On the restricted Chebyshev–Boubaker polynomials

Using the language of Riordan arrays, we study a one-parameter family of orthogonal polynomials that we call the restricted Chebyshev–Boubaker polynomials. We characterize these polynomials in terms of the three term recurrences that they satisfy, and we study certain central sequences defined by their coefficient arrays. We give an integral representation for their moments, and we show that the Hankel transforms of these moments have a simple form. We show that the (sequence) Hankel transform of the row sums of the corresponding moment matrix is defined by a family of polynomials closely related to the Chebyshev polynomials of the second kind, and that these row sums are in fact the moments of another family of orthogonal polynomials.     

A Chebyshev spectral method for solving Korteweg–de Vries equation with hydrodynamical application

It is well known that many hydrodynamical problems appearing in the study of shallow water theory or the theory of rotating fluids, can be reduced to Korteweg–de Vries equation subject to certain initial and boundary conditions. In this work, a Chebyshev spectral method for obtaining a semi-analytical solution to such equation is presented. One numerical application is considered to show how we can apply the presented proposed method. A comparison between our results and the numerical results obtained by the Hopscotch method are made.     

Stochastic Lotka–Volterra systems with Lévy noise

This paper is concerned with stochastic Lotka–Volterra models perturbed by Lévy noise. Firstly, stochastic logistic models with Lévy noise are investigated. Sufficient and necessary conditions for stochastic permanence and extinction are obtained. Then three stochastic Lotka–Volterra models of two interacting species perturbed by Lévy noise (i.e., predator–prey system, competition system and cooperation system) are studied. For each system, sufficient and necessary conditions for persistence in the mean and extinction of each population are established. The results reveal that firstly, both persistence and extinction have close relationships with Lévy noise; Secondly, the interaction rates play very important roles in determining the persistence and extinction of the species.     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

A fourth-order Runge–Kutta method based on BDF-type Chebyshev approximations

In this paper we consider a new fourth-order method of BDF-type for solving stiff initial-value problems, based on the interval approximation of the true solution by truncated Chebyshev series. It is shown that the method may be formulated in an equivalent way as a Runge–Kutta method having stage order four. The method thus obtained have good properties relatives to stability including an unbounded stability domain and large α$\alpha$-value concerning A(α)$A(\alpha )$-stability. A strategy for changing the step size, based on a pair of methods in a similar way to the embedding pair in the Runge–Kutta schemes, is presented. The numerical examples reveals that this method is very promising when it is used for solving stiff initial-value problems.     

The Chebyshev–Shamanskii Method for Solving Systems of Nonlinear Equations

We present a method, based on the Chebyshev third-order algorithm and accelerated by a Shamanskii-like process, for solving nonlinear systems of equations. We show that this new method has a quintic convergence order. We will also focus on efficiency of high-order methods and more precisely on our new Chebyshev–Shamanskii method. We also identify the optimal use of the same Jacobian in the Shamanskii process applied to the Chebyshev method. Some numerical illustrations will confirm our theoretical analysis.     

Debt–equity swap with finite time horizon—variational inequality approach

This paper concerns the finite-horizon optimal reorganization problem under debt–equity swap. The model of equity is formulated as a parabolic variational inequality, or equivalently, a free boundary problem, where the free boundary corresponds to the optimal reorganization boundary. The existence and uniqueness of the solution are proven and the behavior of the free boundary, such as smoothness, monotonicity and boundedness, is studied. To the best of our knowledge, this is the first complete set of results on debt–equity swap for finite maturity obtained using PDE techniques.     

A Fast Nystr?m–Broyden Solver by Chebyshev Compression

When a Nyström–Broyden method is used to solve numerically Urysohn integral equations, the main problem is to evaluate the action of the integral operator at a low cost. Here we suitably approximate the relevant discrete integral operator of dimension n by its m-degree truncated Chebyshev series expansion (with mn), reducing the complexity from the basic O({n ²}) to O(mn). A technique to evaluate cheaply such m is presented. Several linear and nonlinear examples are considered.     

On the orthogonality of the Chebyshev–Frolov lattice and applications

We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev polynomials. If the dimension d of the lattice is a power of two, i.e. \(d=2^m, m \in \mathbb {N}\), the resulting lattice is an admissible lattice in the sense of Skriganov. We prove that these lattices are orthogonal and possess a lattice representation matrix with orthogonal columns and entries not larger than 2 (in modulus). In particular, we clarify the existence of orthogonal admissible lattices in higher dimensions. The orthogonality property allows for an efficient enumeration of these lattices in axis parallel boxes. Hence they serve for a practical implementation of the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates in a broad range of Besov–Lizorkin–Triebel spaces. As an application, we efficiently enumerate the Frolov cubature nodes in the d-cube \([-1/2,1/2]^d\) up to dimension \(d=16\).     

加权Chebyshev逼近

本文考虑了由R. Smarzewski研究的加权Chebyshev逼近问题。首先对Smarzewski。提出的关于K-最大类加权逼近中的一个有意义而又相当困难的问题给出了肯定的回答。其次,解决了A-最大类加权逼近中类似的问题。     

Interpolation of Functions from Besov-type Spaces on Gauß–Chebyshev Grids

In this paper, we give a unified approach to error estimates for interpolation on Gauß–Chebyshev grids for functions from certain Besov-type spaces with dominating mixed smoothness properties.     

Analysis of Convergence for Improved Chebyshev–Halley Methods Under Different Conditions

In this paper, we study a class of improved Chebyshev–Halley methods in Banach spaces and prove the semilocal convergence for these methods. Compared with the super-Halley method, these methods need one less inversion of an operator, and the R-order of these methods is also higher than the one of super-Halley method under the same conditions. Using recurrence relations, we analyze the semilocal convergence for these methods under two different convergence conditions. The convergence theorems are proved to show the existence and uniqueness of a solution. We also give some numerical results to show our approach.