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Re: Sturm-Liouville Problem (differential equation eigenvalue problem)


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
>
>
>





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