An efficient numerical approximation for the linear class of Fredholm integro-differential equations based on Cattani’s method

This paper provides with a generalization of the work by Cattani (Math. Probl. Eng. (2008) 1–24), who has introduced the connection coefficients of the Shannon wavelets. We apply the Shannon wavelets approximation based on Cattani’s connection coefficients together the collocation points for solving the linear Fredholm integro-differential equations. Finally, numerical results are included to demonstrate the validity and applicability of the method and some comparisons are made with existing results.     

Orthogonal and Nonorthogonal Multiresolution Analysis, Scale Discrete and Exact Fully Discrete Wavelet Transform on the Sphere

Based on a new definition of dilation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in the case of band-limited wavelets. June 18, 1996. Date revised: January 14, 1997.     

Numerical solution of high-order differential equations by using periodized Shannon wavelets

In this paper, periodized Shannon wavelets are applied as basis functions in solution of the high-order ordinary differential equations and eigenvalue problem. The first periodized Shannon wavelets are defined. The second the connection coefficients of periodized Shannon wavelets are related by a simple variable transformation to the Cattani connection coefficients. Finally, collocation method is used for solving the high-order ordinary differential equations and eigenvalue problem. Some equations are solved in order to find out advantage of such choice of the basis functions.     

Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets

We present tensor-product divergence-free and curl-free wavelets, and define associated projectors. These projectors enable the construction of an iterative algorithm to compute the Helmholtz decomposition of any vector field, in wavelet domain. This decomposition is localized in space, in contrast to the Helmholtz decomposition calculated by Fourier transform. Then we prove the convergence of the algorithm in dimension two for any kind of wavelets, and in larger dimension for the particular case of Shannon wavelets. We also present a modification of the algorithm by using quasi-isotropic divergence-free and curl-free wavelets. Finally, numerical tests show the validity of this approach for a large class of wavelets.     

Locally supported rational spline wavelets on a sphere

In this paper we construct certain continuous piecewise rational wavelets on arbitrary spherical triangulations, giving explicit expressions of these wavelets. Our wavelets have small support, a fact which is very important in working with large amounts of data, since the algorithms for decomposition, compression and reconstruction deal with sparse matrices. We also give a quasi-interpolant associated to a given triangulation and study the approximation error. Some numerical examples are given to illustrate the efficiency of our wavelets.

     

Haar wavelets method for solving Poisson equations with jump conditions in irregular domain

In this paper, Haar wavelets method is used to solve Poisson equations in the presence of interfaces where the solution itself may be discontinuous. The interfaces have jump conditions which need to be enforced. It is critical for the approximation of the boundaries of the irregular domain. An irregular domain can be treated by embedding the domain into a rectangular domain and Poisson equation is solved by using Haar wavelets method on the rectangle. Firstly, we demonstrate this method in the case of 1-D region, then we consider the solution of the Poisson equations in the case of 2-D region. The efficiency of the method is demonstrated by some numerical examples.     

Wavelet subspaces with an oversampling property

In the classical Shannon sampling theorem, the same sequence of functions is both orthonormal and a sampling sequence. This is not true for most wavelet subspaces in which the sampling functions and the orthonormal bases are different. However by oversampling at double the rate the property of the Shannon wavelets is extended to a much larger class which includes the Meyer wavelets. In fact together with another condition, it characterizez them.     

On Gibbs constant for the Shannon wavelet expansion

Even though the Shannon wavelet is a prototype of wavelets, it lacks condition on decay which most wavelets are assumed to have. By providing a sufficient condition to compute the size of Gibbs phenomenon for the Shannon wavelet series, we can see the overshoot is propotional to the jump at discontinuity. By comparing it with that of the Fourier series, we also see that these two have exactly the same Gibbs constant.     

A New Method of Constructions of Non-Tensor Product Wavelets

In this paper, we give a new method of constructions of non-tensor product wavelets. We start from the one-dimensional scaling functions to directly construct the two-dimensional non-tensor product wavelets. The wavelets constructed by us possess very simple, explicit representations and high regularity, and various symmetry (i.e., axial symmetry, central symmetry, and cyclic symmetry). Using this method, we construct various non-tensor product wavelets and show that there exists a sequence of non-tensor product wavelets with high regularity which tends to the tensor product Shannon wavelet in the L ²-norm.     

ID-WAVELETS METHOD FOR HAMMERSTEIN INTEGRAL EQUATIONS

1.IntroductionInthispaperwewillconsiderthenumericalsolutionsofthenon--linearintegralequationsofHammersteintype:wheref,kandgaregivenfunctionandyistheunknown.TherehasbeenmuchinterestinthisproblemsinceHammersteinintegralequations,whichcamefromtheelectromagneticfluiddynamics,yieldsstrongphysicalbackground.Moreover,theFredholmintegralequationsofsecondkindarethespecialcaseoftheHammersteinintegralequations.In[6,p.700]thestandardcollocationmethodisappliedtoobtaintheapproximationsolutionofEq.(1).Int…     

