Legendre Multiwavelet Research Articles

Numerical Solution of Partial Integrodifferential Equations of Diffusion Type

A collocation method based on linear Legendre multiwavelets is developed for numerical solution of one-dimensional parabolic partial integrodifferential equations of diffusion type. Such equations have numerous applications in many problems in the applied sciences to model dynamical systems. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency, and robustness of the proposed method.

The Application of Legendre Multiwavelet Functions in Image Compression

Legendre multiwavelets are introduced. These functions can be designed in such a way that the properties of orthogonality, polynomial approximation, and symmetry hold at the same time. In this way, they can be effectively deployed in image compression.

An Efficient Decomposition Method for Solving Telegraph Equation Through Quadratic Legendre Multiwavelets

In this paper, we present a new technique based on the quadratic Legendre multiwavelets decomposition for solving telegraph equation. The properties of quadratic Legendre multiwavelets are used to transform the telegraph equation into the solution of ODE’s system and the ODE’s system is converted into the system of algebraic equations. Illustrative examples are included to demonstrate the applicability and validity of the innovative method.

Multiscale approximation of the solution of weakly singular second kind Fredholm integral equation in Legendre multiwavelet basis

Numerical solution of Fredholm integral equation of second kind with weakly singular kernel is obtained in this paper by employing Legendre multi-wavelet basis. The low- and high-pass filters for two-scale relations involving Legendre multiwavelets having four or five vanishing moments of their wavelets have been derived and are used in the evaluation of integrals for the multiscale representation of the integral operator. Explicit expressions for the elements of the matrix associated with the multiscale representation are given. An estimate for the Hölder exponent of the solution of the integral equation at any point in its domain is obtained. A number of examples is provided to illustrate the efficiency of the method developed here.

A Legendre multiwavelets approach to copula density estimation

In this paper, a novel method for copula density estimation using Legendre multiwavelet is proposed. In general, copula density estimation methods based on the multiwavelet benefit from some useful properties, including they are symmetric, orthogonal and have compact support. In particular, the Legendre multiwavelet as a more general and vector-valued polynomial type of wavelets would results a more flexible and accurate approximation for the given copula density. In addition to high ability and nice properties of Legendre multiwavelet in approximation, its support is defined on unit interval, [0,1], as copulas that are normalized to have the support on the unit square and uniform marginal. We further make this approximation method more accurate by using multiresolution techniques. The comparative study reveals that the approximation proposed in this paper is more accurate than a scalar wavelet bases approximation. We eventually apply presented method to approximate multivariate distribution using pair-copula as a flexible multivariate copula to model a dataset of Norwegian financial data.

Approximation Multivariate Distribution with Pair Copula Using the Orthonormal Polynomial and Legendre Multiwavelets Basis Functions

We concentrate on constructing higher dimensional distributions using a fast growing graphical model called Vine/ pair-copula model which has been introduced and developed by Joe, Cooke, Bedford, Kurowica, Daneshkhah, and others. They first construct a n-dimensional copula density by stacking together n(n − 1)/2 bivariate copula density, and they then approximate arbitrarily well these bivariate copulas and the corresponding multivariate distribution using a semi-parametric method. One constructive approach involves the use of minimum information copulas that can be specified to any required degree of precision based on the available data (or possibly based on the experts’ judgments). By using this method, one is able to use a fixed finite dimensional family of copulas to be employed in terms of a vine construction, with the promise of a uniform level of approximation.The basic idea behind this method is to use a two-dimensional ordinary polynomial series to approximate any log-density of a bivariate copula function by truncating the series at an appropriate point. We make this approximation method more accurate and computationally faster by using the orthonormal polynomial and Legendre multiwavelets (LMW) series as the basis functions. We show the derived approximations are more precise and computationally faster with better properties than the one proposed previous method in the literature. We then apply our method to modeling a dataset of Norwegian financial data that was previously analyzed in the series of articles, and finally compare our results by them. At the end, we present a method to simulate from the approximated models, and validate our approximation using the simulation results to recover the same dependency structure of the original data.

Numerical Solution Of Singular Fredholm Integrodifferential Equations Of The Second Kind Via Petrovgalerkin Method By Using Legendre Multiwavelet

It is of interest to know whether we can solve a singular Fredholm integro-differential equation of the second kind with an infinite or semi-infinite range of integrate via Petrov–Galerkin method by using Legendre multiwavelet. For this purpose, we directly deal with infinite range of integrate. We introduce some change of variables for mapping infinite interval into a finite interval. After that, we use Petrov– Galerkin method with Legendre multiwavelet basis that yields linear system. Numerical results of our example will demonstrate accuracy and efficiency of the proposed method.

LEGENDRE MULTIWAVELET GALERKIN METHODS FOR DIFFERENTIAL EQUATIONS

The multiresolution analysis for Legendre multiwavelets are given, anti-derivatives of Legendre multiwavelets are used for the numerical solution of differential equations, a special form of multilevel augmentation method algorithm is proposed to solve the disrete linear system efficiently, convergence rate of the Galerkin methods is given and numerical examples are presented.

