We analyze the Legendre and Chebyshev spectral Galerkin semidiscretizations of a one dimensional homogeneous parabolic problem with nonconstant coefficients. We present error estimates for both smooth and nonsmooth data. In the Chebyshev case a limit in the order of approximation is established. On the contrary, in the Legendre case we find an arbitrary high order of convegence.

The RK5GL3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on a combination of a fifth-order Runge-Kutta method and 3-point Gauss-Legendre quadrature. In this paper we describe an effective local error control algorithm for RK5GL3, which uses local extrapolation with an eighth-order Runge-Kutta method in tandem with RK5GL3, and a ...

In this paper Legendre integral method is proposed to solve optimal control problems governed by higher order ordinary differential equations. Legendre approximat ion method reduced the problem to a constrained optimizat ion problem. Penalty partial quadratic interpolation method is presented to solve the resulting constrained optimization problem. Error estimates for the Legendre approximation...

The RK5GL3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on a combination of a fifth-order Runge-Kutta method and 3-point Gauss-Legendre quadrature. In this paper we describe the propagation of local errors in this method, and show that the global order of RK5GL3 is expected to be six, one better than the underlying RungeKutta m...

The Legendre-Stirling numbers are the coeffi cients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejoh...

in this paper, a new method is investigated for model order reduction of high order systems based on moment matching technique. in this method, at first, full order model is expanded by legendre wavelet function which is included in orthogonal functions. a suitable fixed structure model is considered as reduced order model whose parameters are unknown. these unknown parameters are determined us...

We study the analyticity properties of the free energy fγ(m) of the Kac model at points of first order phase transition, in the van der Waals limit γ ↘ 0. We show that there exists an inverse temperature β0 and γ0 > 0 such that for all β ≥ β0 and for all γ ∈ (0, γ0), fγ(m) has no analytic continuation along the path m ↘ m∗ (m∗ denotes spontaneous magnetization). The proof consists in studying h...