Example 6.  Use Mathematica to find the analytic solution and graph for the I.V.P.  [Graphics:Images/AdamsBashforthMod_gr_91.gif].  

Solution 6.

[Graphics:../Images/AdamsBashforthMod_gr_92.gif]


[Graphics:../Images/AdamsBashforthMod_gr_93.gif]
[Graphics:../Images/AdamsBashforthMod_gr_94.gif]
[Graphics:../Images/AdamsBashforthMod_gr_95.gif]
[Graphics:../Images/AdamsBashforthMod_gr_96.gif]
[Graphics:../Images/AdamsBashforthMod_gr_97.gif]

Dig out the formula for the solution out of the data structure of  solset and put it in  f[t].
Plot the analytic solution at the same sample points that were used for the numerical approximations.

[Graphics:../Images/AdamsBashforthMod_gr_98.gif]

[Graphics:../Images/AdamsBashforthMod_gr_99.gif]

[Graphics:../Images/AdamsBashforthMod_gr_100.gif]
[Graphics:../Images/AdamsBashforthMod_gr_101.gif]

Just for fun, plot the Adams-Bashforth-Moulton solution and the analytic solution. Notice that there is a difference.

[Graphics:../Images/AdamsBashforthMod_gr_102.gif]

[Graphics:../Images/AdamsBashforthMod_gr_103.gif]

[Graphics:../Images/AdamsBashforthMod_gr_104.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004