Example 2.  Use Mathematica to find the analytic solution and graph for the I.V.P.  [Graphics:Images/MilneSimpsonMod_gr_53.gif].  

Solution 2.

[Graphics:../Images/MilneSimpsonMod_gr_54.gif]


[Graphics:../Images/MilneSimpsonMod_gr_55.gif]
[Graphics:../Images/MilneSimpsonMod_gr_56.gif]
[Graphics:../Images/MilneSimpsonMod_gr_57.gif]
[Graphics:../Images/MilneSimpsonMod_gr_58.gif]
[Graphics:../Images/MilneSimpsonMod_gr_59.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_60.gif]

[Graphics:../Images/MilneSimpsonMod_gr_61.gif]

[Graphics:../Images/MilneSimpsonMod_gr_62.gif]
[Graphics:../Images/MilneSimpsonMod_gr_63.gif]

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

[Graphics:../Images/MilneSimpsonMod_gr_64.gif]

[Graphics:../Images/MilneSimpsonMod_gr_65.gif]

[Graphics:../Images/MilneSimpsonMod_gr_66.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004