Search Results (15)

We propose a wavelet collocation method for solving the fractional Riccati equation, using the Müntz–Legendre wavelet basis and its associated operational matrix of fractional integration. The fractional Riccati equation is first transformed into a Volterra integral equation with a weakly singular kernel. By employing the collocation method along with the operational matrix, we reduce the problem to a system of nonlinear algebraic equations, which is then solved using Newton–Raphson’s iterative procedure. The error estimate of the proposed method is analyzed, and numerical simulations are conducted to demonstrate its accuracy and efficiency. The obtained results are compared with existing approaches from the literature, highlighting the advantages of our method in terms of accuracy and computational performance. Full article

Our goal in this work is to solve the fractional Bratu equation, where the fractional derivative is of the Caputo type. As we know, the nonlinearity and derivative of the fractional type are two challenging subjects in solving various equations. In this paper, two approaches based on the collocation method using Müntz–Legendre wavelets are introduced and implemented to solve the desired equation. Three different types of collocation points are utilized, including Legendre and Chebyshev nodes, as well as uniform meshes. According to the experimental observations, we can confirm that the presented schemes efficiently solve the equation and yield superior results compared to other existing methods. Also, the schemes are convergent. Full article

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. Full article

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. Full article

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. Full article

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results. Full article

The stability of a hybrid AC-DC microgrid depends mainly upon the bidirectional interlinking converter (BIC), which is responsible for power transfer, power balance, voltage solidity, frequency and transients sanity. The varying generation from renewable resources, fluctuating loads, and bidirectional power flow from the utility grid, charging station, super-capacitor, and batteries produce various stability issues on hybrid microgrids, like net active-reactive power flow on the AC-bus, frequency oscillations, total harmonic distortion (THD), and voltage variations. Therefore, the control of BIC between AC and DC buses in grid-connected hybrid microgrid power systems is of great importance for the quality/smooth operation of power flow, power sharing and stability of the whole power system. In literature, various control schemes are suggested, like conventional droop control, communication-based control, model predictive control, etc., each addressing different stability issues of hybrid AC-DC microgrids. However, model dependence, single-point-failure (SPF), communication vulnerability, complex computations, and complicated multilayer structures motivated the authors to develop online adaptive neural network (NN) Q-learning-based full recurrent adaptive neurofuzzy nonlinear control paradigms for BIC in a grid-connected hybrid AC-DC microgrid. The proposed strategies successfully ensure the following: (i) frequency stabilization, (ii) THD reduction, (iii) voltage normalization and (iv) negligible net active-reactive power flow on the AC-bus. Three novel adaptive NN Q-learning-based full recurrent adaptive neurofuzzy nonlinear control paradigms are proposed for PQ-control of BIC in a grid-connected hybrid AC-DC microgrid. The control schemes are based on NN Q-learning and full recurrent adaptive neurofuzzy identifiers. Hybrid adaptive full recurrent Legendre wavelet-based Neural Network Q-learning-based full recurrent adaptive NeuroFuzzy control, Hybrid adaptive full recurrent Mexican hat wavelet-based Neural Network Q-learning-based full recurrent adaptive NeuroFuzzy control, and Hybrid adaptive full recurrent Morlet wavelet-based Neural Network Q-learning-based full recurrent adaptive NeuroFuzzy control are modeled and tested for the control of BIC. The controllers differ from each other, based on variants used in the antecedent part (Gaussian membership function and B-Spline membership function), and consequent part (Legendre wavelet, Mexican hat wavelet, and Morlet wavelet) of the full recurrent adaptive neurofuzzy identifiers. The performance of the proposed control schemes was validated for various quality and stability parameters, using a simulation testbench in MATLAB/Simulink. The simulation results were bench-marked against an aPID controller, and each proposed control scheme, for a simulation time of a complete solar day. Full article

In the production process of strip tandem cold rolling mills, the flatness control system is important for improving the flatness quality. The control efficiency of actuators is a pivotal factor affecting the flatness control accuracy. At present, the data-driven methods to intelligently identify the flatness control efficiency have become a research hotspot. In this paper, a wavelet transform longitudinal denoising method, combined with a genetic algorithm (GA-WT), is proposed to handle the big noise of the measured data from each signal channel of the flatness meter, and Legendre orthogonal polynomial fitting is employed to extract the effective flatness features. Based on the preprocessed actual production data, the adaptive moment estimation (Adam) optimization algorithm is applied, to intelligently identify the flatness control efficiency. This paper takes the actual production data of a 1420 mm tandem cold mill as an example, to verify the performance of the new method. Compared with the control efficiency determined by the empirical method, the flatness residual MSE 0.035 is 5.4% lower. The test results indicate that the GA-WT-Legendre-Adam method can effectively reduce the noise, extract the flatness features, and achieve the intelligent determination of the flatness control efficiency. Full article

