Lecture 4.4: Sturm-Liouville theory Matthew Macauley Department of - PowerPoint PPT Presentation

Lecture 4.4: Sturm-Liouville theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics M. Macauley (Clemson) Lecture 4.4: Sturm-Liouville theory Advanced Engineering Mathematics 1 / 8

� Definition A Sturm-Liouville equation is a 2nd order ODE of the following form: − d � p ( x ) y ′ � + q ( x ) y = λ w ( x ) y , where p ( x ), q ( x ), w ( x ) > 0. dx We are usually interested in solutions y ( x ) on a bounded interval [ a , b ], under some homogeneous BCs: α 2 1 + α 2 α 1 y ( a ) + α 2 y ′ ( a ) = 0 2 > 0 β 2 1 + β 2 β 1 y ( b ) + β 2 y ′ ( b ) = 0 2 > 0 . Together, this BVP is called a Sturm-Liouville (SL) problem. Remark 1 − d p ( x ) d � � � � Consider the linear differential operator L = + q ( x ) . w ( x ) dx dx dx + q ( x ) 1 d L 1 = p ( x ) d L 2 = − w ( x ) w ( x ) dx � C ∞ [ a , b ] � C ∞ [ a , b ] C ∞ [ a , b ] + q ( x ) � p ( x ) y ′ ( x ) � − 1 d y � � p ( x ) y ′ ( x ) � w ( x ) y ( x ) w ( x ) dx An SL equation is just an eigenvalue equation: Ly = λ y , and L = L 2 ◦ L 1 is self-adjoint!. M. Macauley (Clemson) Lecture 4.4: Sturm-Liouville theory Advanced Engineering Mathematics 2 / 8

Self-adjointness of the SL operator Theorem 1 − d p ( x ) d � � � � The SL operator L = + q ( x ) is self-adjoint on C ∞ α,β [ a , b ] with w ( x ) dx dx respect to the inner product � b � f , g � = f ( x ) g ( x ) w ( x ) dx . a Proof M. Macauley (Clemson) Lecture 4.4: Sturm-Liouville theory Advanced Engineering Mathematics 3 / 8

Main theorem The Sturm-Liouville problem − ( p ( x ) y ′ ) ′ + q ( x ) y = λ w ( x ) y subject to the homogeneous BCs α 2 1 + α 2 α 1 y ( a ) + α 2 y ′ ( a ) = 0 2 > 0 β 2 1 + β 2 β 1 y ( b ) + β 2 y ′ ( b ) = 0 2 > 0 . has: infinitely many eigenvalues λ 1 < λ 2 < λ 3 · · · → ∞ ; An orthonormal basis of eigenvectors { y n } , so that every f ∈ C ∞ α,β [ a , b ] can be written uniquely as ∞ � f ( x ) = c n y n ( x ) . n =1 Remarks Every 2nd order linear homogeneous ODE, y ′′ + P ( x ) y ′ + Q ( x ) y = 0 can be written as a Sturm-Liouville equation, called its self-adjoint form. Goal Given a Sturm-Liouville problem Ly = λ y (with BCs): Find its eigenvalues. Find its eigenfunctions (which are orthogonal!). M. Macauley (Clemson) Lecture 4.4: Sturm-Liouville theory Advanced Engineering Mathematics 4 / 8

Some familiar examples Definition (recall) A Sturm-Liouville equation is a 2nd order ODE of the following form: − ( p ( x ) y ′ ) ′ + q ( x ) y = λ w ( x ) y , where p ( x ), q ( x ), w ( x ) > 0. We are usually interested in solutions y ( x ) on a bounded interval [ a , b ], under some homogeneous BCs: α 2 1 + α 2 α 1 y ( a ) + α 2 y ′ ( a ) = 0 2 > 0 β 2 1 + β 2 β 1 y ( b ) + β 2 y ′ ( b ) = 0 2 > 0 . Together, this BVP is called a Sturm-Liouville (SL) problem. Example 1 (Dirichlet BCs) − y ′′ = λ y , y (0) = 0, y ( L ) = 0 is a SL problem. Here, p ( x ) = 1, q ( x ) = 0, w ( x ) = 1, α 1 = β 1 = 1, and α 2 = β 2 = 0. � n π � 2 , Eigenvalues: λ n = n = 1 , 2 , 3 , . . . L Eigenfunctions: y n ( x ) = sin( n π x L ). M. Macauley (Clemson) Lecture 4.4: Sturm-Liouville theory Advanced Engineering Mathematics 5 / 8

