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Euler Method

Euler Method. Mechanical Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Euler Method http://numericalmethods.eng.usf.edu. y. True value. y 1 , Predicted value. x 0 ,y 0. Φ.

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Euler Method

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

  2. Euler Methodhttp://numericalmethods.eng.usf.edu

  3. y True value y1, Predicted value x0,y0 Φ Step size, h x Euler’s Method Slope Figure 1 Graphical interpretation of the first step of Euler’s method http://numericalmethods.eng.usf.edu

  4. y True Value yi+1, Predicted value yi Φ h Step size x xi xi+1 Euler’s Method Figure 2. General graphical interpretation of Euler’s method http://numericalmethods.eng.usf.edu

  5. 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 http://numericalmethods.eng.usf.edu

  6. Example A solid steel shaft at room temperature of 27°C 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 −33°C. The rate of change of temperature of the solid shaft q is given by Using Euler’s method, find the temperature of the steel shaft after 86400 seconds. Take a step size of h = 43200 seconds. http://numericalmethods.eng.usf.edu

  7. Solution Step 1: is the approximate temperature at http://numericalmethods.eng.usf.edu

  8. Solution Cont Step 2: is the approximate temperature at http://numericalmethods.eng.usf.edu

  9. Solution Cont The solution to this nonlinear equation at t=86400s is http://numericalmethods.eng.usf.edu

  10. Comparison of Exact and Numerical Solutions Figure 3. Comparing exact and Euler’s method http://numericalmethods.eng.usf.edu

  11. Effect of step size Table 1 Temperature at 86400 seconds as a function of step size, h http://numericalmethods.eng.usf.edu

  12. Comparison with exact results Figure 4 Comparison of Euler’s method with exact solution for different step sizes http://numericalmethods.eng.usf.edu

  13. Effects of step size on Euler’s Method Figure 5. Effect of step size in Euler’s method. http://numericalmethods.eng.usf.edu

  14. Errors in Euler’s Method It can be seen that Euler’s 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 Euler’s method. The true error in the approximation is given by http://numericalmethods.eng.usf.edu

  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_method.html

  16. THE END http://numericalmethods.eng.usf.edu

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