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