Error analysis for numerical solution of fractional differential equation by Haar wavelets method

In this paper, an exact upper bound is presented through the error analysis to solve the numerical solution of fractional differential equation with variable coefficient. The fractional differential equation is solved by using Haar wavelets. From the exact upper bound, we can draw a conclusion easily that the method is convergent. Finally, we also give some numerical examples to demonstrate the validity and applicability of the method.     

Periodic Wavelets from Scratch

Periodic wavelets can be constructed from most standard wavelets by periodization. In this work we first derive some of their properties and then construct the periodic wavelets directly from their Fourier series without reference to standard wavelets. Several examples are given some of which are not constructable from the usual wavelets on the real line.     

Frame Wavelets with Compact Supports for L^2（R^n）

The construction of frame wavelets with compact supports is a meaningful problem in wavelet analysis. In particular, it is a hard work to construct the frame wavelets with explicit analytic forms. For a given n × n real expansive matrix A, the frame-sets with respect to A are a family of sets in R^n. Based on the frame-sets, a class of high-dimensional frame wavelets with analytic forms are constructed, which can be non-bandlimited, or even compactly supported. As an application, the construction is illustrated by several examples, in which some new frame wavelets with compact supports are constructed. Moreover, since the main result of this paper is about general dilation matrices, in the examples we present a family of frame wavelets associated with some non-integer dilation matrices that is meaningful in computational geometry.     

A new approach by two‐dimensional wavelets operational matrix method for solving variable‐order fractional partial integro‐differential equations

In this paper, a new computational scheme based on operational matrices (OMs) of two‐dimensional wavelets is proposed for the solution of variable‐order (VO) fractional partial integro‐differential equations (PIDEs). To accomplish this method, first OMs of integration and VO fractional derivative (FD) have been derived using two‐dimensional Legendre wavelets. By implementing two‐dimensional wavelets approximations and the OMs of integration and variable‐order fractional derivative (VO‐FD) along with collocation points, the VO fractional partial PIDEs are reduced into the system of algebraic equations. In addition to this, some useful theorems are discussed to establish the convergence analysis and error estimate of the proposed numerical technique. Furthermore, computational efficiency and applicability are examined through some illustrative examples.     

A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

In this paper, a fast numerical algorithm based on the Taylor wavelets is proposed for finding the numerical solutions of the fractional integro‐differential equations with weakly singular kernels. The properties of Taylor wavelets are given, and the operational matrix of fractional integration is constructed. These wavelets are utilized to reduce the solution of the given fractional integro‐differential equation to the solution of a linear system of algebraic equations. Also, convergence of the proposed method is studied. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Gegenbauer wavelet collocation method for the extended Fisher-Kolmogorov equation in two dimensions

Gegenbauer wavelets operational matrices play an important role in the numeric solution of differential equations. In this study, operational matrices of rth integration of Gegenbauer wavelets are derived and used to obtain an approximate solution of the nonlinear extended Fisher-Kolmogorov (EFK) equation in two-space dimension. Nonlinear EFK equation is converted into the linear partial differential equation by quasilinearization technique. Numerical examples have shown that present method is convergent even in the case of a small number of grid points. The results of the presented method are in a good agreement with the results in literature.     

Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets

In this work, we present a computational method for solving nonlinear Fredholm integral equations of the second kind which is based on the use of Haar wavelets. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples.     

一类非线性微分方程组的有理化Haar小波解法

利用有理化Haar小波性质和方法,建立了一类非线性微分方程组在任意区间[a,b）的求解算法．基于该算法,运用计算机代数系统Maple,给出了求解非线性微分方程组的程序．并运用此程序给出了一类微分方程组的计算实例,从数值模拟来看可以达到较高的精度,并对方程组的动力学行为给出较好的描述．     

Designing Multiresolution Analysis-type Wavelets and Their Fast Algorithms

Often, the Dyadic Wavelet Transform is performed and implemented with the Daubechies wavelets, the Battle-Lemarie wavelets, or the splines wavelets, whereas in continuous-time wavelet decomposition a much larger variety of mother wavelets is used. Maintaining the dyadic time-frequency sampling and the recursive pyramidal computational structure, we present various methods for constructing wavelets ψ_wanted, with some desired shape and properties and which are associated with semi-orthogonal multiresolution analyses. We explain in detail how to design any desired wavelet, starting from any given multiresolution analysis. We also explicitly derive the formulae of the filter bank structure that implements the designed wavelet. We illustrate these wavelet design techniques with examples that we have programmed with Matlab routines.     

