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Euler Method
Civil Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM
Undergraduates
1/10/2010
http://numericalmethods.eng.usf.edu
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Euler Method
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Euler’s Method
y
dy
= f ( x, y ), y (0 ) = y 0
dx
Slope
=
Rise
Run
=
y1 − y 0
x1 − x0
= f ( x0 , y 0 )
y1 = y 0 + f ( x0 , y 0 )( x1 − x0 )
= y0 + f (x0 , y0 )h
3
True value
Φ
x0,y0
y1, Predicted
value
Step size, h
x
Figure 1 Graphical interpretation of the first step of Euler’s method
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Euler’s Method
y
yi +1 = yi + f ( xi , yi )h
True Value
h = xi +1 − xi
yi+1, Predicted value
Φ
yi
h
Step size
xi
xi+1
x
Figure 2. General graphical interpretation of Euler’s method
4
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How to write Ordinary Differential
Equation
How does one write a first order differential equation in the form of
dy
= f ( x, y )
dx
Example
dy
+ 2 y = 1.3e − x , y (0 ) = 5
dx
is rewritten as
dy
= 1.3e − x − 2 y, y (0 ) = 5
dx
In this case
f (x, y ) = 1.3e − x − 2 y
5
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Example
A polluted lake with an initial concentration of a bacteria is 107
parts/m3, while the acceptable level is only 5×106 parts/m3. The
concentration of the bacteria will reduce as fresh water enters the
lake. The differential equation that governs the concentration C of
the pollutant as a function of time (in weeks) is given by
dC
+ 0.06C = 0, C (0) = 107
dt
Find the concentration of the pollutant after 7 weeks. Take a step
size of 3.5 weeks.
dC
dt
= −0.06C
f (t , C ) = −0.06C
C i +1 = C i + f (t i , C i )h
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Solution
Step 1:
For i = 0, t0 = 0, C0 = 107
C1 = C0 + f (t0 , C0 )h
(
)
= 107 + f 0, 10 7 3.5
( ( ))
+ (− 6 × 10 )3.5
= 107 + − 0.06 10 7 3.5
= 107
5
= 7.9 × 106 parts / m 3
C1
is the approximate concentration of bacteria at
t = t1 = t0 + h = 0 + 3.5 = 3.5 weeks
C (3.5) ≈ C1 = 7.9 ×106 parts/m 3
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Solution Cont
Step 2:
For i = 1, t1 = 3.5, C1 = 7.9 ×106
C2 = C1 + f (t1 , C1 )h
(
)
+ (− 0.06(7.9 ×10 ))3.5
+ (− 4.74 ×10 )3.5
= 7.9 ×106 + f 3.5, 7.9 ×106 3.5
= 7.9 × 106
= 7.9 × 106
6
5
= 6.241× 106 parts/m 3
C 2 is the approximate concentration of bacteria at
t = t 2 = t1 + h = 3.5 + 3.5 = 7 weeks
C (7 ) ≈ C2 = 6.241×106 parts/m 3
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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
C (t ) = 1×107 e
 −3t 


 50 
The solution to this nonlinear equation at t=7 weeks is
C (7) = 6.5705 ×106 parts/m 3
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Comparison of Exact and
Numerical Solutions
Figure 3. Comparing exact and Euler’s method
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Effect of step size
Table 1 Concentration of bacteria after 7 weeks as a
function of step size
Step size
h
7
3.5
1.75
0.875
0.4375
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C (7 )
5.8 106
6.241 106
6.4164 106
6.4959 106
6.5337 106
Et
770470
329470
154060
74652
36763
|∈t | %
11.726
5.0144
2.3447
1.1362
0.55952
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Comparison with exact results
Figure 4. Comparison of Euler’s method with exact
solution for different step sizes
12
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Effects of step size on Euler’s
Method
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.
1 d2y
1 d3y
dy
2
3
(xi +1 − xi ) +
(
)
(
)
−
+
−
+ ...
x
x
x
x
y i +1 = y i +
1
1
i
+
i
i
+
i
2
3
2! dx x , y
3! dx x , y
dx xi , yi
i
yi +1 = yi + f ( xi , yi )( xi +1 − xi ) +
i
i
i
1
1
2
3
f ' ( xi , yi )( xi +1 − xi ) + f ' ' ( xi , yi )( xi +1 − xi ) + ...
2!
3!
As you can see the first two terms of the Taylor series
yi +1 = yi + f ( xi , yi )h are the Euler’s method.
The true error in the approximation is given by
Et =
14
f ′(xi , yi ) 2 f ′′(xi , yi ) 3
h +
h + ...
2!
3!
Et ∝ h 2
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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
THE END
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