Legendre spectral method for the fractional Bratu problem

In this paper, the Legendre spectral collocation method (LSCM) is applied for the solution of the fractional Bratu's equation. It shows the high accuracy and low computational cost of the LSCM compared with some other numerical methods. The fractional Bratu differential equation is transformed into a nonlinear system of algebraic equations for the unknown Legendre coefficients and solved with some spectral collocation methods. Some illustrative examples are also given to show the validity and applicability of this method, and the obtained results are compared with the existing studies to highlight its high efficiency and neglectable error.     

An improved estimate of PSWF approximation and approximation by Mathieu functions

In this paper, an error estimate of spectral approximations by prolate spheroidal wave functions (PSWFs) with explicit dependence on the bandwidth parameter and optimal order of convergence is derived, which improves the existing result in [Chen et al., Spectral methods based on prolate spheroidal wave functions for hyperbolic PDEs, SIAM J. Numer. Anal. 43 (5) (2005) 1912-1933]. The underlying argument is applied to analyze spectral approximations of periodic functions by Mathieu functions, which leads to new estimates featured with explicit dependence on the intrinsic parameter.     

The choice of spectral element basis functions in domains with an axis of symmetry

New spectral element basis functions are constructed for problems possessing an axis of symmetry. In problems defined in domains with an axis of symmetry there is a potential problem of degeneracy of the system of discrete equations corresponding to nodes located on the axis of symmetry. The standard spectral element basis functions are modified so that the axial conditions are satisfied identically. The modified basis is employed only in spectral elements that are adjacent to the axis of symmetry. This modification of the spectral element method ensures that the nodes are the same in each element, which is not the case in other methods that have been proposed to tackle the problem along the axis of symmetry, and that there are no nodes along the axis of symmetry. The problems of Stokes flow past a confined cylinder and sphere are considered and the performance of the original and modified basis functions are compared.     

Sobolev orthogonal Legendre rational spectral methods for problems on the half line

Modified Legendre rational spectral methods for solving second-order differential equations on the half line are proposed. Some Sobolev orthogonal Legendre rational basis functions are constructed, which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series. Numerical results demonstrate the effectiveness and the spectral accuracy of this approach.     

Spectral methods based on Hermite functions for linear hyperbolic equations

We consider the approximation by spectral and pseudo‐spectral methods of the solution of the Cauchy problem for a scalar linear hyperbolic equation in one space dimension posed in the whole real line. To deal with the fact that the domain of the equation is unbounded, we use Hermite functions as orthogonal basis. Under certain conditions on the coefficients of the equation, we prove the spectral convergence rate of the approximate solutions for regular initial data in a weighted space related to the Hermite functions. Numerical evidence of this convergence is also presented. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Padé–Jacobi Filtering for Spectral Approximations of Discontinuous Solutions

Spectral type methods for the discretization of partial differential equations rely on the approximation of the solution by polynomials of high degree. These methods are proven, both theoretically and numerically, to be of infinite order of accuracy. This infinite order is achieved if the solution is very regular. On the other hand, the Gibbs phenomenon prevents – a priori – the good convergence if the solution is discontinuous. Nevertheless, for systems of conservation laws, the spectral vanishing viscosity method leads to numerical solutions that are spectrally close to the projection of the exact solution on the set of polynomials. The idea is then to postprocess the numerical solution in order to extract pertinent physical information. The aim of this paper is to propose and analyse such a postprocessing method based on rational approximants that allows to circumvent the Gibbs phenomenon and can be used as an acceleration device for spectral numerical solution.     

Spectral division methods for block generalized Schur decompositions

We provide a different perspective of the spectral division methods for block generalized Schur decompositions of matrix pairs. The new approach exposes more algebraic structures of the successive matrix pairs in the spectral division iterations and reveals some potential computational difficulties. We present modified algorithms to reduce the arithmetic cost by nearly 50%, remove inconsistency in spectral subspace extraction from different sides (left and right), and improve the accuracy of subspaces. In application problems that only require a single-sided deflating subspace, our algorithms can be used to obtain a posteriori estimates on the backward accuracy of the computed subspaces with little extra cost.

