Euler Method of solving Initial value problems

This document is helpful to study Euler method of Numerical solutions for Ordinary differential equations of first order with Initial value problems.

1. 5/7/2024 http://numericalmethods.eng.usf.edu 1
Euler Method
Major: All Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates

2. Euler Method
http://numericalmethods.eng.usf.edu

3. http://numericalmethods.eng.usf.edu3
Euler’s Method
Φ
Stepsize,h
x
y
x
0
,y
0
Truevalue
y
1
,Predicted
value
00,, yyyxf
dx
dy

SlopeRun
Rise
 01
01
xx
yy


  
00,yxf   
010001 , xxyxfyy   hyxfy
000 ,
Figure 1Graphical interpretation of the first step of Euler’s method

4. http://numericalmethods.eng.usf.edu4
Euler’s Method
Φ
Step size
h
True Value
y
i+1,Predicted value
y
i
x
y
x
i x
i+1
Figure 2.General graphical interpretation of Euler’s method hyxfyy
iiii ,
1 
 iixxh 
1

5. http://numericalmethods.eng.usf.edu5
How to write Ordinary Differential
Equation
Example50,3.12 

yey
dx
dy x
isrewrittenas50,23.1 

yye
dx
dy x
Inthiscase yeyxf
x
23.1, 

How does one write a first order differential equation in the form ofyxf
dx
dy
,

6. http://numericalmethods.eng.usf.edu6
Example
Aballat1200Kisallowedtocooldowninairatanambienttemperature
of300K.Assumingheatislostonlyduetoradiation,thedifferential
equationforthetemperatureoftheballisgivenby  K
dt
d
12000,1081102067.2
8412




Findthetemperatureat480t secondsusingEuler’smethod.Assumeastepsizeof240h
seconds.

7. http://numericalmethods.eng.usf.edu7
Solution

 
  
 
K
f
htf
htf
iiii
09.106
2405579.41200
24010811200102067.21200
2401200,01200
,
,
8412
0001
1










Step1:1
istheapproximatetemperatureat2402400
01  httt  K09.106240
1  
8412
1081102067.2 


dt
d   
8412
1081102067.2, 

tf

8. http://numericalmethods.eng.usf.edu8
Solution Cont
For09.106,240,1
11  ti 
 
  
 
K
f
htf
32.110
240017595.009.106
240108109.106102067.209.106
24009.106,24009.106
,
8412
1112






 Step2:2
istheapproximatetemperatureat480240240
12  httt  K32.110480
2

9. http://numericalmethods.eng.usf.edu9
Solution Cont
Theexactsolutionoftheordinarydifferentialequationisgivenbythe
solutionofanon-linearequationas  9282.21022067.000333.0tan8519.1
300
300
ln92593.0
31




t


Thesolutiontothisnonlinearequationatt=480secondsisK57.647)480(

10. http://numericalmethods.eng.usf.edu10
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)
Temperature,
h=240
Exact Solution
θ(K)

11. Step, h(480) E
t |є
t|%
480
240
120
60
30
−987.81
110.32
546.77
614.97
632.77
1635.4
537.26
100.80
32.607
14.806
252.54
82.964
15.566
5.0352
2.2864
http://numericalmethods.eng.usf.edu11
Effect of step size
Table1.Temperatureat480secondsasafunctionofstepsize,hK57.647)480(
(exact)

12. http://numericalmethods.eng.usf.edu12
Comparison with exact results-1500
-1000
-500
0
500
1000
1500
0 100 200 300 400 500
Tim e, t (sec)
Temperature,
Exact solution
h=120
h=240
h=480
θ(K)
Figure 4.Comparison of Euler’s method with exact solution for different step sizes

13. http://numericalmethods.eng.usf.edu13
Effects of step size on Euler’s
Method-1200
-800
-400
0
400
800
0 100 200 300 400 500
Step size, h (s)
Temperature,
θ(K)
Figure 5.Effect of step size in Euler’s method.

14. http://numericalmethods.eng.usf.edu14
Errors in Euler’s Method
ItcanbeseenthatEuler’smethodhaslargeerrors.Thiscanbeillustratedusing
Taylorseries.     ...
!3
1
!2
1 3
1
,
3
3
2
1
,
2
2
1
,
1

 ii
yx
ii
yx
ii
yx
ii
xx
dx
yd
xx
dx
yd
xx
dx
dy
yy
iiii
ii      ...),(''
!3
1
),('
!2
1
),(
3
1
2
111 
 iiiiiiiiiiiiii xxyxfxxyxfxxyxfyy
AsyoucanseethefirsttwotermsoftheTaylorserieshyxfyy
iiii ,
1 

Thetrueerrorintheapproximationisgivenby 
...
!3
,
!2
,
32




 h
yxf
h
yxf
E
iiii
t
aretheEuler’smethod.2
hE
t

15. Additional Resources
For 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_meth
od.html

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