This paper analyzes a method for solving the thirdand fifth-order differential equations with constant coefficients using a Jacobi dual-Petrov–Galerkin method, which is more reasonable than the standard Galerkin one. The spatial approximation is based on Jacobi polynomials P (α,β) n with α, β ∈ (−1, ∞) and n is the polynomial degree. By choosing appropriate base functions, the resulting system ...

We generalize existing Jacobi–Gauss–Lobatto collocation methods for variable-order fractional differential equations using a singular approximation basis in terms of weighted Jacobi polynomials of the form (1 ± x)μP a,b j (x), where μ > −1. In order to derive the differentiation matrices of the variable-order fractional derivatives, we develop a three-term recurrence relation for both integrals...

When one approximates elliptic equations by the spectral collocation method on the Chebyshev-Gauss-Lobatto (CGL) grid, the resulting coefficient matrix is dense and illconditioned. It is known that a good preconditioner, in the sense that the preconditioned system becomes well conditioned, can be constructed with finite difference on the CGL grid. However, there is a lack of an efficient solver...

Article history: Received 18 July 2013 Received in revised form 30 December 2013 Accepted 3 January 2014 Available online 8 January 2014

There are two parts in this paper. In the first part we consider an overdetermined system of differential-algebraic equations (DAEs). We are particularly concerned with Hamiltonian and Lagrangian systems with holonomic constraints. The main motivation is in finding methods based on Gauss coefficients, preserving not only the constraints, symmetry, symplecticness, and variational nature of traje...

We analyze the consistency, stability, and convergence of an hp discontinuous Galerkin spectral element method of Kopriva [J. Comput. Phys., 128 (1996), pp. 475–488] and Kopriva, Woodruff, and Hussaini [Internat. J. Numer. Methods Engrg., 53 (2002), pp. 105–122]. The analysis is carried out simultaneously for acoustic, elastic, coupled elastic-acoustic, and electromagnetic wave propagation. Our...

This paper gives an ecient numerical method for solving the nonlinear systemof Volterra-Fredholm integral equations. A Legendre-spectral method based onthe Legendre integration Gauss points and Lagrange interpolation is proposedto convert the nonlinear integral equations to a nonlinear system of equationswhere the solution leads to the values of unknown functions at collocationpoints.