Trigonometric-Fitted Explicit Numerov-Type Method with Vanishing Phase-Lag and Its First and Second Derivatives

An effectively four-stage, sixth-order, hybrid explicit Numerov-type method is presented for the solution of the special second-order initial value problem. The new method uses variable step, is trigonometric fitted and the phase-lag is nullified along with its first and second derivative. Extensive numerical tests illustrate the superiority of our proposal over similar methods found in the relevant literature.     

Error analysis of the finite-strip method for parabolic equations

The finite-strip method (FSM) is a hybrid technique which combines spectral and finite-element methods. Finite-element approximations are made for each mode of a finite Fourier series expansion. The Galerkin formulated method is set apart from other weighted-residual techniques by the selection of two types of basis functions, a piecewise linear interpolating function and a trigonometric function. The efficiency of the FSM is due in part to the orthogonality of the complex exponential basis: The linear system which results from the weak formulation is decoupled into several smaller systems, each of which may be solved independently. An error analysis for the FSM applied to time-dependent, parabolic partial differential equations indicates the numerical solution error is O(h² + M^?r). M represents the Fourier truncation mode number and h represents the finite-element grid mesh. The exponent r ≥ 2 increases with the exact solution smoothness in the respective dimension. This error estimate is verified computationally. Extending the result to the finite-layer method, where a two-dimensional trigonometric basis is used, the numerical solution error is O(h² + M^?r + N^?q). The N and q represent the trucation mode number and degree of exact solution smoothness in the additional dimension. © 1993 John Wiley & Sons, Inc.     

Interpolation Wavelets in Boundary Value Problems

We propose and validate a simple numerical method that finds an approximate solution with any given accuracy to the Dirichlet boundary value problem in a disk for a homogeneous equation with the Laplace operator. There are many known numerical methods that solve this problem, starting with the approximate calculation of the Poisson integral, which gives an exact representation of the solution inside the disk in terms of the given boundary values of the required functions. We employ the idea of approximating a given 2π-periodic boundary function by trigonometric polynomials, since it is easy to extend them to harmonic polynomials inside the disk so that the deviation from the required harmonic function does not exceed the error of approximation of the boundary function. The approximating trigonometric polynomials are constructed by means of an interpolation projection to subspaces of a multiresolution analysis (approximation) with basis 2π-periodic scaling functions (more exactly, their binary rational compressions and shifts). Such functions were constructed by the authors earlier on the basis of Meyer-type wavelets; they are either orthogonal and at the same time interpolating on uniform grids of the corresponding scale or only interpolating. The bounds on the rate of approximation of the solution to the boundary value problem are based on the property ofMeyer wavelets to preserve trigonometric polynomials of certain (large) orders; this property was used for other purposes in the first two papers listed in the references. Since a numerical bound of the approximation error is very important for the practical application of the method, a considerable portion of the paper is devoted to this issue, more exactly, to the explicit calculation of the constants in the order bounds of the error known earlier.     

求解混合三角多项式方程组的直接GBQ方法

许多科学与工程领域,我们经常需要求混合三角多项式方程组的全部解.一般来说,混合三角多项式方程组可以通过变量替换及增加二次多项式转化为多项式方程组,进而利用数值方法进行求解,但这种转化会增大问题的规模从而增加计算量.在本文中,我们不将问题转化,考虑利用直接同伦方法求解,并给出基于GBQ方法构造的初始方程组及同伦定理的证明.数值实验结果表明我们构造的直接同伦方法较已有的直接同伦方法更加有效.     

The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational principle (Galerkin’s approach) with a choice of global basis functions which are combinations of trigonometric functions. Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is treated by Newton’s iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced. Numerical results of solving the problems for polygonal and ferroconcrete plates are presented. Communicated by A. Zhou.     

Hybrid numerical methods for periodic initial value problems involving second-order differential equations

Computer simulation of problems in celestial mechanics often leads to the numerical solution of the system of second-order initial value problems with periodic solutions. When conventional methods are applied to obtain the solution, the time increment must be limited to a value of the order of the reciprocal of the frequency of the periodic solution.In this paper hybrid methods of orders four and six which are P-stable are developed. Further, the adaptive hybrid methods of polynomial order four and trigonometric order one have also been discussed. The numerical results for the undamped Duffing equation with a forced harmonic function are listed.     

