Re: Sturm-Liouville Problem (differential equation eigenvalue problem)
- To: mathgroup at smc.vnet.net
- Subject: [mg21880] Re: Sturm-Liouville Problem (differential equation eigenvalue problem)
- From: "Andrew" <bzhang at ee.cityu.edu.hk>
- Date: Wed, 2 Feb 2000 22:54:36 -0500 (EST)
- Organization: City University of Hong Kong
- References: <86r7p8$ba0@smc.vnet.net> <8710uh$a6b$6@dragonfly.wolfram.com>
- Sender: owner-wri-mathgroup at wolfram.com
Hi, Kuska:
It's so glad to receive your response.
How to transform it to homogeneous boundary conditions .
Can you tell me references about the *transformation*.
Best Regards
Andrew
Jens-Peer Kuska <kuska at informatik.uni-leipzig.de> wrote in message
news:8710uh$a6b$6 at dragonfly.wolfram.com...
> Hi Andrew,
>
> a) if you have inhomogen boundary conditions like
> u[a]=va and u[b]=vb you *must* transform the
> equation to get homogen boundary conditions
> Otherwise you can't determine the eigenvalue.
> b) Shooting for eigenvalue problems is a hard task
> because you have either no solution (EIG is not an
> eigenvalue) or a infinte number of solutions because
> c*u[x] is an solution for every c. A multiple
> shooting method works relative good in the most cases.
>
> Hope that helps
>
> Andrew wrote:
> >
> > 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
>
>
>