeated calculations of the improved Euler method are best handled

with a computer. As before, you can use whatever tool you find most appropriate, but the simplest


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is probably the spreadsheet.

Open up your favorite spreadsheet. We set things up similarly to the Euler's method, but with a

couple of extra columns now for the extra computations. Enter x in cell A1, l ef t y' in cell B1, y

t i l de in cell C1, r i ght y' in cell D1, y in cell E1, and h in cell H1. Next enter the initial values 0 in

cell A2, 1 in cell E2, and 0. 1 in cell H2. Then enter the formulas =A2+$H$2 in cell A3, =2*A2- E2 in

cell B3, =E2+$H$2* B3 in cell C3, =2*A3- C3 in cell D3, and =E2+$H$2*( B3+D3) / 2 in cell E3. Now click

and drag to select A3..E3 and then copy the formula down the sheet to repeat the calculations as

far as you want.

As before, you can apply this method for other equations just by editing the formulas for the slopes

in columns B and D. And as before, you can add columns for the exact values and the relative error.

Enter y exact in cell F1 and rel error in cell G1. Enter the formulas =3*exp( - A2)+2*A2- 2 in cell

F2 and =( F2- E2) / F2 in cell G2. Click and drag to select F2..G2 and then copy those formulas down

the sheet to show the exact values and the relative error. As before it looks nicer if you set the

format of the cells in column G to Percentage.

Note that the improved Euler's method is twice as much work as the original Euler's method, since

you do an extra improvement step each row. On the other hand, the error for is now just

0.2%, which is much closer than the 2.85% error we got by doing twice as much work with a

smaller step size with the original Euler's method. We can improve the accuracy of our

approximation further by using a smaller step size of . The result of this calculation is off by

only 0.05% of the true value. While halving the step size (hence doubling the work) halved the

error for Euler's method, it reduces the error in the improved Euler's method by about a factor of 4.

This is another advantage of the improved Euler's method.


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Concluding Discussion

We could repeat the process of improving the estimate by using both and from the improved

Euler's method to get a New And Improved Euler's method. And of course we could then improve

the estimate another time and yet another time. This leads to a class of methods called

Runge-Kutta methods. To find the best approximations of the slope by taking weighted averages of

different simpler approximations at different points requires that we replace the simple tangent line

approximation by a Taylor series approximation and do some more algebra. But since that's all been

done, you can just look up the formulas and enter them into a spreadsheet in the same manner as

we've done here.

Of course, applying these methods requires you to pick the correct step size. Looking back at the

geometry you should realize there is no reason you have to always use the same step size. Indeed,

you will be better off if you vary the step size to be small where the solution is curving quickly and a

larger step size (hence fewer steps so quicker execution and less roundoff) where the solution is

moving in more of a straight line. In developing code to track the motion of space probes (which

have long straight paths between planets and complicated curves as the swing around planets),

Fehlberg developed a further refinement using two different Runge-Kutta methods where you

compare their answers to get an approximation for the error. This approximation can then be used

to adjust the step size automatically. These are called Runge-Kutta-Fehlberg methods and are

perhaps the best of the common techniques for finding careful approximations to initial value

problems.

While the RKF techniques are recommended for careful approximations, the methods we've learned

here, besides introducing the basic ideas, are also useful when you need rough approximations

quickly. Being so simple, Euler's method and the improved Euler's method will run quicker than

more accurate techniques. The physics model of many video games is implemented using thesemethods. To update the screen 60 times a second, you need to compute the position of many

different objects as quickly as possible. And you just need things to look and feel right to the

players; you don't need precise answers (for example, any error less than one pixel is irrelevant). In

this situation, using Euler's method or the improved Euler's method is the best choice.

Exercises

Approximate for each of the initial value problems using Euler's method, first with a step size

of and then with a step size of .

1.

2.


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3.

4.

5.

6.

Approximate for each of the initial value problems using the improved Euler's method, first

with a step size of and then with a step size of .

7.

8.

9.

10.

11.

12.

Find the exact solution to each of these equations and use that to find an exact value for for

each of the initial value problems and compare the errors in the different methods used above.

13.

14.

15.

16.

17.

18.

If you have any problems with this page, please contact [email protected].

©2010, 2014 Andrew G. Bennett


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EulerMethod Excel Implementation Elementary Differential Equations