Numerical solution of Fredholm integral equations of first kind by two-dimensional trigonometric wavelets in holder space <Emphasis Type="Italic">C</Emphasis><Superscript>α</Superscript>([<Emphasis Type="Italic">a,b</Emphasis>])

In this article, we employ trigonometric wavelet bases to numerical solution of Fredholm integral equations of first kind in Holder space. Employment of Galerkin method for trigonometric wavelets in Fredholm integral equations of first kind has resulted in occurrence of two-dimensional trigonometric wavelets. Here, we present the convergence of two-dimensional trigonometric wavelets in numerical solution in Holder space C ^α([a, b]).     

Guaranteed bounds on homogenized periodic media by FFT-based Galerkin method

An FFT-based method, introduced by Moulinec and Suquet [1] in 1994, is an effective alternative to conventional Finite Element Method (FEM) for numerical homogenization of periodic media. Here, we summarize the recent variational reformulation and discretizations by Vondřejc et al. [2–6], which are based on conforming Galerkin approximations with trigonometric polynomials as basis functions. This insight, naturally leading to guaranteed bounds on homogenized matrix, opens a wide area of further investigations, which are also briefly discussed here. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimized Local Trigonometric Bases

This paper generalizes Malvar–Coifman–Meyer (MCM) wavelets by extending the choice of bell functions. We dispense with the orthonormality of MCM wavelets to produce a family of smooth local trigonometric bases that efficiently compress trigonometric functions. Any such basis is, in general, not orthogonal, but any element of the dual basis differs from the corresponding element of the original basis only by the shape of the bell. Furthermore, in our scheme the bell functions are bounded by 1 and the dual bell functions are bounded by (2^1/2+ 1)/2 ≈1.2. These bounds ensure the numerical stability of the forward and the inverse transformations in these bases. Numerical examples demonstrate that in many cases the proposed bases provide substantially better (up to a factor of two) compression than the standard MCM wavelets.     

Trigonometric wavelet-based method for elastic thin plate analysis

Dealing with the boundary conditions is one of the difficult problems when using wavelet function as trial function to carry out structural analysis. In this paper, the two-dimensional tensor product trigonometric Hermite wavelet that has both good approximation characteristics of trigonometric function and multi-resolution, local characteristics of wavelet is proposed as trial function, and the united formulation of elastic bending, vibration and buckling of rectangle thin plate (on elastic foundation) with different boundary conditions is derived based on the principle of minimum potential energy. Two approaches, hierarchical and multi-resolution approach, are presented to improve calculation accuracy. The impact of proposed method is discussed by different numerical examples. Due to the Hermite interpolation properties, the proposed trigonometric wavelet method can process all kinds of boundary conditions conveniently. The solution accuracy of hierarchical method can be increased steadily with raising the order of wavelet, while the solution accuracy of multi-resolution method can be improved along with increasing the scale of wavelet.     

A new symmetric linear eight-step method with fifth trigonometric order for the efficient integration of the Schrödinger equation

On the basis of a classical symmetric eight-step method, an optimized method with fifth trigonometric order for the numerical solution of the Schrödinger equation is developed in this work. The local truncation error analysis of the method proves the decrease of the maximum power of the energy in relation to the corresponding classical method, which renders the method highly efficient. This is confirmed by comparing the method to other methods from the literature while integrating the equation. The superiority of the method is strengthened by the existence of a larger interval of periodicity of the new method in comparison to the corresponding classical method.     

A basis function for approximation and the solutions of partial differential equations

In this article, we introduce a type of basis functions to approximate a set of scattered data. Each of the basis functions is in the form of a truncated series over some orthogonal system of eigenfunctions. In particular, the trigonometric eigenfunctions are used. We test our basis functions on recovering the well‐known Franke's and Peaks functions given by scattered data, and on the extension of a singular function from an irregular domain onto a square. These basis functions are further used in Kansa's method for solving Helmholtz‐type equations on arbitrary domains. Proper one level and two level approximation techniques are discussed. A comparison of numerical with analytic solutions is given. The numerical results show that our approach is accurate and efficient. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A method for the numerical resolution of Abel-type integral equations of the first kind

A numerical method to solve Abel-type integral equations of first kind is given. In this paper we suggest the research of a numerical solution for Abel-type integral equations of the first kind, by using a collocation method employing an interpolatory product-quadrature formula with a trigonometric polynomial of the first order. Some results of numerical examples are reported.     

