Euler Method

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Euler Method

• Mechanical Engineering Majors
• Authors Autar Kaw, Charlie Barker
• http//numericalmethods.eng.usf.edu
• Transforming Numerical Methods Education for STEM
  Undergraduates

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Euler Method http//numericalmethods.eng.us
f.edu
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Eulers Method
Slope
Figure 1 Graphical interpretation of the first
step of Eulers method
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Eulers Method
Figure 2. General graphical interpretation of
Eulers method
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How to write Ordinary Differential Equation
How does one write a first order differential
equation in the form of
Example
is rewritten as
In this case
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Example
A solid steel shaft at room temperature of 27C
is needed to be contracted so that it can be
shrunk-fit into a hollow hub. It is placed in a
refrigerated chamber that is maintained at -33C.
The rate of change of temperature of the solid
shaft q is given by
Using Eulers method, find the temperature of the
steel shaft after 86400 seconds. Take a step
size of h 43200 seconds.
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Solution
Step 1
is the approximate temperature at
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Solution Cont
Step 2
is the approximate temperature at
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Solution Cont
The solution to this nonlinear equation at
t86400s is
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Comparison of Exact and Numerical Solutions
Figure 3. Comparing exact and Eulers method
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Effect of step size
Table 1 Temperature at 86400 seconds as a
function of step size, h
Step size,
86400 43200 21600 10800 5400 -153.52 -463.32 -29.542 -27.795 -26.958 127.42 437.22 3.4421 1.6962 0.85870 488.21 1675.2 14.189 6.4988 3.2902
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Comparison with exact results
Figure 4 Comparison of Eulers method with exact
solution for different step sizes
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Effects of step size on Eulers Method
Figure 5. Effect of step size in Eulers method.
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Errors in Eulers Method
It can be seen that Eulers method has large
errors. This can be illustrated using Taylor
series.
As you can see the first two terms of the Taylor
series
are the Eulers method.
The true error in the approximation is given by
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Additional Resources

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  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

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