No abstract

No abstract

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of O(N 4 ) (N is the number of retained modes of polynomial approximations). This paper presents some efficient spectral algorithms, which have a condition number of O(N 2 ), based on the ultraspherical-Galerkin methods for second-order elliptic equations in one and two space variables.The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. The complexities of the algorithms are a small multiple of N d+1 operations for a d-dimensional domain with (N − 1) d unknowns, while the convergence rates of the algorithms are exponential for problems with smooth solutions. AMS subject classifications. 65N35, 65N22, 65F05, 35J05PII. S1064827500378933 1. Introduction. This paper aims to develop some efficient spectral algorithms based on the ultraspherical-Galerkin methods (UGM) for elliptic second-order differential equations in one and two space variables.Spectral methods (see, for instance, Canuto et al.[2], Doha [3,4,5], and Gottlieb and Orszag [8]) involve representing the solution to a problem in terms of a truncated series of smooth global functions. They give very accurate approximations for a smooth solution with relatively few degrees of freedom. For Dirichlet problems, Heinrichs [10] uses the Chebyshev polynomials as basis functions. It turns out that for the well-known standard spectral methods (tau, Galerkin, collocation) the condition number is very large and grows as O(N 4 ) (N is the number of retained modes of the approximation; see Orszag [14]). Heinrichs [11] proposes a spectral method based on a subclass of orthogonal ultraspherical polynomials (λ = 3 2 ) for solving the Helmholtz equation in two dimensions subject to Dirichlet homogeneous boundary conditions with a symmetric and sparse matrix, whose condition number grows only as O(N 2 ); and he shows in [12] that certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system.In this paper we are concerned with the direct solution techniques for the secondorder elliptic equations, using the ultraspherical-Galerkin approximations. We present appropriate bases for the UGM applied to Helmholtz elliptic equations with various boundary conditions. This leads to discrete systems with specially structured matrices that can be efficiently inverted. We note that the improved technique of Heinrichs [10] and the two efficient Legendre-Galerkin and Chebyshev-Galerkin approximations developed by Shen [15,16], respectively, and some other very interesting cases, can be obtained directly as special cases from our proposed ultraspherical-Galerkin approximations. This motivates our interest for making such a generalization.

The basic aim of this paper is to develop new numerical algorithms for solving some linear and nonlinear fractional-order differential equations. We have developed a new type of Chebyshev polynomials, namely, Chebyshev polynomials of sixth kind. This type of polynomials is a special class of symmetric orthogonal polynomials, involving four parameters that were constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. The proposed algorithms are basically built on reducing the fractional-order differential equations with their initial/boundary conditions to systems of algebraic equations which can be efficiently solved. The new proposed algorithms are supported by a detailed study of the convergence and error analysis of the sixth-kind Chebyshev expansion. New connection formulae between Chebyshev polynomials of the second and sixth kinds were established for this study. Some examples were displayed to illustrate the efficiency of the proposed algorithms compared to other methods in literature. The proposed algorithms have provided accurate results, even using few terms of the proposed expansion.