Quadrature formulae with multiple nodes and a maximal trigonometric degree of exactness

In this paper we consider interpolatory quadrature formulae with multiple nodes, which have the maximal trigonometric degree of exactness. Our approach is based on a procedure given by Ghizzeti and Ossicini (Quadrature formulae, Academie-Verlag, Berlin, 1970). We introduce and consider the so-called σ-orthogonal trigonometric polynomials of semi-integer degree and give a numerical method for their construction. Also, some numerical examples are included. The authors were supported in part by the Serbian Ministry of Science and Technological Development (Project: Orthogonal Systems and Applications, grant number #144004) and the Swiss National Science Foundation (SCOPES Joint Research Project No. IB7320-111079 “New Methods for Quadrature”).     

New Trigonometric Basis Possessing Exponential Shape Parameters

Four new trigonometric Bernstein-like basis functions with two exponential shape parameters are constructed, based on which a class of trigonometric Bézier-like curves, analogous to the cubic Bézier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bézier-like curves is given. Any arc of an ellipse or a parabola can be represented exactly by using the trigonometric Bézier-like curves. The corresponding trigonometric Bernstein-like operator is presented and the spectral analysis shows that the trigonometric Bézier-like curves are closer to the given control polygon than the cubic Bézier curves. Based on the new proposed trigonometric Bernstein-like basis, a new class of trigonometric B-spline-like basis functions with two local exponential shape parameters is constructed. The totally positive property of the trigonometric B-spline-like basis is proved. For different values of the shape parameters, the associated trigonometric B-spline-like curves can be $C^2$ ∩ $FC^3$ continuous for a non-uniform knot vector, and $C^3$ or $C^5$ continuous for a uniform knot vector. A new class of trigonometric Bézier-like basis functions over triangular domain is also constructed. A de Casteljau-type algorithm for computing the associated trigonometric Bézier-like patch is developed. The conditions for $G^1$ continuous joining two trigonometric Bézier-like patches over triangular domain are deduced.     

Fast approximation to Poisson integral based on trigonometric periodic multiresolution analysis

In this article, a novel fast numerical computational algorithm for Poisson integral is developed by means of periodic trigonometric multiresolution analysis (PTMRA). The approximation formula of Poisson integral is derived. Subsequently, we establish some error estimates of approximation Poisson integral. Finally, several numerical results are given. Comparing with the existing wavelet-based method, the proposed method gives superior results.     

三次Gauss和及二项指数和的混合均值

我们用三角和的性质研究一类三次Gauss和与两项指数和混合均值的计算问题,并给出一个精确的计算公式.     

Biorthogonal Smooth Local Trigonometric Bases

In this paper we discuss smooth local trigonometric bases. We present two generalizations of the orthogonal basis of Malvar and Coifman-Meyer: biorthogonal and equal parity bases. These allow natural representations of constant and, sometimes, linear components. We study and compare their approximation properties and applicability in data compression. This is illustrated with numerical examples.     

A wavelets method for plane elasticity problem

A Neumann boundary value problem of plane elasticity problem in the exterior circular domain is reduced into an equivalent natural boundary integral equation and a Poisson integral formula with the DtN method. Using the trigonometric wavelets and Galerkin method, we obtain a fast numerical method for the natural boundary integral equation which has an unique solution in the quotient space. We decompose the stiffness matrix in our numerical method into four circulant and symmetrical or antisymmetrical submatrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform (FFT) and the inverse fast Fourier transform (IFFT) instead of the inverse matrix. Examples are given for demonstrating our method has good accuracy of our method even though the exact solution is almost singular.     

Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments. AMS subject classification (2000)  65F10, 65L60, 65M60, 65N22