Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

and Applied Analysis 3 nonlinear third-order differential equations. Section 5 gives the corresponding results for those obtained in Sections 2, 3, and 4, but for the fifth-order differential equations. In Section 6, we present some numerical results exhibiting the accuracy and efficiency of our numerical algorithms. 2. SJG Method for Third-Order Differential Equations with Constant Coefficients Let w α,β x 1 − x α 1 x , then we define the weighted space L2 w α,β −1, 1 as usual, equipped with the following inner product and norm, u, v w α,β ∫1 −1 u x v x w α,β x dx, ‖v‖w α,β v, v 1/2 w α,β . 2.1 The set of Jacobi polynomials forms a complete L2 wα,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 ) . 2.2 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 properties of Jacobi polynomials 14, 19 , then it can be easily shown that P α,β L,k 0 −1 k Γ ( k β 1 ) Γ ( β 1 ) k! , 2.3 DP α,β L,k 0 −1 k−qΓ(k β 1)(k α β 1)q LqΓ ( k − q 1)Γ(q β 1) . 2.4 Next, let w α,β L x L − x xβ, then we define the weighted space L2w α,β L 0, L in the usual way, with the following inner product and norm,