# TAMU MATH 412 - Lecture2-7web (16 pages)

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## Lecture2-7web

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Lecture Notes

- Pages:
- 16
- School:
- Texas A&M University
- Course:
- Math 412 - Theory Of Pdes

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Math 412 501 Theory of Partial Differential Equations Lecture 2 7 Sturm Liouville eigenvalue problems Sturm Liouville differential equation d d p q 0 a x b dx dx where p p x q q x x are known functions on a b and is an unknown constant The Sturm Liouville equation is a linear homogeneous ODE of the second order Sturm Liouville eigenvalue problem Sturm Liouville differential equation linear homogeneous boundary conditions J C F Sturm 1803 1855 J Liouville 1809 1882 The Sturm Liouville equation usually arises after separation of variables in a linear homogeneous PDE of the second order Examples 0 heat wave Laplace s equations dh d 2h r h dr 2 dr Laplace s equation in polar coordinates r2 standard notation x 2 x 0 canonical form x x 1 0 Heat flow in a nonuniform rod u u c K0 Q t x x K0 K0 x c c x x Q Q u x t The equation is linear homogeneous if Q x t u We assume that x u u c K0 u t x x Separation of variables u x t x G t Substitute this into the heat equation d d dG K0 G G c dt dx dx Divide both sides by c x x x G t c u 1 d d 1 dG K0 const G dt c dx dx c The variables have been separated dG G 0 dt d d K0 c 0 dx dx Sturm Liouville differential equation d d p q 0 a x b dx dx Examples of boundary conditions a b 0 Dirichlet conditions a b 0 von Neumann conditions a 2 a b 3 b Robin conditions a 0 b 0 mixed conditions a b a b periodic conditions a b 0 singular conditions d d p q 0 dx dx a x b The equation is regular if p q are real and continuous on a b and p 0 on a b The Sturm Liouville eigenvalue problem is regular if the equation is regular and boundary conditions are of the form 1 a 2 a 0 3 b 4 b 0 where i R 1 2 6 0 3 4 6 0 This includes Dirichlet Neumann and Robin conditions but excludes periodic and singular ones Regular Sturm Liouville eigenvalue problem d d p q 0 a x b dx dx 1 a 2 a 0 3 b 4 b 0 Eigenfunction nonzero solution of the boundary value problem Eigenvalue corresponding value of Eigenvalues and eigenfunctions of a regular SturmLiouville eigenvalue problem

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