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.     

Construction of wavelets with composite dilations

In order to overcome classical wavelets’ shortcoming in image processing problems, people developed many producing systems, which built up wavelet family. In this paper, the notion of AB-multiresolution analysis is generalized, and the corresponding theory is developed. For an AB-multiresolution analysis associated with any expanding matrices, we deduce that there exists a singe scaling function in its reducing subspace. Under some conditions, wavelets with composite dilations can be gotten by AB-multiresolution analysis, which permits the existence of fast implementation algorithm. Then, we provide an approach to design the wavelets with composite dilations by classic wavelets. Our way consists of separable and partly nonseparable cases. In each section, we construct all kinds of examples with nice properties to prove our theory.     

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.

     

Wavelet analysis method for solving linear and nonlinear singular boundary value problems

In this paper, a robust and accurate algorithm for solving both linear and nonlinear singular boundary value problems is proposed. We introduce the Chebyshev wavelets operational matrix of derivative and product operation matrix. Chebyshev wavelets expansions together with operational matrix of derivative are employed to solve ordinary differential equations in which, at least, one of the coefficient functions or solution function is not analytic. Several examples are included to illustrate the efficiency and accuracy of the proposed method.     

紧支半正交小波的构造

§ 1　IntroductionIt is well known that wavelets with dilation factor two can be constructed from amultiresolution analysis and lots of their applications have been found.It is also knownthat wavelets with general dilation factor M may be constructed from the multiresolutionanalysis with dilation factor M≥ 2 [1— 3] .Wavelets are closely related to M-channel filterbanks[4,5] ,so some important applications such as in audio coding and communication aredeveloped.Semi-orthogonal wavelets const…     

Algorithms for wavelet construction on Vilenkin groups

In this paper, some algorithms for constructing orthogonal and biorthogonal compactly supported wavelets on Vilenkin groups are suggested. As application, several examples of p-adic wavelets, which correspond to the refinable functions presented recently by the first author, are given.     

区间上的双正交小波的一种构造方法

小波分析是近十几年来十分热门的课题,早期的小波都是定义在无穷区间上的．而实际问题常常是有限区间,常用的方法是将有限区间上的数据向区间外延拓,但这样做容易产生边界误差,如何构造区间上的具有良好性质的小波,如光滑性、对称性等是非常有意义的．在文献［２］中,Cohen,Darbechies与Vial在总结了构造区间周期小波,折叠小波等方法的基础上,提出了一种新的构造方法,并把无穷区间上的Daubechies小波改造成区间上的Daubechies小波．但是,Daubechies小波没有对称性,光滑性也差．本文用类似的方法把无穷区间上的双正交小波改造…     

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.     

Numerical solutions to initial and boundary value problems for linear fractional partial differential equations

In this article, Haar wavelets have been employed to obtain solutions of boundary value problems for linear fractional partial differential equations. The differential equations are reduced to Sylvester matrix equations. The algorithm is novel in the sense that it effectively incorporates the aperiodic boundary conditions. Several examples with numerical simulations are provided to illustrate the simplicity and effectiveness of the method.     

Existence and Uniqueness of Positive Solutions for a System of Multi-order Fractional Differential Equations

In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equa-tions. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.     

Legendre wavelets approach for numerical solutions of distributed order fractional differential equations

In this study, a new numerical method for the solution of the linear and nonlinear distributed fractional differential equations is introduced. The fractional derivative is described in the Caputo sense. The suggested framework is based upon Legendre wavelets approximations. For the first time, an exact formula for the Riemann–Liouville fractional integral operator for the Legendre wavelets is derived. We then use this formula and the properties of Legendre wavelets to reduce the given problem into a system of algebraic equations. Several illustrative examples are included to observe the validity, effectiveness and accuracy of the present numerical method.