Module for Adams Bashforth Moulton Method for O.D.E.'s

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Algorithm (Adams-Bashforth-Moulton Method).  To approximate the solution of the initial value problem with   over by using the predictor formula

      

and  the corrector formula

    .  

Animations (Adams-Bashforth-Moulton Method  Adams-Bashforth-Moulton Method).  
Internet hyperlinks to animations.

Mathematica Subroutine (Adams-Bashforth-Moulton Method).

Example 1.  Solve the I.V.P.    with    over  .
Plot the solution. Report the last point which is the numerical approximation to  (10, y(10)).

Solution 1.

Example 2.   Solve    with    over  .
Use 65 subintervals of [0,10] to get your answer.  Plot the solution.
Use 37 subintervals of [0,10] to get your answer.  Plot the solution.

Solution 2.

Example 3.  Use Mathematica or techniques learned in calculus to obtain the analytic solution to the D.E.

Solution 3.

Example 4.  Graph the analytic solution to the D.E. found in problem 5.

Solution 4.

Example  5.  Why does the second solution in problem 2 exhibit a chaotic behavior ?  How can this be remedied ?

Solution 5.

Adams-Bashforth-Moulton Method (Fixed a little).  The fix for Adams Bashforth Moulton for the stiff differential equation     with    over   is iterating the corrector more than once.  They tend to send the solution into chaos.

Mathematica Subroutine (Adams-Bashforth-Moulton #2).

Example 6.  Solve    with    over  .  Use the ABM2 subroutine with iteration on the corrector.
Use 37 subintervals of [0,10] to get your answer.  Plot the solution.

Solution 6.

Old Lab Project  (Adams-Bashforth-Moulton Method  Adams-Bashforth-Moulton Method).  
Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

Adams-Bashforth-Moulton Method  Adams-Bashforth-Moulton Method  
Internet hyperlinks to web sites and a bibliography of articles.
  

Downloads (Adams-Bashforth-Moulton Method Adams-Bashforth-Moulton Method).  
Download this Mathematica notebook.  

(c) John H. Mathews 2003