Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet

In this paper, we first construct the second kind Chebyshev wavelet. Then we present a computational method based on the second kind Chebyshev wavelet for solving a class of nonlinear Fredholm integro-differential equations of fractional order. The second kind Chebyshev wavelet operational matrix of fractional integration is derived and used to transform the equation to a system of algebraic equations. The method is illustrated by applications and the results obtained are compared with the existing ones in open literature. Moreover, comparing the methodology with the known technique shows that the present approach is more efficient and more accurate.     

Discretizing continuous wavelet transforms using integrated wavelets

We present integrated wavelets as a method for discretizing the continuous wavelet transform. Using the language of group theory, the results are presented for wavelet transforms over semidirect product groups. We obtain tight wavelet frames for these wavelet transforms. Further integrated wavelets yield tight families of convolution operators independent of the choice of discretization of scale and orientation parameters. Thus these families can be adapted to specific problems. The method is more flexible than the well-known dyadic wavelet transform. We state an exact algorithm for implementing this transform. As an application the enhancement of digital mammograms is presented.     

Linearity measure for curve segments

In this paper, we consider linearity measure for a bounded length curves. First, we define a new linearity measure for open curve segments, and then extend method to closed curves (contours). The derived measures (for both, open curve segments and closed curves) are invariant with respect to similarity transformations. The linearity measure for open curve segments picks the value 1 if and only if the measured open line segment is a perfect straight line segment while the established linearity measures for closed curves never reach 1, as preferred.     

Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

In this paper, a new numerical method for solving fractional differential equations is presented. The fractional derivative is described in the Caputo sense. The method is based upon Bernoulli wavelet approximations. The Bernoulli wavelet is first presented. An operational matrix of fractional order integration is derived and is utilized to reduce the initial and boundary value problems to system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

偏微分方程的区间小波自适应精细积分法

利用插值小波理论构造了拟Shannon区间小波,并结合外推法给出了一种求解非线性常微分方程组的时间步长自适应精细积分法,在此基础上构造了求解非线性偏微分方程的区间小波自适应精细积分法（AIWPIM）．数值结果表明,该方法在计算精度上优于将小波和四阶Runge-Kutta法组合得到的偏微分方程的数值求解方法,而计算量则相差不大．该文方法通过Burgers方程给出,但适用于一般情形．     

基于四点插值细分小波的SAR图像去噪

针对SAR图像去噪过程中存在降低相干斑与保持有效细节这一矛盾,提出了一种基于四点插值细分小波的SAR图像去噪算法,该方法将小波和细分方法相融合,将四点插值细分规则应用到细分小波中,提出了图像去噪的新方法.该算法先用四点插值细分小波对原始图像进行分解,然后用Bayes自适应阈值及阈值函数对图像进行去噪,最后对去噪的小波系数进行重构,并通过等效视数、边缘保持指数等评价指标对去噪结果进行了评价.实验结果表明,算法的等效视数、边缘保持指数都有所提高,去噪效果得到了优化.     

Legendre Wavelets Direct Method for the Numerical Solution of Time-Fractional Order Telegraph Equations

In this paper, a Legendre wavelet collocation method for solving a class of time-fractional order telegraph equation defined by Caputo sense is discussed. Fractional integral formula of a single Legendre wavelet in the Riemann–Liouville sense is derived by means of shifted Legendre polynomials. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence analysis and error analysis of the proposed method are investigated. Several examples are presented to show the applicability and accuracy of the proposed method.     

Frame wavelet sets in

In this paper, we try to answer an open question raised by Han and Larson, which asks about the characterization of frame wavelet sets. We completely characterize tight frame wavelet sets. We also obtain some necessary conditions and some sufficient conditions for a set to be a (general) frame wavelet set. Some results are extended to frame wavelet functions that are not defined by frame wavelet set. Several examples are presented and compared with some known results in the literature.

     

Wavelet moment method for the Cauchy problem for the Helmholtz equation

The paper is concerned with the problem of reconstruction of acoustic or electromagnetic field from inexact data given on an open part of the boundary of a given domain. A regularization concept is presented for the moment problem that is equivalent to a Cauchy problem for the Helmholtz equation. A method of regularization by projection with application of the Meyer wavelet subspaces is introduced and analyzed. The derived formula, describing the projection level in terms of the error bound of the inexact Cauchy data, allows us to prove the convergence and stability of the method.     

A wavelet theory for local fields and related groups

Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = ℚ_p, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = ℤ_p, the ring of padic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient Ĝ/H^⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to socalled wavelet sets in the dual group Ĝ. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.     

