286bvphandout - 286-E1 Spring 2010Boundary Value Problems1Three major boundary conditions we have considered for the homogeneous second order

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Unformatted text preview: 286-E1, 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 fixed-endpoint 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 aλ-eigenfunction of the given BVP....
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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.

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286bvphandout - 286-E1 Spring 2010Boundary Value Problems1Three major boundary conditions we have considered for the homogeneous second order

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