In this paper, the Legendre wavelet neural network with extreme learning machine is proposed for the numerical solution of the time fractional Black–Scholes model. In this way, the operational matrix of the fractional derivative based on the two-dimensional Legendre wavelet is derived and employed to solve the European options pricing problem. This scheme converts this problem into the calculation of a set of algebraic equations. The Legendre wavelet neural network is constructed; meanwhile, the extreme learning machine algorithm is adopted to speed up the learning rate and avoid the over-fitting problem. In order to evaluate the performance of this scheme, a comparative study with the implicit differential method is constructed to validate its feasibility and effectiveness. Experimental results illustrate that this scheme offers a satisfactory numerical solution compared to the benchmark method. Full article

In this article, we solve fractional Integro differential equations (FIDEs) through a well-known technique known as the Chebyshev Pseudospectral method. In the Caputo manner, the fractional derivative is taken. The main advantage of the proposed technique is that it reduces such types of equations to linear or nonlinear algebraic equations. The acquired results demonstrate the accuracy and reliability of the current approach. The results are compared to those obtained by other approaches and the exact solution. Three test problems were used to demonstrate the effectiveness of the proposed technique. For different fractional orders, the results of the proposed technique are plotted. Plotting absolute error figures and comparing results to some existing solutions reveals the accuracy of the proposed technique. The comparison with the exact solution, hybrid Legendre polynomials, and block-pulse functions approach, Reproducing Kernel Hilbert Space method, Haar wavelet method, and Pseudo-operational matrix method confirm that Chebyshev Pseudospectral method is more accurate and straightforward as compared to other methods. Full article

An efficient algorithm is proposed to find an approximate solution via the wavelet collocation method for the fractional Fredholm integro-differential equations (FFIDEs). To do this, we reduce the desired equation to an equivalent linear or nonlinear weakly singular Volterra–Fredholm integral equation. In order to solve this integral equation, after a brief introduction of Müntz–Legendre wavelets, and representing the fractional integral operator as a matrix, we apply the wavelet collocation method to obtain a system of nonlinear or linear algebraic equations. An a posteriori error estimate for the method is investigated. The numerical results confirm our theoretical analysis, and comparing the method with existing ones demonstrates its ability and accuracy. Full article

The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or Besov-type norms. The purpose of the present paper was twofold. Firstly, we proved wavelet criteria for a natural extended directional scaling function expressed in terms of directional Sobolev or Besov spaces. Secondly, we performed the directional multifractal formalism, i.e., we computed or estimated directional Hölder spectra, either directly or via some Legendre transforms on either directional scaling function or anisotropic scaling functions. We obtained general upper bounds for directional Hölder spectra. We also showed optimal results for two large classes of examples of deterministic and random anisotropic self-similar tools for possible modeling turbulence (or cascades) and textures in images: Sierpinski cascade functions and fractional Brownian sheets. Full article

This paper introduces a new numerical approach to solving a system of fractional differential equations (FDEs) using the Legendre wavelet operational matrix method (LWOMM). We first formulated the operational matrix of fractional derivatives in some special conditions using some notable characteristics of Legendre wavelets and shifted Legendre polynomials. Then, the system of fractional differential equations was transformed into a system of algebraic equations by using these operational matrices. At the end of this paper, several examples are presented to illustrate the effectivity and correctness of the proposed approach. Comparing the methodology with several recognized methods demonstrates that the advantages of the Legendre wavelet operational matrix method are its accuracy and the understandability of the calculations. Full article

This paper presents a new efficient method for the numerical solution of a linear time-dependent partial differential equation. The proposed technique includes the collocation method with Legendre wavelets for spatial discretization and the three-step Taylor method for time discretization. This procedure is third-order accurate in time. A comparative study between the proposed method and the one-step wavelet collocation method is provided. In order to verify the stability of these methods, asymptotic stability analysis is employed. Numerical illustrations are investigated to show the reliability and efficiency of the proposed method. An important property of the presented method is that unlike the one-step wavelet collocation method, it is not necessary to choose a small time step to achieve stability. Full article

Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among these nonlinear phenomena’s. To tackle with the nonlinearity arising, in these phenomena’s we recommend a new method. In the proposed method, Picard’s iteration is used to convert the nonlinear fractional order oscillation equation into a fractional order recurrence relation and then Legendre wavelets method is applied on the converted problem. In order to check the efficiency and accuracy of the suggested modification, we have considered three problems namely: fractional order force-free Duffing–van der Pol oscillator, forced Duffing–van der Pol oscillator and higher order fractional Duffing equations. The obtained results are compared with the results obtained via other techniques. Full article