Approximate Solution of Fractional Nonlinear Partial Differential Equations by the Legendre Multiwavelet Galerkin Method

The Legendre multiwavelet Galerkin method is adopted to give the approximate solution for the nonlinear fractional partial differential equations (NFPDEs). The Legendre multiwavelet properties are presented. The main characteristic of this approach is using these properties together with the Galerkin method to reduce the NFPDEs to the solution of nonlinear system of algebraic equations. We presented the numerical results and a comparison with the exact solution in the cases when we have an exact solution to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense.

Legendre multiwavelet collocation method for solving the linear fractional time delay systems

Abstract In this article the Legendre multiwavelet basis with aid of collocation method has been applied to give approximate solution for fractional delay systems. The properties of Legendre multiwavelet are presented. These properties together with the collocation method are then utilized to reduce the problem to the solution of algebraic system. Numerical results and comparison with exact solutions in the cases when we have exact solution are given in test examples in order to demonstrate the applicability and efficiency of the method.

Solving Differential Equations of Second Order using Quadratic Legendre Multi-wavelets (QLMW) with Operational Matrix of Integration

In this paper is suggested an efficient method to solve differential equations. Using quadratic Legendre multi-wavelets approximation method, differential equations are converted into the system of algebraic equations with the help of operational matrix of integration and its product. Some illustrative examples are included to show the efficiency and applicability of the method.

Mixed discontinuous Legendre wavelet Galerkin method for solving elliptic partial differential equations

By incorporating the Legendre multiwavelet into the mixed discontinuous Galerkin method, in this paper, we present a novel method for solving second-order elliptic partial differential equations (PDEs), which is known as the mixed discontinuous Legendre multiwavelet Galerkin method, derive an adaptive algorithm for the method and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. The proposed method is also applicable to some other kinds of PDEs.

Ritz Legendre Multiwavelet Method for the Damped Generalized Regularized Long-Wave Equation

In this paper, a numerical method is proposed to approximate the solution of the nonlinear damped generalized regularized long-wave (DGRLW) equation with a variable coefficient. The method is based upon Ritz Legendre multiwavelet approximations. The properties of Legendre multiwavelet are first presented. These properties together with the Galerkin method are then utilized to reduce the nonlinear DGRLW equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems

In this article the Legendre multiwavelet basis with the aid of a collocation method has been applied to give the approximate solution for the fractional optimal control problems (FOCPs). The properties of the Legendre multiwavelet are presented. These properties together with the collocation method are then utilized to reduce the problem to the solution of an algebraic system. Numerical results and a comparison with the exact solution in the cases when we have an exact solution are given to demonstrate the applicability and efficiency of the new method.

Suboptimal control of linear systems with delays in state and input by orthonormal basis

This paper presents a method for finding the optimal control of linear systems with delays in state and input and the quadratic cost functional using an orthonormal basis for square integrable functions. The state and control variables and their delays are expanded in an orthonormal basis with unknown coefficients. Using operational matrices of integration, delay and product, the relation between coefficients of variables is provided. Then, necessary condition of optimality is driven as a linear system of algebraic equations in terms of the unknown coefficients of state and control variables. As an application of this method, the linear Legendre multiwavelets as an orthonormal basis for L 2[0, 1] are used. Two time-delayed optimal control problems are approximated to establish the usefulness of the method.

Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets

An effective method based upon Legendre multiwavelets is proposed for the solution of Fredholm weakly singular integro-differential equations. The properties of Legendre multiwavelets are first given and their operational matrices of integral are constructed. These wavelets are utilized to reduce the solution of the given integro-differential equation to the solution of a sparse linear system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain the solution to this system of algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation.

Optimal Control of Linear Time-Delayed Systems by Linear Legendre Multiwavelets

This paper introduces the application of linear Legendre multiwavelets to the optimal control synthesis for linear time-delayed systems. Based on some useful properties of linear Legendre multiwavelets, integration, product and delay operational matrices are proposed to solve the linear time-delayed systems first. Then, a quadratic cost functional is approximated by those properties. By using Lagrange multipliers, the quadratic cost functional is minimized subject to the solution of the linear time-delayed system and an explicit formula for the optimal control is obtained. The effectiveness of the method and accuracy of the solution are shown in comparison with some other methods by illustrative examples.

Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equation

AbstractRecently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion for such branches of sciences. In this article a numerical method for solving the one‐dimensional hyperbolic telegraph equation is presented. The method is based upon Legendre multiwavelet approximations. The properties of Legendre multiwavelet are first presented. These properties together with Galerkin method are then utilized to reduce the telegraph equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Numerical evaluation of the Hankel transform by using linear Legendre multi-wavelets

An efficient algorithm for evaluating the Hankel transform F n ( p ) of order n of a function f ( r ) is given. As the continuous Legendre multi-wavelets forms an orthonormal basis for L 2 ( R ) ; we expand the part r f ( r ) of the integrand in its wavelet series reducing the Hankel transform integral as a series of Bessel functions multiplied by the wavelet coefficients of the input function. Numerical examples are given to illustrate the efficiency of the proposed method.

Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets

In this paper, we suggest an efficient method for solving Fredholm integral equations of the first kind. The continuous Legendre multi-wavelets constructed on [0, 1] are utilized as a basis in Galerkin method to reduce the solution of linear integral equations to a system of algebraic equations. Illustrative examples are included to demonstrate the efficiency and the application of the technique.