     

Numerical Approaches to Compute Spectra of Non-Self Adjoint Operators and Quadratic Pencils

In this article we are interested in the numerical computation of spectra of non-self adjoint quadratic operators. This leads to solve nonlinear eigenvalue problems. We begin with a review of theoretical results for the spectra of quadratic operators, especially for the Schrödinger pencils. Then we present the numerical methods developed to compute the spectra: spectral methods and finite difference discretization, in infinite or in bounded domains. The numerical results obtained are analyzed and compared with the theoretical results. The main difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are very unstable.     

Efficient and Accurate Legendre Spectral Element Methods for One-Dimensional Higher Order Problems

Efficient and accurate Legendre spectral element methods for solving one-dimensional higher order differential equations with high oscillatory or steep gradient solutions are proposed. Some Sobolev orthogonal/biorthogonal basis functions corresponding to each subinterval are constructed, which reduce the non-zero entries of linear systems and computational cost. Numerical experiments exhibit the effectiveness and accuracy of the suggested approaches.     

On the multidimensional zeta functions associated with theta functions,and the multidimensional Appell polynomials

We define and study the multidimensional Appell polynomials associated with theta functions. For the trivial theta functions, we obtain the various well-known Appell polynomials. Many other interesting examples are given. To push our study, by Mellin transform, we introduce and investigate the multidimensional zeta functions associated with thetas functions and prove that the multidimensional Appell polynomials are special values at the nonpositive integers of these zeta functions. Using zeta functions techniques, among others, we prove an induction formula for multidimensional Appell polynomials. The last part of this paper is devoted to spectral zeta functions and its generalization associated with Laplacians on compact Riemannian manifolds. From this generalization, we construct new Appell polynomials associated with Riemannan manifolds of finite dimensions.     

A class of nonlinear equations in Hilbert space and its application to completeness problems

Nonlinear equations arising in the spectral theory of self-adjoint operator functions and related completeness problems for eigenvectors are studied. A separation theorem about the values of the Rayleigh functional on solutions of a nonlinear equation is proved. This theorem is used, as a new approach to establish completeness of eigenvectors for some classes of self-adjoint operator functions. Examples from matrix pencils are given.     

复变函数中由调和函数求解析函数的方法

本文给出了由解析函数的实部或虚部求该解析函数的几个定理,再以例题说明其应用。这些方法及应用,解决了复变函数中这一问题的理论与计算,而且所给方法能广泛进行推广。     

On nonstandard Padé approximants suitable for effective properties of two-phase composite materials

This article investigates the existence of the nonstandard Padé approximants introduced by Cherkaev and Zhang [D.-L. Zhang and E. Cherkaev, Reconstruction of spectral function from effective permittivity of a composite material using rational function approximations, J. Comput. Phys. 228 (2009), pp. 5390–5409] for approximating the spectral function of composites from effective properties at different frequencies. The spectral functions contain information on microstructure of composites. Since this reconstruction problem is ill-posed Cherkaev [Inverse homogenization for evaluation of effective properties of a mixture, Inverse Probl. 17 (2001), pp. 1203–1218], the well-performed Padé approach is noteworthy and deserves further investigations. In this article, we validate the assumption that the effective dielectric component of interest of all two-phase composites can be approximated by Padé approximants whose denominator has nonzero power one term. We refer to this as the nonstandard Padé approximant, in contrast to the standard approximants whose denominators have nonzero constant terms. For composites whose spectral function assumes infinitely many different values such as the checkerboard microstructure, the proof is carried by using classical results for Markov–Stieltjes functions (also referred to as Stieltjes functions) Golden and Papanicolaou [Bounds on effective parameters of heterogeneous media by analytic continuation, Commun. Math. Phys. 90 (1983), pp. 473–491] and Cherkaev and Ou [De-homogenization: Reconstruction of moments of the spectral measure of the composite, Inverse Probl. 24 (2008), p. 065008]. However, it is well-known that spectral functions for microstructure such as rank-n laminates assume only finitely many different values, i.e. the measure in the Markov–Stieltjes function is supported at only finitely many points. For this case, we cannot find any existence results for nonstandard Padé approximants in the literature. The proof for this case is the focus of this article. It is done by utilizing a special product decomposition of the coefficient matrix of the Padé system. The results in this article can be considered as an extension of the Padé theory for Markov–Stieltjes functions whose spectral function take infinitely many different values to those taking only finitely many values. In the literature, the latter is usually excluded from the definition of Markov–Stieltjes functions because they correspond to rational functions, hence convergence of their Padé approximants is trivial. However, from an inverse problem point of view, we need to assure both the existence and convergence of the nonstandard Padé approximants, for all microstructures. The results in this article provide a mathematical foundation for applying the Padé approach for reconstructing the spectral functions of composites whose microstructure is not a priori known.     

