pde_slides_sturm-liouville

pde_slides_sturm-liouville - Summary of Chapter 5(Why we...

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Summary of Chapter 5 (Why we have orthogonal eigenfunctions for so many physical problems) Key: A Sturm-Lioville problem has orthogonal eigenfunctions Remarks: (1) The forms of the ODE and b.c.'s above are general enough that many physical problems can be converted to a standard Sturm-Liouville problem => Orthogonality of eigenfunctions (2) It is crucial that the b.c.'s are homogeneous. If they are not, there may not be orthogonal eigenfunctions for the system. Sturm-Liouville (eigenvalue) problem: d dx [ P x du d x ] Q x u −  R x u = 0 , (1) for u ( x ) defined on x [ a , b ], plus homogeneous b.c.'s A u ( a ) + B u '( a ) = 0 (u' du / dx ) (I) C u ( b ) + D u '( b ) = 0 . (II)
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Proof of orthogonality. .. Step 1 : Define the operator L as L { u } d dx [ P x du d x ] Q x u , such that the original Eq. (1) in the Sturm-Liouville system can be written as L { u } =  R ( x ) u (2) Let u and v be two solutions (need not be eigenfunctions at this point) to the Sturm-Liouville
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  • Spring '08
  • SivabalSivaloganathan
  • Eigenvalue, eigenvector and eigenspace, dx, Eigenfunction, Sturm-Liouville problem, orthogonal eigenfunctions

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pde_slides_sturm-liouville - Summary of Chapter 5(Why we...

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