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Euler Method

Euler Method. Computer 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 Computer 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 rectifier-based power supply requires a capacitor to temporarily store power when the rectified waveform from the AC source drops below the target voltage. To properly size this capacitor a first-order ordinary differential equation must be solved. For a particular power supply, with a capacitor of 150 μF, the ordinary differential equation to be solved is Find voltage across the capacitor at t= 0.00004s. Use step size h=0.00002 http://numericalmethods.eng.usf.edu

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

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

  9. Solution Cont The solution to this nonlinear equation at t=0.00004 seconds 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 Voltage at 0.00004 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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