Orthogonal Functions: The Legendre, Laguerre, and Hermite - PowerPoint PPT Presentation

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Orthogonal Functions: The Legendre, Laguerre, and Hermite Polynomials Thomas Coverson 1 Savarnik Dixit 3 Alysha Harbour 2 Tyler Otto 3 1 Department of Mathematics Morehouse College 2 Department of Mathematics University of Texas at Austin 3 Department of Mathematics Louisiana State University SMILE REU Summer 2010 Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Outline General Orthogonality 1 Legendre Polynomials 2 3 Sturm-Liouville Conclusion 4 Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Overview When discussed in R 2 , vectors are said to be orthogonal when the dot product is equal to 0. w · ˆ ˆ v = w 1 v 1 + w 2 v 2 = 0 . Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Overview Definition � b We define an inner product ( y 1 | y 2 ) = a y 1 ( x ) y 2 ( x ) dx where y 1 , y 2 ∈ C 2 [ a , b ] . Definition Two functions are said to be orthogonal if ( y 1 | y 2 ) = 0. Definition A linear operator L is self-adjoint if ( Ly 1 | y 2 ) = ( y 1 | Ly 2 ) for all y 1 , y 2 . Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Trigonometric Functions and Fourier Series Orthogonality of the Sine and Cosine Functions Expansion of the Fourier Series ∞ f ( x ) = a 0 � 2 + ( a k cos kx + b k sin kx ) k = 1 Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Legendre Polynomials Legendre Polynomials are usually derived from differential equations of the following form: ( 1 − x 2 ) y ′′ − 2 xy ′ + n ( n + 1 ) y = 0 We solve this equation using the standard power series method. Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Legendre Polynomials Suppose y is analytic. Then we have ∞ � a k x k y ( x ) = k = 0 ∞ y ′ ( x ) = � a k + 1 ( k + 1 ) x k k = 0 ∞ y ′′ ( x ) = � a k + 2 ( k + 1 )( k + 2 ) x k k = 0 Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Recursion Formula After implementing the power series method, the following recursion relation is obtained. a k + 2 ( k + 2 )( k + 1 ) − a k ( k )( k − 1 ) − 2 a k ( k ) − n ( n + 1 ) a k = 0 a k + 2 = a k [ k ( k + 1 ) − n ( n + 1 )] ( k + 2 )( k + 1 ) Using this equation, we get the coefficients for the Legendre polynomial solutions. Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Legendre Polynomials L 0 ( x ) = 1 L 1 ( x ) = x L 2 ( x ) = 1 2 ( 3 x 2 − 1 ) L 3 ( x ) = 1 2 ( 5 x 3 − 3 x ) L 4 ( x ) = 1 8 ( 35 x 4 − 30 x 2 + 3 ) L 5 ( x ) = 1 8 ( 63 x 5 − 70 x 3 + 15 x ) Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Legendre Graph Figure: Legendre Graph Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Sturm-Liouville A Sturm-Liouville equation is a second-order linear differential equation of the form ( p ( x ) y ′ ) ′ + q ( x ) y + λ r ( x ) y = 0 p ( x ) y ′′ + p ′ ( x ) y ′ + q ( x ) y + λ r ( x ) y = 0 which allows us to find solutions that form an orthogonal system. Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Sturm-Liouville cont. We can define a linear operator by Ly = ( p ( x ) y ′ ) ′ + q ( x ) y which gives the equation Ly + λ r ( x ) y = 0 . Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Self-adjointness To obtain orthogonality, we want L to be self-adjoint. ( Ly 1 | y 2 ) = ( y 1 | Ly 2 ) which implies 0 = ( Ly 1 | y 2 ) − ( y 1 | Ly 2 ) 1 ) ′ + qy 1 | y 2 ) − ( y 1 | ( py ′ 2 ) ′ + qy 2 ) = (( py ′ � b ( p ′ y ′ 1 y 2 + py ′′ 1 y 2 + qy 1 y 2 − y 1 p ′ y ′ 2 − y 1 py ′′ = 2 − y 1 q 1 y 2 ) dx a Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Self-adjointness � b ( p ′ y ′ 1 y 2 + py ′′ 1 y 2 − y 1 p ′ y ′ 2 − y 1 py ′′ = 2 ) dx a � b [ p ( y ′ 2 y 1 )] ′ dx 1 y 2 − y ′ = a = p ( b )( y ′ 1 ( b ) y 2 ( b ) − y ′ 2 ( b ) y 1 ( b )) − p ( a )( y 1 ( a ) y 2 ( a ) − y ′ 2 ( a ) y 1 ( a )) Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Orthogonality Theorem Theorem If ( y 1 , λ 1 ) and ( y 2 , λ 2 ) are eigenpairs and λ 1 � = λ 2 then ( y 1 | y 2 ) r = 0 . Proof. ( Ly 1 | y 2 ) = ( y 1 | Ly 2 ) ( − λ 1 ry 1 | y 2 ) = ( y 1 | − λ 2 ry 2 ) � b � b y 1 y 2 rdx = λ 2 y 1 y 2 rdx λ 1 a a λ 1 ( y 1 | y 2 ) r = λ 2 ( y 1 | y 2 ) r ( y 1 | y 2 ) r = 0 Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Legendre Polynomials - Orthogonality Recall the Legendre differential equation ( 1 − x 2 ) y ′′ − 2 xy ′ + n ( n + 1 ) y = 0 . So Ly = (( 1 − x 2 ) y ′ ) ′ λ = n ( n + 1 ) r ( x ) = 1 . We want L to be self-adjoint, so we must determine necessary boundary conditions. Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Sturm-Liouville Problem - Legendre For any two functions f , g ∈ C [ − 1 , 1 ] , by the general theory, we get � 1 Lf ( x ) g ( x ) − f ( x ) Lg ( x ) dx − 1 � 1 (( 1 − x 2 ) f ′ ) ′ g ( x ) − f ( x )(( 1 − x 2 ) g ′ ) ′ dx = − 1 = [( 1 − x 2 )( f ′ g − g ′ f )] 1 − 1 = 0 . Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Legendre Polynomials - Orthogonality Because ( 1 − x 2 ) = 0 when x = − 1 , 1 we know that L is self-adjoint on C [ − 1 , 1 ] .Hence we know that the Legendre polynomials are orthogonal by the orthogonality theorem stated earlier. Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Hermite Polynomials For a Hermite Polynomial, we begin with the differential equation y ′′ − 2 xy ′ + 2 ny = 0 Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Hermite Orthogonality First, we need to arrange the differential equation so it can be written in the form ( p ( x ) y ′ ) ′ + ( q ( x ) + λ r ( x )) y = 0 . We must find some r ( x ) by which we will multiply the equation. For the Hermite differential equation, we use r ( x ) = e − x 2 to get ( e − x 2 y ′ ) ′ + 2 ne − x 2 y = 0 ⇒ e − x 2 y ′′ − 2 xe − x 2 y ′ + 2 ne − x 2 y = = 0 Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Hermite Orthogonality Sturm-Liouville problems can be written in the form Ly + λ r ( x ) y = 0 . In our case, Ly = ( e − x 2 y ′ ) ′ and λ r ( x ) = 2 ne − x 2 y . � ∞ 0 = ( Lf | g ) − ( f | Lg ) = Lf ( x ) g ( x ) − f ( x ) Lg ( x ) dx −∞ Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Hermite Orthogonality So we get from the general theory that � ∞ ( e − x 2 f ′ ( x )) ′ g ( x ) − f ( x )( e − x 2 g ′ ( x )) ′ dx −∞ � ∞ [( e − x 2 )( f ′ ( x ) g ( x ) − g ′ ( x ) f ( x ))] ′ dx = −∞ Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Hermite Orthogonality With further manipulation we obtain a →−∞ [( e − x 2 )( f ′ ( x ) g ( x ) − g ′ ( x ) f ( x ))] 0 lim a b →∞ [( e − x 2 )( f ′ ( x ) g ( x ) − g ′ ( x ) f ( x ))] b + lim 0 Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.

General Orthogonality Legendre Polynomials Sturm-Liouville Conclusion Hermite Orthogonality We want x →±∞ e − x 2 f ( x ) g ′ ( x ) = 0 lim for all f , g ∈ BC 2 ( −∞ , ∞ ) . So we impose the following conditions on the space of functions we consider x →±∞ e − x 2 / 2 h ( x ) = 0 lim and x →±∞ e − x 2 / 2 h ′ ( x ) = 0 lim for all h ∈ C 2 ( −∞ , ∞ ) . Coverson, Dixit, Harbour, Otto Orth.Funct. Leg., Lag. Hermite.