Spectral Shifted Jacobi Tau and Collocation Methods for Solving Fifth-Order Boundary Value Problems

and Applied Analysis 3 2. Preliminaries Let w α,β x 1 − x α 1 x β be a weight function in the usual sense for α, β > −1. The set of Jacobi polynomials {P α,β k x } k 0 forms a complete L 2 w α,β −1, 1 -orthogonal system, and ∥ ∥ ∥P α,β k ∥ ∥ ∥ 2 w α,β h α,β k 2 β 1Γ k α 1 Γ ( k β 1 ) ( 2k α β 1 ) Γ k 1 Γ ( k α β 1 ) . 2.1 Here, L2 w α,β −1, 1 is a weighted space defined by L2 w α,β −1, 1 {v : v is measurable and ‖v‖w α,β < ∞}, 2.2 equipped with the norm ‖v‖w α,β ∫1 −1 |v x |w α,β dx )1/2 , 2.3 and the inner product u, v w α,β ∫1 −1 u x v x w α,β x dx, ∀u, v ∈ L2 w α,β −1, 1 . 2.4 It is well known that P α,β k −x −1 P β,α k x , P α,β k −1 −1 kΓk β 1 k!Γ ( β 1 ) , P α,β k 1 Γ k α 1 k!Γ α 1 , DP α,β k x 2 −m Γ ( m k α β 1 ) Γ ( k α β 1 ) P α m,β m k−m x . 2.5 If we define the shifted Jacobi polynomial of degree k by P α,β L,k x P α,β k 2x/L − 1, L > 0, and in virtue of 2.5 , then it can easily be shown that DP α,β L,k 0 −1 k−qΓk β 1k α β 1)q LqΓ ( k − q 1Γq β 1 , DP α,β L,k L Γ k α 1 ( k α β 1 ) q LqΓ ( k − q 1Γq α 1 . 2.6 4 Abstract and Applied Analysis Now, let w α,β L x L − x xβ. The set of shifted Jacobi polynomials {P α,β L,k x } ∞ k 0 forms a complete L2 w α,β L 0, L -orthogonal system. Moreover, and due to 2.1 , we get ∥ ∥ ∥P α,β L,k ∥ ∥ ∥ 2 w α,β L ( L 2 )α β 1 h α,β k h α,β L,k , 2.7 where L2 w α,β L 0, L is a weighted space defined by L2 w α,β L 0, L { v : v is measurable and ‖v‖ w α,β L < ∞ } , 2.8 equipped with the norm