Partial differential equations and Bessel's ODE
By Thomas Mooney (P3048) on Wednesday, October 11, 2000 - 05:08 pm :

What is the purpose of describing whether a Partial Differential equation is parabolic, Hyperbolic etc? Also in in bessels ODE how does one derive the second solution?

By Sean Hartnoll (Sah40) on Saturday, October 14, 2000 - 03:03 pm :
The classification of the PDEs has to do with the shape of the something called characteristic curves, which can be more or less understood as the lines along which information is propagated. There is a lot of theory here, but it is important because the uniqueness, existence of solutions, and the type of boundary conditions required depends on the type of equation.

As for Bessel's eqn. there are many ways. For some values of the parameter nu, the two solutions both fall out by searching for series solutions. There is a general technique for linear differential equations which is to substitute


f(z)= å
an zn

and find recurrence relations for the an. When this doesn't give two solutions you can always find another one by trying


g(z)=log(z) å
an zn + å
bn zn

Sean


By Thomas Mooney (P3048) on Saturday, October 14, 2000 - 03:26 pm :

Thanks Shaun, but could you please show me how to get the second solution. I only know the first.

By Sean Hartnoll (Sah40) on Saturday, October 14, 2000 - 06:43 pm :
You take the equation satisfied by w(z), for example it may be the Bessel equation. You write
w(z)= ¥
å
0 
 an zn

. You substitute this into the equation, differentiating where appropriate. Each term in the resulting sum is grouped into powers of z. So you now have one big sum of the form:


¥
å
0 
 (

some expression) zn=0

You require each term to vanish. So

some expression =0.

This will be an equation that will give you the ans. And hence you have the series. It may give you two solutions, in which case you have all the solutions. Now, if it doesn't you need to find a second solution of the form I mentioned above. Do exactly the same thing, but now you will have an additional log term.

Try working through this for the Bessel equation...

Sean