Perturbation Solution of
Find an approximate solution of
as
The equation obviously has an analytical solution, so the purpose
here is to demonstrate the perturbation method on an equation with
a known solution, so that we can see that the method works.
First, define the equation:
Find the exact solutions, which comes out as a list of 2 replacment rules:
Show how to access the element of a list (in case you do not know):
Check that these solutions really do satisfy qe:
Define functions x(ε)for the two exact, analytical solutions
Now prepare to find an approximate perturbation solution. Note
the O[ε] symbol is powerful in Mathematica, try changing {n,0,2} to {n,0,10) and see what happens...
Substitute the series solution for xinto the equation qe:
LogicalExpand is really cool, and works because of the presence of the O[ε] symbol :
The "logical and" might just as well be a list of equations that need to be true. In fact the equation can be accessed as list, for example:
So here are the two approximate solutions to qe:
Compare the exact (green) and approximate (red) solutions:
