Legendre Multiwavelet Research Articles

Approximating Partial Differential Equations with Physics-Informed Legendre Multiwavelets CNN

The purpose of this paper is to leverage the advantages of physics-informed neural network (PINN) and convolutional neural network (CNN) by using Legendre multiwavelets (LMWs) as basis functions to approximate partial differential equations (PDEs). We call this method Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), which can continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any precomputed training data, simultaneously providing fast and continuous solutions that generalize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, the incompressible Navier–Stokes (N-S) equation, and the two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs.

A Novel Feature Extraction Method Based on Legendre Multi-Wavelet Transform and Auto-Encoder for Steel Surface Defect Classification

A Novel Feature Extraction Method Based on Legendre Multi-Wavelet Transform and Auto-Encoder for Steel Surface Defect Classification

Legendre Multiwavelet Transform and Its Application in Bearing Fault Detection

Bearing failures often result from compound faults, where the characteristics of these compound faults span across multiple domains. To tackle the challenge of extracting features from compound faults, this paper proposes a novel fault detection method based on the Legendre multiwavelet transform (LMWT) combined with envelope spectrum analysis. Additionally, to address the issue of identifying suitable wavelet decomposition coefficients, this paper introduces the concept of relative energy ratio. This ratio assists in identifying the most sensitive wavelet coefficients associated with fault frequency bands. To assess the performance of the proposed method, the results obtained from the LMWT method are compared with those derived from the empirical wavelet transform (EWT) method using different datasets. Experimental findings demonstrate that the proposed method exhibits more effective frequency spectrum segmentation and superior detection performance across various experimental conditions.

A New Method for Dynamical System Identification by Optimizing the Control Parameters of Legendre Multiwavelet Neural Network

Wavelet neural networks have been widely applied to dynamical system identification fields. The most difficult issue lies in selecting the optimal control parameters (the wavelet base type and corresponding resolution level) of the network structure. This paper utilizes the advantages of Legendre multiwavelet (LW) bases to construct a Legendre multiwavelet neural network (LWNN), whose simple structure consists of an input layer, hidden layer, and output layer. It is noted that the activation functions in the hidden layer are adopted as LW bases. This selection if based on the its rich properties of LW bases, such as piecewise polynomials, orthogonality, various regularities, and more. These properties contribute to making LWNNs more effective in approximating the complex characteristics exhibited by uncertainties, step, nonlinear, and ramp in the dynamical systems compared to traditional wavelet neural networks. Then, the number of selection LW bases and the corresponding resolution level are effectively optimized by the simple Genetic Algorithm, and the improved gradient descent algorithm is implemented to learn the weight coefficients of LWNN. Finally, four nonlinear dynamical system identification problems are applied to validate the efficiency and feasibility of the proposed LWNN-GA method. The experiment results indicate that the LWNN-GA method achieves better identification accuracy with a simpler network structure than other existing methods.

LW-BPNN: A Novel Feature Extraction Method for Rolling Bearing Fault Diagnosis

Efficiently diagnosing bearing faults is of paramount importance to enhance safety and reduce maintenance costs for rotating machinery. This paper introduces a novel bearing fault diagnosis method (LW-BPNN), which combines the rich properties of Legendre multiwavelet bases with the robust learning capabilities of a BP neural network (BPNN). The proposed method not only addresses the limitations of traditional deep networks, which rely on manual feature extraction and expert experience but also eliminates the complexity associated with designing and training deep network architectures. To be specific, only two statistical parameters, root mean square (RMS) and standard deviation (SD), are calculated on different Legendre multiwavelet decomposition levels to thoroughly represent more salient and comprehensive fault characteristics by using several scale and wavelet bases with various regularities. Then, the mapping relation between the extracted features and the health conditions of the bearing is automatically learned by the simpler BPNN classifier rather than the complex deep network structure. Finally, a few experiments on a popular bearing dataset are implemented to verify the effectiveness and robustness of the presented method. The experimental findings illustrate that the proposed method exhibits a high degree of precision in diagnosing various fault patterns. It outperforms other methods in terms of diagnostic accuracy, making it a viable and promising solution for real-world industrial applications in the field of rotating machinery.

Application of the Variational Iteration Method for the time-fractional Kaup-Kupershmidt Equation and the Boussinesq-Burger equation

The variational iteration method is used to deal with linear and nonlinear differential equations. The main characteristics of the method lie in its flexibility and ability to accurately and easily solve nonlinear equations. In this work, a general framework is presented for a variational iteration method for the analytical treatment of partial differential equations in fluid mechanics. The Caputo sense is used to describe fractional derivatives. The time-fractional Kaup-Kupershmidt (KK) equation is investigated, as it is the solution of the system of partial differential equations via the Boussinesq-Burger equation. By comparing the results that are obtained by the variational iteration method with those obtained by the two-dimensional Legendre multiwavelet, the optimal homotopy asymptotic method (OHAM), the q-homotopy analysis transform method, the Laplace Adomian Decomposition Method, and the homotopy perturbation method, the first method proved to be very effective and convenient. The main methodology in this work is anticipated to be applied to various fractional calculus, linear, and nonlinear problems.

Gearbox Compound Fault Diagnosis in Edge-IoT Based on Legendre Multiwavelet Transform and Convolutional Neural Network

The application of edge computing combined with the Internet of Things (edge-IoT) has been rapidly developed. It is of great significance to develop a lightweight network for gearbox compound fault diagnosis in the edge-IoT context. The goal of this paper is to devise a novel and high-accuracy lightweight neural network based on Legendre multiwavelet transform and multi-channel convolutional neural network (LMWT-MCNN) to fast recognize various compound fault categories of gearbox. The contributions of this paper mainly lie in three aspects: The feature images are designed based on the LMWT frequency domain and they are easily implemented in the MCNN model to effectively avoid noise interference. The proposed lightweight model only consists of three convolutional layers and three pooling layers to further extract the most valuable fault features without any artificial feature extraction. In a fully connected layer, the specific fault type of rotating machinery is identified by the multi-label method. This paper provides a promising technique for rotating machinery fault diagnosis in real applications based on edge-IoT, which can largely reduce labor costs. Finally, the PHM 2009 gearbox and Paderborn University bearing compound fault datasets are used to verify the effectiveness and robustness of the proposed method. The experimental results demonstrate that the proposed lightweight network is able to reliably identify the compound fault categories with the highest accuracy under the strong noise environment compared with the existing methods.

On a new method for finding numerical solutions to integro-differential equations based on Legendre multi-wavelets collocation

In this article, a wavelet collocation method based on linear Legendre multi-wavelets is proposed for the numerical solution of the first as well as higher orders Fredholm, Volterra and Volterra–Fredholm integro-differential equations. The presented numerical method has the capability to tackle the solutions of both linear and nonlinear problems of these model equations. In order to endorse accuracy and efficiency of the method, it is tested on various numerical problems from literature with the aid of maximum absolute errors and rates of convergence. L∞ norms are used to compare the numerical results with other available methods such as Multi-Scale-Galerkin’s method, Haar wavelet collocation method and Meshless method from literature. The comparability of the presented method with other existing numerical methods demonstrates superior efficiency and accuracy.

A minimal search method for solving fractional integro-differential equations based on modified Legendre multiwavelets

A minimal search method for solving fractional integro-differential equations based on modified Legendre multiwavelets

Legendre multi-wavelets collocation method for numerical solution of linear and nonlinear integral equations

In this article, a new collocation technique for numerical solution of Fredholm, Volterra and mixed Volterra-Fredholm integral equations of the second kind is introduced and also developed a numerical integration formula on the basis of linear Legendre multi-wavelets. We also use the linear Legendre multi-wavelets basis for the proposed method. In this technique, the unknown function is approximated by truncated linear Legendre multi-wavelets series. The newly developed numerical technique is applied to both linear and nonlinear benchmark test models from the literature including models with discontinues and non-differentiable exact solutions. The numerical results are compared with the other existing numerical techniques from literature. The comparison among the discussed methods with Monte Carlo method, rationalized Haar functions method, Operational matrix with block-pulse functions method, Bernstein operational matrices method and P-order spline direct method divulge that the present technique is authentic and valid for other physical and engineering problems. The proposed numerical method handle the discontinuity and non-differentiability in the exact solutions very well, whereas the other existence numerical methods are failed to capture these problems. Further, more the comparison of exact and approximate in term of L∞ norms which demonstrate the accuracy, flexibility and robustness of the newly proposed numerical method.

Numerical Solution Of Partial Integro-Differential Equation Using Legendre Multi Wavelets

It is very important to state the initial assumptions for the description of physical phenomenon in the case of partial integro-differential equation. The parabolic equations and boundary conditions can be used to define the time-dependent diffusion process. The integro-differential equation is the combination of integration and derivatives. It is part of the technology, which includes science and engineering. Various models that cover the area of science and engineering are available. Moreover, variable techniques are accessible to solve the integro-differential equations. Numerical method is an important way to solve the challenges in the field of science and industry. To improve efficiency, the companies were working on computer simulation. For reliability, flexibility, and inexpensiveness, the numerical methods are preferred. Linear Legendre multi wavelets form a collocated method based on the numerical solution of one-dimensional parabolic partial integro-differential equation of diffusion type. In this study, we aim to study the diffusion method of numerical solution for the integro-partial differential equation. The diffusion method, its basic concept, and other methods used to solve integro-partial differential equations are also studied in detail. The proposed numerical method is useful to different benchmark problems and provides efficient, accurate, and robust results.

Legendre multiwavelet functions for numerical solution of multi-term time-space convection–diffusion equations of fractional order

In this study, the Ritz–Galerkin method based on Legendre multiwavelet functions is introduced to solve multi-term time-space convection–diffusion equations of fractional order with variable coefficients and initial-boundary conditions. This method reduces the problem to a set of algebraic equations. The coefficients of approximate solutions are obtained from the coefficients of this system. A convergence analysis for function approximations is also presented together with an upper bound for the error of estimates. Numerical examples are included to demonstrate the validity and applicability of the technique.

Numerical Integration of Functions from Holder Classes Hs [0, 1] by Linear Legendre Multi Wavelets

Numerical Integration of Functions from Holder Classes Hs [0, 1] by Linear Legendre Multi Wavelets

Efficient Quadrature Rules for Numerical Integration Based on Linear Legendre Multi-Wavelets

In this work, generalize solution based on linear Legendre multi-wavelets are proposed for single, double and triple integrals with variable limits. To obtain the numerical approximations for integrals, an algorithm with the properties of linear Legendre multi-wavelets are applied. The main benefits of this method are its simple applicable and efficient. Furthermore, the error analysis for single, double and triple integrals is worked out to show the efficiency of the method. Numerical examples for the integrals are conducted by using linear Legendre multi-wavelets in order to validate the error estimation.

Approximate solution of first kind singular integral equation with generalized kernel using Legendre multiwavelets

In this study, numerical solution of a class of Cauchy type singular integral equations of the first kind with generalized kernels (CSIEFKGK) is obtained using Legendre multiwavelets (LMW). Instead of using complex function theory, the appropriate weight function of the solution is obtained by applying LMW-based numerical scheme. We evaluate the matrix representation of generalized kernel with and without the weight factor using recurrence relations and some elementary methods. Using the multiscale representation of the integral operator, the equation is converted into a system of linear equations. The numerical solution of CSIEFKGK is obtained by solving this systems. Finally, a number of examples including applications of crack problems are considered to illustrate the efficiency of the method developed here.

Numerical Integration Based on Linear Legendre Multi Wavelets

In the present work, a new direct computational method for solving definite integrals based on linear Legendre multi wavelets is introduced. This approach is an improvement of previous methods which are based on Haar wavelets functions. An algorithm using properties of the linear Legendre multi wavelets is developed in order to find numerical approximations for double, triple and improper integrals. The main advantage of this method is its efficiency and simple applicability. To validate the algorithm, numerical experiments are conducted to illustrate the accuracy of the method.

A new approach for solving fractional RL circuit model through quadratic Legendre multi-wavelets

The aim of present work is to obtain the approximate solution of fractional model for the electrical RL circuit by using quadratic Legendre multiwavelet method (QLMWM). The beauty of the paper is convergence theorem and mean square error analysis, which shows that our approximate solution converges very rapidly to the exact solution and the numerical solution is compared with the classical solution and Legendre wavelets method (LWM) solution, which is much closer to the exact solution. The fractional integration is described in the Riemann-Liouville sense. The results are shows that the method is very effective and simple. In addition, using plotting tools, we compare approximate solutions of each equation with its classical solution and LWM .

Use of Legendre multiwavelets to solve Carleman type singular integral equations

• Multiscale representation of Cauchy singular integral operator and weight ω ( x ) ( = x α ( 1 − x ) β , α , β > − 1 ) is given in LMW bases. • Estimation formula of local Holder exponent helps to recast the equation with bounded solution throughout the domain. • A posteriori L 2 error of approximate solution u J M S ( x ) is given in terms of wavelet coefficients of resolutions J 2, J 1. • Numerical examples from the theory of elasticity having real as well as complex exponents of the solution are presented. • The stress intensity factor at the tip of the crack can be efficiently estimated with the aid of the approximate solution. This paper is concerned with obtaining approximate numerical solutions of Carleman type singular integral equations by using Legendre multiwavelet basis. The properties of Legendre multiwavelets are first given and the low- and high-pass filters for two-scale relations involving Legendre multiwavelets have been derived. The scheme proposed here consists of two basic steps. In the first step the multiscale representation of the given integral operator is obtained and explicit expressions for the elements of the matrix associated with the multiscale representation are derived. In the second step the given integral equation is converted into a system of linear algebraic equations. The convergence of the method is proved in L 2 space. Two illustrative examples from elasticity are given to show the efficiency of the method developed here.

A numerical method for pricing discrete double barrier option by Legendre multiwavelet

In this Article, a fast numerical numerical algorithm for pricing discrete double barrier option is presented. According to Black-Scholes model, the price of option in each monitoring date can be evaluated by a recursive formula upon the heat equation solution. These recursive solutions are approximated by using Legendre multiwavelets as orthonormal basis functions and expressed in operational matrix form. The most important feature of this method is that its CPU time is nearly invariant when monitoring dates increase. Besides, the rate of convergence of presented algorithm was obtained. The numerical results verify the validity and efficiency of the numerical method.

A New Spectral-Collocation Method Using Legendre Multi-wavelets for Solving of Nonlinear Fractional Differential Equations

A New Spectral-Collocation Method Using Legendre Multi-wavelets for Solving of Nonlinear Fractional Differential Equations