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 > > >