Sturm-Liouville Problem (differential equation eigenvalue problem)
- To: mathgroup at smc.vnet.net
- Subject: [mg21796] Sturm-Liouville Problem (differential equation eigenvalue problem)
- From: "Andrew" <bzhang at ee.cityu.edu.hk>
- Date: Thu, 27 Jan 2000 22:57:39 -0500 (EST)
- Organization: City University of Hong Kong
- Sender: owner-wri-mathgroup at wolfram.com
Hello,
The trouble I met is about Sturm-Liouville problems, it's a
differential equation eigenvalue problem.
If we know function values at each endpoints, say u(a)=VA,
u(b)=VB, can we solve the Sturm-Liouville problem by shooting
method? Can we solve it by scaled Pr(u)fer transformation?
The 'regular' Sturm-Liouville problem is described as follows:
On interval [a,b],
- (pu')' + qu = EIG* r* u (1)
where real coefficient functons p,q,r are continues, a and b are
finite real number. EIG is the unknow eigenvalue, u is the
unknow eigenfunction. u' is the first order derivative.
However, as we know the standard boundary condition is:
A1*u(a)+A2*p(a)u'(a)=0 at point a (2)
B1*u(b)+B2*p(b)u'(b)=0 at point b (3)
How can we meet the requirement of (2) and (3) when we
only know: u(a)=VA, u(b)=VB?
Or, is it true we need not know boundary conditions to find
EIG only?
Actually, it seems ridiculous to me to let A2 and B2 equal to 0
in Eq.(2) and (3), because it make u(a)=u(b)=0.
Thank you
Andrew