Example 6.  Use Mathematica to find the analytic solution and graph for the I.V.P.  [Graphics:Images/MilneSimpsonMod_gr_94.gif].  

Solution 6.

[Graphics:../Images/MilneSimpsonMod_gr_95.gif]


[Graphics:../Images/MilneSimpsonMod_gr_96.gif]
[Graphics:../Images/MilneSimpsonMod_gr_97.gif]
[Graphics:../Images/MilneSimpsonMod_gr_98.gif]
[Graphics:../Images/MilneSimpsonMod_gr_99.gif]
[Graphics:../Images/MilneSimpsonMod_gr_100.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/MilneSimpsonMod_gr_101.gif]

[Graphics:../Images/MilneSimpsonMod_gr_102.gif]

[Graphics:../Images/MilneSimpsonMod_gr_103.gif]
[Graphics:../Images/MilneSimpsonMod_gr_104.gif]

Just for fun, plot the Milne-Simpson solution and the analytic solution. Notice that there is a difference.

[Graphics:../Images/MilneSimpsonMod_gr_105.gif]

[Graphics:../Images/MilneSimpsonMod_gr_106.gif]

[Graphics:../Images/MilneSimpsonMod_gr_107.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004