# ME205: Numerical Methods for Engineers

Unit 8: Numerical Solution of Ordinary Differential Equations   Ordinary differential equations often appear in mechanical engineering  through the study of dynamics amongst other areas. Often, the equations do not permit easy solutions because of their form or the variable coefficients.  In these cases, numerical solutions may be a convenient way to tackle the problem.*

*This unit reviews some concepts of ordinary differential equations and introduces the Euler’s method, the Runge-Kutta method, and the shooting method (for boundary-value problems).

This unit will take you approximately 13 hours to complete.

☐    Subunit 8.1: 2 hours

☐    Subunit 8.2: 6 hours

☐    Web Media: 1.5 hours

☐    Assessment: 3 hours

☐    Subunit 8.3: 3 hours

☐    Subunit 8.4: 2 hours

Unit8 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Define and distinguish between ordinary and partial differential equations.
• Implement Euler’s methods for solving ordinary differential equations.
• Investigate how step size affects accuracy in Euler’s method.
• Implement and use the Runge-Kutta 2nd order method for solving ordinary differential equations.
•  Apply the shooting method to solve boundary-value problems.

8.1 Review of Ordinary Differential Equations   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Ordinary Differential Equations” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Ordinary Differential Equations” (YouTube)

Instructions: View all 5 videos under the heading “Digital Audiovisual Lectures.”  To access each video, click on the YouTube link after the title.  The approximate run time is 41 minutes.

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “A Primer on Ordinary Differential Equations” Link: University of South Florida:  Holistic Numerical Methods Institute’s “A Primer on Ordinary Differential Equations” (PDF)

Instructions: Please read the entire chapter (39 pages).  What is the difference between an ordinary differential equation and a partial differential equation?  What constitutes initial and or boundary conditions?

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

8.2 Euler’s Method   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on Euler’s Method Link: University of South Florida:  Holistic Numerical Methods Institute’s Lectures on Euler’s Method (YouTube)

Instructions: View all 4 videos under the heading “Digital Audiovisual Lectures:” “Euler’s Method of Solving ODEs: Derivation;” “Euler’s Method of Solving ODEs: Example;” “Euler’s Method of Estimating Integrals: Theory;” and “Euler’s Method of Estimating Integrals: Example.”  To access each video, click on the YouTube link after the video’s title.  The run time is approximately 37 minutes.

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Euler’s Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Euler’s Method” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Euler’s Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Euler’s Method” (PDF, DOC, HTML, or FLASH)

Instructions: Under “Multiple Choice Test,” click on the link for your preferred format (PDF, DOC, HTML, or FLASH) to download the quiz.  Please complete the entire multiple choice quiz.  Please note that the solutions only appear in the Flash version.

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Chemical Engineering Example of Euler’s Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Industrial Engineering Example of Euler’s Method” (PDF)

Instructions: Read the example in its entirety.  Try to reproduce the calculations as you read.

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Lecture: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.086: “Finite Differences, Accuracy, Stability, Convergence” and Mathematics 18.085 “Exam Review” Links: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.086: “Finite Differences, Accuracy, Stability, Convergence” (FLASH, MP4, or iTunes) and Mathematics 18.085: “Exam Review” (FLASH, MP4, or iTunes)

Also available in:

Instructions: Please view the entire video lecture titled “Finite Differences, Accuracy, Stability, Convergence” (about 55 minutes).  Then, please view the entire “Exam Review” video (52:29 minutes).

• Assessment: The Saylor Foundation’s “ME205: Unit 8 Exercise Euler’s Method in Scilab” Link: The Saylor Foundation’s “ME205: Unit 8 Exercise” (PDF)

Instructions:  Please perform this exercise.  When you are done, check your work against The Saylor Foundation’s “ME304: Unit 8 Exercise Solution Guide" (PDF).  This exercise should require less than 3 hours to complete.

8.3 Runge-Kutta Methods   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Runge-Kutta 2nd Order Method” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Runge-Kutta 2nd Order Method” (YouTube)

Instructions: Please view all 8 videos under the heading “Digital Audiovisual Lectures” in their entirety (run time: about 75 minutes).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Runge-Kutta 2nd Order Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Runge-Kutta 2nd Order Method” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Runge-Kutta 2nd Order Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Runge-Kutta 2nd Order Method” (PDF)

Instructions: Attempt all of the questions on the multiple choice quiz.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Industrial Engineering Example of Runge-Kutta 2nd Order Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Industrial Engineering Example of Runge-Kutta 2nd Order Method” (PDF)

Instructions: Read through the entire example, and try to reproduce the calculations.

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

8.4 Shooting Method   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Shooting Method” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Shooting Method” (YouTube)

Instructions: View all 6 videos under the heading “Digital Audiovisual Lectures” in their entirety (about 35 minutes).  To access each video, click on the YouTube link following the video’s title.