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MathGroup Archive 2000

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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








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