This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 286E1, Spring 2010Boundary Value Problems1Three major boundary conditions we have considered for the homogeneous second order ODEx00+λx= 0 are:x(0) =x(L) = 0(Dirichlet or fixedendpoint condition)(1)x(0) =x(L) = 0(Neumann or insulated endpoint condition)(2)x(L) =x(L), x(L) =x(L)(Fourier or periodic condition)(3)Two basic common features of these and similar boundary value problems (an example would be mixed boundaryconditionsx(0) =x(L) = 0 etc.) are:1.Fredholm Alternative:Given a BVP as above, for eachλ∈Rexactly one of the following two alternativesholds:(a)λis not an eigenvalue  meaning that the equationx00+λx= 0 with given boundary conditions has nosolutionsx(t) (except the trivial, identically zero, solution). In this case, for anyf(t) the nonhomogeneousBVPx00+λx=f(t) has a unique solution.(b)λis an eigenvalue  meaning that the equationx00+λx= 0 with given boundary conditions has a solutionx(t) which is not identically zero. In this case, any scalar multiple ofx(t) is also a solution and we saythatx(t) is aλeigenfunction of the given BVP....
View
Full
Document
This note was uploaded on 10/05/2010 for the course ECE 329 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Kim
 Electromagnet

Click to edit the document details