Chapter_9_Systems_of_ODES_and_BVP_102009

# Chapter_9_Systems_of_ODES_and_BVP_102009 - Chapter 9...

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1 Chapter 9 Systems of ODES and Boundary Value Problems Contents: Introduction . ........................................................................................................................ 3 9.1 Second Order Initial Value Problems: Integrating a System of Two Simultaneous 1 st Order ODE . ................................................................................................................... 3 Example 9.1 A Second Order BVP Solved with Heun’s Method . ............................... 9 9.2 Integrating Systems with an Arbitrary Number of First Order ODEs . ....................... 12 Example 9.2 Solution of a Second Order ODE with the RK4 Algorithm . ................. 14 Example 9.3 Solution of Three Simultaneous First Order ODEs . .............................. 16 9.3 Second Order Boundary Value Problems . .................................................................. 18 Example 9.4 Solution of a Second Order Boundary Value Problem Using the Shooting Method . .................................................................................... 21 9.4 Varieties of Initial Value Problems and Boundary Value Problems for Second Order ODEs . .......................................................................................................................... 25 9.4.1 Possible Initial Value Problem Constraints . ...................................................... 26 9.4.2 Possible Boundary Value Problem Constraints . ................................................ 26 Example 9.5 Boundary Value Problems with Multiple Roots . ................................... 28 Chapter 9 Summary . ......................................................................................................... 33 Exercises . .......................................................................................................................... 34 Table of Figures: Figure 9.1 Simultaneous Integration of Two ODE with Euler's Method . .......................... 6 Figure 9.2 VB Code for a User Defined Function to Integrate Two ODE Simultaneously 7 Figure 9.3 Heun's Method for Two Simultaneous ODE Showing Stage-by-Stage Completion of a Single Step from to ࢐ା૚ ...................................................................... 8 Figure 9.4 VB Code for Heun's Method as a User Defined Function for Two Simultaneous ODE. ............................................................................................................. 8 Figure 9.5 Heated Metal Rod Loses Heat to the Surroundings . ......................................... 9 Figure 9.6 Column Headings for Simultaneous Integration of Two ODE in Example 9.1 ........................................................................................................................................... 10 Figure 9.7 Using the Function Arguments Window to use Heun_2 to Integrate f1 and f2 from ࢞ ൌ ૙ to ࢞ൌ૙ .૞ ..................................................................................................... 11 Figure 9.8 Numerical Solution of the ODE in Example 9.1 using Heun's Method . ......... 12 Figure 9.9 VB Code for a Heun's Method Function for Simultaneous Integration of m- ODE . ................................................................................................................................. 13 Figure 9.10 VB Code for Derivatives Subroutine for use with User Defined Function Heun_m in Example 9.1 . .................................................................................................. 14 Figure 9.11 VB Code for User Function RK4_m which Simultaneously Integrates - ODE . ................................................................................................................................. 15 Figure 9.12 Solution to the Harmonic Oscillator Equation using the RK4 Method . ........ 16 Figure 9.13 Numerical Integration of Three Simultaneous First Order ODE using the RK4 Method. ..................................................................................................................... 17 Figure 9.14 Plot of Solution for 3 Simultaneous ODE Solved with RK4 method . .......... 18

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2 Figure 9.15 Heated Metal Rod in Example 9.4 . ............................................................... 21 Figure 9.16 Results of Guessing y2=10: y2(10)= 235.04. Note that for this grid size and 2 steps per grid point we have at least 4 significant figures using the RK4 method. . ......... 22 Figure 9.17 Improving the Estimate of the Unknown Boundary Value using the Secant Method with ࢌሺ࢟ ሺ૙ሻሻ ൌ ૛૙૙ െ ࢟ ሺ૚૙ሻ ........................................................................ 23 Figure 9.18 Using Excel Solver to Find the Initial Value Satisfying the Boundary Condition in Example 9.4 . ................................................................................................ 24 Figure 9.19 Using Solver and the RK4 Method to Find the Solution to the BVP in Example 9.4 . ..................................................................................................................... 24 Figure 20 Using the RK4 Method to Scan for Starting Values for the Initial Conditions Required to Solve the Second Order BVP in Example 9.4. .............................................. 25 Figure 9.21 Spreadsheet for Scanning for Starting Values for Nonlinear BVP . .............. 30 Figure 9.22 Saving the Module that Contains User Defined Function RK4_m from Example 9.4 . ..................................................................................................................... 31 Figure 9.23 Graphing the Boundary Value as a Function of the Guessed Initial Value.
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## This note was uploaded on 01/26/2011 for the course CH E 2112 taught by Professor Dr.harwell during the Spring '10 term at The University of Oklahoma.

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Chapter_9_Systems_of_ODES_and_BVP_102009 - Chapter 9...

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