Second Kind Chebyshev Wavelet Galerkin Method for Stochastic Itô-Volterra Integral Equations

In this paper, an efficient wavelet Galerkin method based on the stochastic operational matrix of second kind Chebyshev wavelet is proposed for solving stochastic Itô-Volterra integral equations. Convergence and error analysis of the presented wavelets method are investigated. The numerical results are compared with exact solution and those of other existing methods.     

过抽样M通道小波星图去噪方法研究

小波图像去噪已经成为目前图像去噪的主要方法之一,在分析了小波变换的基本理论和小波变换的多尺度分析基础上,根据多尺度小波变换的多分辨特性,提出了过抽样M通道小波变换去噪方法,并将此方法用于星图降噪处理中,收到良好的效果.     

A general construction of nonseparable multivariate orthonormal wavelet bases

The construction of nonseparable and compactly supported orthonormal wavelet bases of L ²(R ⁿ ); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating multidimensional multiwavelet bases.      

A fast adaptive spectral graph wavelet method for the viscous Burgers' equation on a star-shaped connected graph

Water wave propagation in an open channel network can be described by the viscous Burgers' equation on the corresponding connected graph, possibly with small viscosity. In this paper, we propose a fast adaptive spectral graph wavelet method for the numerical solution of the viscous Burgers' equation on a star-shaped connected graph. The vital feature of spectral graph wavelets is that they can be constructed on any complex network using the graph Laplacian. The essence of the method is that the same operator can be used for the construction of the spectral graph wavelet and the approximation of the differential operator involved in the Burgers' equation. In this paper, two test problems are considered with homogeneous Dirichlet boundary condition. The numerical results show that the method accurately captures the evolution of the localized patterns at all the scales, and the adaptive node arrangement is accordingly obtained. The convergence of the given method is verified, and efficiency is shown using CPU time.     

小波分频在重力异常界面反演计算过程中的应用

重力资料是研究地质结构的重要资料，通常测得的重力异常数据是地下各界面造成的异常的失加，如何把这种叠加效应剥离出来一直是一个难以解决的问题，如果能从综合信息中离出能反映各界面信息的重力异常值，我们就能将问题归结为单界面重力异常的反演问题（关于单界面重力异常的反演可参见文献[1]），小波分析提供了这类问题的新思路。     

Constrained variational refinement

A non-uniform, variational refinement scheme is presented for computing piecewise linear curves that minimize a certain discrete energy functional subject to convex constraints on the error from interpolation. Optimality conditions are derived for both the fixed and free-knot problems. These conditions are expressed in terms of jumps in certain (discrete) derivatives. A computational algorithm is given that applies to constraints whose boundaries are either piecewise linear or spherical. The results are applied to closed periodic curves, open curves with various boundary conditions, and (approximate) Hermite interpolation.     

多元向量值正交小波包的刻画

In this paper,the notion of orthogonal vector-valued wavelet packets of space L2(Rs,Cn)is introduced.A procedure for constructing the orthogonal vector-valued wavelet packets is presented.Their properties are characterized by virtue of time-frequency analysis method,matrix theory and finite group theory,and three orthogonality formulas are obtained.Finally,new orthonormal bases of space L2(Rs,Cn)are extracted from these wavelet packets.     

一种基于小波分析的各向异性图象去噪方法

本文利用非线性各向异性扩散方程结合小波变换提出一种图象去噪的方法。首先对图像进行离散小波变换,然后对其各个分量分别用各向异性的方法实现去噪。实验结果表明,该方法能够较好的去除噪声的同时,很好的保留边缘信息。     

Construction of multivariate compactly supported tight wavelet frames

Two simple constructive methods are presented to compute compactly supported tight wavelet frames for any given refinable function whose mask satisfies the QMF or sub-QMF conditions in the multivariate setting. We use one of our constructive methods in order to find tight wavelet frames associated with multivariate box splines, e.g., bivariate box splines on a three or four directional mesh. Moreover, a construction of tight wavelet frames with maximum vanishing moments is given, based on rational masks for the generators. For compactly supported bi-frame pairs, another simple constructive method is presented.     

An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations

In this paper, a novel single-term Haar wavelet series (STHWS) method is implemented for the solution of the Duffing equation and Painleve’s transcendents (PI and PII). The results, in the form of a block pulse and a discrete solution, are presented. Unlike classical numerical schemes, the STHWS method has no restrictions on the coefficients of the Duffing equation as regards its solution. PI and PII are analysed as regards their solutions, up to nearest singularities (poles), using the STHWS. Also, an efficient computational implementation shows the remarkable features of wavelet based techniques.