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MA221: Differential Equations

Unit 6: Numerical Methods   Until now, we have studied particular kinds of equations for which we have methods to find analytical solutions.  In most cases, however, it is impossible to find such solutions, and we must accordingly rely on methods of numerical approximation. 

Reasonably enough, the technique for obtaining a solution for an ODE that can be written as a system of first-order ODEs is numerical integration.  Roughly speaking, the value for the derivative of the desired solution is numerically integrated in order to obtain a solution to the ODE.   As numerical integration is a rather stable process (meaning that the result can be made arbitrarily close to the value of the actual integral), automated solver programs can numerically integrate a wide range of ODEs.  Note that nonlinear ODEs can be solved by numerical integration!  Despite the scope of numerical methods, however, it is important to appreciate their limitations.

Unit6 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Find the approximate solution for an ordinary differential equation using Verlet integration. - Find the approximate solution for an ordinary differential equation using Predictor-Corrector methods. - Find the approximate solution for an ordinary differential equation using the Runge-Kutta methods. - Find the approximate solution for an ordinary differential equation using the Adams-Bashforth and Adams-Moulton methods.

6.1 Verlet Integration   - Reading: Wikipedia’s “Verlet Integration” Link: Wikipedia’s “Verlet Integration” (PDF)
 
Instructions: Click on the link above and read the entire article.
 
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6.2 Predictor-Corrector Methods   - Reading: Wikipedia’s “Predictor-Corrector Method” Link: Wikipedia’s “Predictor-Corrector Method” (PDF)
 
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6.3 Runge-Kutta Methods   - Reading: MIT: “Honors Differential Equations”: Ernest Ngaruiya’s “Numerical Approximations in Differential Equations: The Runge-Kutta Method” Link: MIT: “Honors Differential Equations”: Ernest Ngaruiya’s “Numerical Approximations in Differential Equations: The Runge-Kutta Method” (PDF)
 
Instructions: Click on the link above.  Scroll down and click on the link for “18034 Honors Differential Equations.”  Read the entire paper.
 
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  • Reading: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.1: Numerical Solutions of Differential Equations” Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.1: Numerical Solutions of Differential Equations” (PDF)
     
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    Instructions: Click on the link above and scroll down to “Chapter 8: Differential Equations.”  Then click on the link for “Numerical Solutions of Differential Equations” and read the entire section.

    Terms of Use: The work above is released under a Creative Commons Attribution-Share-Alike License 1.0 (HTML).  It is attributed to Dan Sloughter, and the original version can be found here (PDF).

     

  • Reading: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “Chapter 8: Differential Equations”: “Section 8.1: Numerical Solutions of Differential Equations”: “Problems” Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations:“Chapter 8: Differential Equations”: “Section 8.1: Numerical Solutions of Differential Equations”: “Problems” (PDF)
     
    Also available in:

    EPUB

    Instructions: Click on the link above and scroll down to “Chapter 8: Differential Equations.”  Then click on the link for “Numerical Solutions of Differential Equations.”    Go to page 10 and work with problems: 4-a, c, e, g; 5-a, c, e, g; 7-b, c, d; 9-b, c, d.  After you finish, check your work against the solutions provided here.

    Terms of Use: The work above is released under a Creative Commons Attribution-Share-Alike License 1.0 (HTML).  It is attributed to Dan Sloughter, and the original version can be found here (PDF).

6.4 Adams-Bashforth and Adams-Moulton Methods   - Reading: Wapedia’s “Wiki: Linear Multistep Method” Link: Wapedia’s “Wiki: Linear Multistep Method” (PDF)
 
Instructions: Click on the link above and read the entire article.
 
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML).  It attributed to Wikipedia, and the original versin can be found here (HTML).