Some familiar examples Definition (recall) A Sturm-Liouville equation is a 2nd order ODE of the following form: − ( p ( x ) y ′ ) ′ + q ( x ) y = λ w ( x ) y , where p ( x ), q ( x ), w ( x ) > 0. We are usually interested in solutions y ( x ) on a bounded interval [ a , b ], under some homogeneous BCs: α 2 1 + α 2 α 1 y ( a ) + α 2 y ′ ( a ) = 0 2 > 0 β 2 1 + β 2 β 1 y ( b ) + β 2 y ′ ( b ) = 0 2 > 0 . Together, this BVP is called a Sturm-Liouville (SL) problem. Example 2 (Neumann BCs) − y ′′ = λ y , y ′ (0) = 0, y ′ ( L ) = 0 is a SL problem. Here, p ( x ) = 1, q ( x ) = 0, w ( x ) = 1, α 1 = β 1 = 0, and α 2 = β 2 = 1. � n π � 2 , Eigenvalues: λ n = n = 0 , 1 , 2 , 3 , . . . L Eigenfunctions: y n ( x ) = cos( n π x L ). M. Macauley (Clemson) Lecture 4.4: Sturm-Liouville theory Advanced Engineering Mathematics 6 / 8

Some familiar examples Definition (recall) A Sturm-Liouville equation is a 2nd order ODE of the following form: − ( p ( x ) y ′ ) ′ + q ( x ) y = λ w ( x ) y , where p ( x ), q ( x ), w ( x ) > 0. We are usually interested in solutions y ( x ) on a bounded interval [ a , b ], under some homogeneous BCs: α 2 1 + α 2 α 1 y ( a ) + α 2 y ′ ( a ) = 0 2 > 0 β 2 1 + β 2 β 1 y ( b ) + β 2 y ′ ( b ) = 0 2 > 0 . Together, this BVP is called a Sturm-Liouville (SL) problem. Example 3 (Mixed BCs) − y ′′ = λ y , y (0) = 0, y ′ ( L ) = 0 is a SL problem. Here, p ( x ) = 1, q ( x ) = 0, w ( x ) = 1, α 1 = β 2 = 1, and α 2 = β 1 = 0. � ( n +0 . 5) π � 2 Eigenvalues: λ n = , n = 0 , 1 , 2 , 3 , . . . L � ( n +0 . 5) π x � Eigenfunctions: y n ( x ) = sin . L M. Macauley (Clemson) Lecture 4.4: Sturm-Liouville theory Advanced Engineering Mathematics 7 / 8

Some familiar examples Definition (recall) A Sturm-Liouville equation is a 2nd order ODE of the following form: − ( p ( x ) y ′ ) ′ + q ( x ) y = λ w ( x ) y , where p ( x ), q ( x ), w ( x ) > 0. We are usually interested in solutions y ( x ) on a bounded interval [ a , b ], under some homogeneous BCs: α 2 1 + α 2 α 1 y ( a ) + α 2 y ′ ( a ) = 0 2 > 0 β 2 1 + β 2 β 1 y ( b ) + β 2 y ′ ( b ) = 0 2 > 0 . Together, this BVP is called a Sturm-Liouville (SL) problem. Example 4 (Robin BCs) − y ′′ = λ y , y (0) = 0, y ( L ) + y ′ ( L ) = 0 is a SL problem. Here, p ( x ) = 1, q ( x ) = 0, w ( x ) = 1, α 1 = β 1 = β 2 = 1, and α 2 = 0. Eigenvalues: λ n = ω 2 n , n = 1 , 2 , . . . [ ω n ’s are the positive roots of y ( x ) = x − tan Lx ]. Eigenfunctions: y n ( x ) = sin( ω n x ). M. Macauley (Clemson) Lecture 4.4: Sturm-Liouville theory Advanced Engineering Mathematics 8 / 8