Chapter 8 Ordinary Differential Equations and First Order IVP

Chapter 8 Ordinary Differential Equations and First Order IVP

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Unformatted text preview: - 1 - Chapter 8 Ordinary Differential Equations and 1 st Order Initial Value Problems Contents Introduction ..................................................................................................................... - 3 - Initial Value Problems versus Boundary Value Problems .......................................... - 3 - The First Order IVP .................................................................................................... - 4 - Runge-Kutta Methods for IVPs .................................................................................. - 5 - 8.1 Eulers Method.......................................................................................................... - 5 - Example 8.1 Using Eulers Method ........................................................................... - 8 - 8.2 Heuns Method........................................................................................................ - 13 - Example 8.2 Using Heuns Method .......................................................................... - 15 - 8.3 Classical 4 th Order Runge-Kutta Method ................................................................ - 17 - Example 8.3 The Classical 4 th Order Runge-Kutta Method ..................................... - 22 - 8.4 Other 4 th Order and Higher Order Runge-Kutta Methods ...................................... - 24 - 8.5 Adjustable Parameters in a User Defined Function ................................................ - 26 - Example 8.4 A Differential Equation with an Adjustable Parameter ....................... - 27 - 8.6 Step Size Stability Limit ......................................................................................... - 28 - 8.7 Automatic Step Size Adjustment ............................................................................ - 29 - 8.8 Discontinuities ........................................................................................................ - 30 - 8.9 Multistep Methods .................................................................................................. - 31 - 8.10 Summary ............................................................................................................... - 32 - Exercises ....................................................................................................................... - 33 -- 2 - List of Figures Figure 8.1 Graphical Interpretation of a Single Step Using Euler's Method, from x j to x j+1 ....................... - 7 - Figure 8.2 Comparison of Analytical Solution for Example 8.1 with Euler's Method Integration Using h=0.1 ........................................................................................................................................ - 9 - Figure 8.3 Cell Formulas for Spreadsheet in Figure 8.2 ........................................................................... - 10 - Figure 8.4 Spreadsheet Set-up for Converging the Solution to Example 8.1 with User Defined Function Figure 8....
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Chapter 8 Ordinary Differential Equations and First Order IVP

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