8/8/2015 1 euler method major: all engineering majors authors: autar kaw, charlie barker
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04/19/23http://
numericalmethods.eng.usf.edu 1
Euler Method
Major: All Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
Euler Method
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Euler’s Method
Φ
Step size, h
x
y
x0,y0
True value
y1, Predicted
value
00,, yyyxfdx
dy
Slope Run
Rise
01
01
xx
yy
00 , yxf
010001 , xxyxfyy
hyxfy 000 ,Figure 1 Graphical interpretation of the first step of Euler’s method
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Euler’s Method
Φ
Step size
h
True Value
yi+1, Predicted value
yi
x
y
xi xi+1
Figure 2. General graphical interpretation of Euler’s method
hyxfyy iiii ,1
ii xxh 1
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How to write Ordinary Differential Equation
Example
50,3.12 yeydx
dy x
is rewritten as
50,23.1 yyedx
dy x
In this case
yeyxf x 23.1,
How does one write a first order differential equation in the form of
yxfdx
dy,
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ExampleA ball at 1200K is allowed to cool down in air at an ambient temperature of 300K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by
Kdt
d12000,1081102067.2 8412
Find the temperature at 480t seconds using Euler’s method. Assume a step size of
240h seconds.
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Solution
K
f
htf
htf iiii
09.106
2405579.41200
24010811200102067.21200
2401200,01200
,
,
8412
0001
1
Step 1:
1 is the approximate temperature at 240240001 httt
K09.106240 1
8412 1081102067.2
dt
d
8412 1081102067.2, tf
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Solution ContFor 09.106,240,1 11 ti
K
f
htf
32.110
240017595.009.106
240108109.106102067.209.106
24009.106,24009.106
,
8412
1112
Step 2:
2 is the approximate temperature at 48024024012 httt
K32.110480 2
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Solution Cont
The exact solution of the ordinary differential equation is given by the solution of a non-linear equation as
9282.21022067.000333.0tan8519.1300
300ln92593.0 31
t
The solution to this nonlinear equation at t=480 seconds is
K57.647)480(
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Comparison of Exact and Numerical Solutions
Figure 3. Comparing exact and Euler’s method
0
200
400
600
800
1000
1200
1400
0 100 200 300 400 500
Time, t(sec)
Te
mp
era
ture
,
h=240
Exact Solution
θ(K
)
Step, h (480) Et |єt|%
4802401206030
−987.81110.32546.77614.97632.77
1635.4537.26100.8032.60714.806
252.5482.96415.5665.03522.2864
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Effect of step size
Table 1. Temperature at 480 seconds as a function of step size, h
K57.647)480( (exact)
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Comparison with exact results
-1500
-1000
-500
0
500
1000
1500
0 100 200 300 400 500
Time, t (sec)Tem
per
atu
re,
Exact solution
h=120h=240
h=480
θ(K
)
Figure 4. Comparison of Euler’s method with exact solution for different step sizes
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Effects of step size on Euler’s Method
-1200
-800
-400
0
400
800
0 100 200 300 400 500
Step size, h (s) Te
mp
era
ture
, θ(K
)
Figure 5. Effect of step size in Euler’s method.
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Errors in Euler’s Method
It can be seen that Euler’s method has large errors. This can be illustrated using Taylor series.
...!3
1
!2
1 31
,
3
32
1
,
2
2
1,
1 ii
yx
ii
yx
iiyx
ii xxdx
ydxx
dx
ydxx
dx
dyyy
iiiiii
...),(''!3
1),('
!2
1),( 3
12
111 iiiiiiiiiiiiii xxyxfxxyxfxxyxfyy
As you can see the first two terms of the Taylor series
hyxfyy iiii ,1
The true error in the approximation is given by
...
!3
,
!2
, 32
hyxf
hyxf
E iiiit
are the Euler’s method.
2hEt
Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit
http://numericalmethods.eng.usf.edu/topics/euler_method.html