Wavelets collocation methods for the numerical solution of elliptic BV problems

Based on collocation with Haar and Legendre wavelets, two efficient and new numerical methods are being proposed for the numerical solution of elliptic partial differential equations having oscillatory and non-oscillatory behavior. The present methods are developed in two stages. In the initial stage, they are developed for Haar wavelets. In order to obtain higher accuracy, Haar wavelets are replaced by Legendre wavelets at the second stage. A comparative analysis of the performance of Haar wavelets collocation method and Legendre wavelets collocation method is carried out. In addition to this, comparative studies of performance of Legendre wavelets collocation method and quadratic spline collocation method, and meshless methods and Sinc–Galerkin method are also done. The analysis indicates that there is a higher accuracy obtained by Legendre wavelets decomposition, which is in the form of a multi-resolution analysis of the function. The solution is first found on the coarse grid points, and then it is refined by obtaining higher accuracy with help of increasing the level of wavelets. The accurate implementation of the classical numerical methods on Neumann’s boundary conditions has been found to involve some difficulty. It has been shown here that the present methods can be easily implemented on Neumann’s boundary conditions and the results obtained are accurate; the present methods, thus, have a clear advantage over the classical numerical methods. A distinct feature of the proposed methods is their simple applicability for a variety of boundary conditions. Numerical order of convergence of the proposed methods is calculated. The results of numerical tests show better accuracy of the proposed method based on Legendre wavelets for a variety of benchmark problems.     

基于小波分析的空间机械臂运动规划的最优控制

讨论了空间机械臂系统非完整运动规划的最优控制问题.利用小波分析方法,将离散正交小波函数引入最优控制,由小波级数展开式逼近替代传统的Fourier基函数,提出基于小波分析的最优控制数值算法.仿真结果表明,该方法对求解空间机械臂非完整运动规划问题是有效的.     

A new sequential approach for solving the integro-differential equation via Haar wavelet bases

In this work, we present a method for numerical approximation of fixed point operator, particularly for the mixed Volterra–Fredholm integro-differential equations. The main tool for error analysis is the Banach fixed point theorem. The advantage of this method is that it does not use numerical integration, we use the properties of rationalized Haar wavelets for approximate of integral. The cost of our algorithm increases accuracy and reduces the calculation, considerably. Some examples are provided toillustrate its high accuracy and numerical results are compared with other methods in the other papers.     

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.     

对称反对称多重尺度函数的构造

１　多重小波的定义和双尺度相似变换 作为一种分析工具,小波已经运用在各种领域,并取得了显著的成果．近年来,多重小波成为小波研究的热点．Ｉ．Ｄａｕｂｅｃｈｉｅｓ［１］已经证明,对单重小波,除Ｈａｒｒ基外不存在对称和反对称的有紧支集的小波正交基．而多重小波则不受这一限制． 利用分形插值的方法,Ｇｅｒｏｎｉｍｏ、Ｈａｒｄｉｎ和 Ｍａｓｓｏｐｕｓｔ［２］等构造出了ＧＨＭ多重小波,相应的多重尺度函数和多重小波函数如图１和图２所示．ＧＨＭ多重小波的两个尺度函数都是对称的,相应的小波函数则一个对称另一个反对称;…     

The construction of plane elastomechanics and Mindlin plate elements of B-spline wavelet on the interval

Based on two-dimensional tensor product B-spline wavelet on the interval (BSWI), a class of C0 type plate elements is constructed to solve plane elastomechanics and moderately thick plate problems. Instead of traditional polynomial interpolation, the scaling functions of two-dimensional tensor product BSWI are employed to form the shape functions and construct BSWI elements. Unlike the process of direct wavelets adding in the previous work, the elemental displacement field represented by the coefficients of wavelets expansions is transformed into edges and internal modes via the constructed transformation matrix in this paper. The method combines the versatility of the conventional finite element method (FEM) with the accuracy of B-spline functions approximation and various basis functions for structural analysis. Some numerical examples are studied to demonstrate the proposed method and the numerical results presented are in good agreement with the closed-form or traditional FEM solutions.     

Asymptotic error expansion of wavelet approximations of smooth functions II

Summary. We generalize earlier results concerning an asymptotic error expansion of wavelet approximations. The properties of the monowavelets, which are the building blocks for the error expansion, are studied in more detail, and connections between spline wavelets and Euler and Bernoulli polynomials are pointed out. The expansion is used to compare the error for different wavelet families. We prove that the leading terms of the expansion only depend on the multiresolution subspaces and not on how the complementary subspaces are chosen. Consequently, for a fixed set of subspaces , the leading terms do not depend on the fact whether the wavelets are orthogonal or not. We also show that Daubechies' orthogonal wavelets need, in general, one level more than spline wavelets to obtain an approximation with a prescribed accuracy. These results are illustrated with numerical examples. Received May 3, 1993 / Revised version received January 31, 1994     

Degenerate kernel schemes by wavelets for nonlinear integral equations on the real line

A degenerate kernel scheme is developed for nonlinear integral equations on the real line by approximation of kernels by wavelets. The rate of convergence of the approximate solutions is established in terms of the decay rate of the kernels andthe numbers of dilations and shifts used in approximation of the kernels. For linear equations, the Haar wavelet approximation is used and a numerical example is included.     

Pointwise convergence of wavelets of generalized Shannon type

In this paper, a new result on pointwise convergence of wavelets of generalized Shannon type is proved, which improves a theorem established by Zayed.     

How to Get the Original Discrete Approximation for the Pyramid Algorithm of Multiresolution Decomposition

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｕｓｅｔｈｅｎｏｔａｔｉｏｎｓＺ,Ｒ,Ｌ２（Ｒ）ａｎｄｌ２ｆｏｒｔｈｅｓｅｔｏｆｉｎｔｅｇｅｒｓ,ｒｅ－ａｌｓ,ｓｑｕａｒｅｉｎｔｅｇｒａｂｌｅｆｕｎｃｔｉｏｎｓａｎｄｓｑｕａｒｅｓｕｍｍａｂ...