Chebyshev-Fourier Spectral Methods for Nonperiodic Boundary Value Problems

Abstract

A new class of spectral methods for solving two-point boundary value problems for linear ordinary differential equations is presented in the paper. Although these methods are based on trigonometric functions, they can be used for solving periodic as well as nonperiodic problems. Instead of using basis functions periodic on a given interval−1,1, we use functions periodic on a wider interval. The numerical solution of the given problem is sought in terms of the half-range Chebyshev-Fourier (HCF) series, a reorganization of the classical Fourier series using half-range Chebyshev polynomials of the first and second kind which were first introduced by Huybrechs (2010) and further analyzed by Orel and Perne (2012). The numerical solution is constructed as a HCF series via differentiation and multiplication matrices. Moreover, the construction of the method, error analysis, convergence results, and some numerical examples are presented in the paper. The decay of the maximal absolute error according to the truncation numberNfor the new class of Chebyshev-Fourier-collocation (CFC) methods is compared to the decay of the error for the standard class of Chebyshev-collocation (CC) methods.

Receding Horizon Pseudospectral Control for Energy Maximization With Application to Wave Energy Devices

“…The remainder of this paper is organized as follows. Based on [15], [32], and [33], the definition and application of the HRCF basis for pseudospectral methods is presented in Section III. A direct application of the control algorithm for a WEC is presented in IV.…”

“…Half-range Chebyshev polynomials of the first and the second kind of order k, T h k and U h k , respectively, are orthogonal with lower order monomials with respect to the weights 1/(1 − y 2 ) 1/2 , for the first kind, and (1 − y 2 ) 1/2 , for the second kind, on the interval [0, 1]. Definitions of T h k and U h k are given in [32] based on [15].…”

“…As was shown in [32], the sets of basis functions {T h k (cos(π/2)τ )} n k=0 and {U h k (cos(π/2)τ ) sin(π/2)τ } n−1 k=0 can be employed in the resolution of a nonperiodic boundary value problem using pseudospectral methods.…”

A New Numerical Algorithm for Two-Point Boundary Value Problems

“…Rashidinia et al [5] presented a parametric spline method for (1). For these references, please see [6][7][8][9][10].…”