Polynomial particular solutions for certain partial differential operators

In this article, we consider a variant of the Dual Reciprocity Method (DRM) for solving boundary value problems based on approximating source terms by polynomials other than the traditional basis functions. The use of pseudo‐spectral approximations and symbolic methods enables us to obtain highly accurate results without solving the often ill‐conditioned equations that occur when radial basis function approximations are used. When the given partial differential equation is either Poisson's equation or an inhomogeneous Helmholtz‐type equation, we are able to obtain either closed form particular solutions or efficient recursive algorithms. Using the particular solutions, we convert the inhomogeneous equations to homogeneous. The resulting homogeneous equations are then amenable to solution by boundary‐type methods such as the Boundary Element Method (BEM) or the Method of Fundamental Solutions (MFS). Using the MFS, we provide numerical solutions to a variety of boundary value problems in R² and R³ . Using this approach, we can achieve high accuracy with a modest number of interpolation and collocation points. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 112–133, 2003     

Modified cramer-von mises goodness-of-fit tests for spectral distribution functions

Modifications to the Cramer-von Mises goodness-of-fit test statistic for spectral distributions are discussed. The modifications consist of inserting weight functions into the usual sto¬chastic integral for the test statistic. Conditions on the weight function are given under which the integral of the weighted square of the difference between the empirical and theoretical spectral distribution functions converges in distribution to the corresponding integral of a process related to Brownian Motion. The distributions of the test statistic under certain alternatives to the null hypothesis are also discussed. A discussion is given of the large sample distributions for weight function of the form ψ(t) = at ^k ,k < –2.     

A functional limit theorem for tapered empirical spectral functions

Using convolution properties of frequency-kernels and their upper bounds we obtain some new upper bounds for the cumulants of time series statistics. From these results we derive the asymptotic normality of some spectral estimates and the tightness of tapered empirical spectral functions in the space of Lipschitz-continuous functions. It follows that tapering increases the asymptotic variance of the estimates by a constant factor. All results are proved under integrability conditions on the spectra. A functional limit theorem for the empirical spectral function is also given without assuming all moments of the underlying process to exist.     

P-凸函数及其性质

给出P-凸函数的定义及其性质,将一个有关凸函数的命题推广到P-凸函数情形,并指出赵海清等人关于该命题的证明过程是错误的.     

Spectral domain embedding for elliptic PDEs in complex domains

Spectral methods are a class of methods for solving partial differential equations (PDEs). When the solution of the PDE is analytic, it is known that the spectral solutions converge exponentially as a function of the number of modes used. The basic spectral method works only for regular domains such as rectangles or disks. Domain decomposition methods/spectral element methods extend the applicability of spectral methods to more complex geometries. An alternative is to embed the irregular domain into a regular one. This paper uses the spectral method with domain embedding to solve PDEs on complex geometry. The running time of the new algorithm has the same order as that for the usual spectral collocation method for PDEs on regular geometry. The algorithm is extremely simple and can handle Dirichlet, Neumann boundary conditions as well as nonlinear equations.     

Petrov-Galerkin Spectral Element Method for Mixed Inhomogeneous Boundary Value Problems on Polygons

The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons. As examples of applications, spectral element methods for two model problems, with the spectral accuracy in certain Jacobi weighted Sobolev spaces, are proposed. The techniques developed in this paper are also applicable to other higher order methods.     

Empirical Gramian-based spatial basis functions for model reduction of nonlinear distributed parameter systems

Correct selection of spatial basis functions is crucial for model reduction for nonlinear distributed parameter systems in engineering applications. To construct appropriate reduced models, modelling accuracy and computational costs must be balanced. In this paper, empirical Gramian-based spatial basis functions were proposed for model reduction of nonlinear distributed parameter systems. Empirical Gramians can be computed by generalizing linear Gramians onto nonlinear systems, which results in calculations that only require standard matrix operations. Associated model reduction is described under the framework of Galerkin projection. In this study, two numerical examples were used to evaluate the efficacy of the proposed approach. Lower-order reduced models were achieved with the required modelling accuracy compared to linear Gramian-based combined spatial basis function- and spectral eigenfunction-based methods.