Wavelet Approximation Research Articles

Approximation of one and two dimensional nonlinear generalized Benjamin-Bona-Mahony Burgers' equation with local fractional derivative

This study presents, a numerical method for the solutions of the generalized nonlinear Benjamin-Bona-Mahony-Burgers' equation, with variable order local time fractional derivative. This derivative is expressed as a product of two functions, the usual integer order time derivative, and a function of time having a fractional exponent. Then, forward difference approximation is used for time derivative. The unknown solution of the differential problem and corresponding derivatives are estimated using Haar wavelet approximations (HWA). The collocation procedure is then implemented in HWA, to transform the given model to the system of linear algebraic equations for the determination of unknown constant coefficient of the Haar wavelet series, which update the derivatives and the numerical solutions. The sufficient condition is established for the stability of the proposed technique, and then verified computationally. To check the performance of the scheme, few illustrative examples in one and two dimensions along with l∞ and l2 error norms are also given. Besides this, the computational convergence rate is calculated for both type equations. Additionally, computed solutions are compared with available results in literature. Simulations and graphical data discloses, that suggested scheme works well for such complex problems.

Analysis of nonlinear Burgers equation with time fractional Atangana-Baleanu-Caputo derivative

This paper demonstrates, a numerical method to solve the one and two dimensional Burgers' equation involving time fractional Atangana-Baleanu Caputo (ABC) derivative with a non-singular kernel. The numerical stratagem consists of a quadrature rule for time fractional (ABC) derivative along with Haar wavelet (HW) approximations of one and two dimensional problems. The key feature of the scheme is to reduce fractional problems to the set of linear equations via collocation procedure. Solving the system gives the approximate solution of the given problem. To verify the effectiveness of the developed method five numerical examples are considered. Besides this, the obtained simulations are compared with some published work and identified that proposed technique is better. Moreover, computationally the convergence rate in spatiotemporal directions is presented which shows order two convergence. The stability of the proposed scheme is also described via Lax-Richtmyer criterion. From simulations it is obvious that the scheme is quite useful for the time fractional problems.

An analysis of best wavelet approximation problem of a function using Hermite wavelet

In this paper, we introduce two novel wavelet approximations tailored for functions exhibiting a restricted second derivative and a bounded derivative. Employing the Hermite wavelet method, we derive these approximations to address the need for effective representations of such functions. Our findings reveal that these new wavelet approximations offer enhanced accuracy and efficiency in capturing the underlying structure of , making them valuable tools in various applications requiring precise function approximations with limited derivative constraints.

Brightness Aware Pixel Stretching for Perceptually Invisible Images Using Wavelet Approximation Balancing

Brightness Aware Pixel Stretching for Perceptually Invisible Images Using Wavelet Approximation Balancing

Edge Machine Learning-Enabled Predictive Fault Detection System for Conveyor Belt Maintenance Optimization in Industrial Settings

Digital transformation is essential for industrial and manufacturing sectors, especially in conveyor belt systems. On the other hand, Innovative electronic components and software applications offer opportunities for advanced fault prediction, minimizing operational disruptions, reducing inefficiencies, and enhancing overall industrial efficiency. Efficient conveyor belt systems are crucial for industrial operations, requiring early fault detection to prevent costly downtime and ensure safety. The utilization of embedded machine learning via the Edge Impulse platform enhances data collection and spectral feature extraction using wavelet approximation. This article presents a novel methodology for predictive fault detection in conveyor belt systems employing edge machine learning. Leveraging the ESP32 microcontroller and accelerometer sensor for real-time vibration data capture, a four-layer Artificial Neural Network (ANN) model, trained on 336 feature samples, is deployed via Edge Impulse and TensorFlow scripts. TensorFlow Lite compresses the model for microcontroller integration. The ESP32 microcontroller, acting as an edge device, achieves real-time predictions with 97.5% accuracy and 0.20 minimal loss. This work promises significant strides in predictive maintenance, offering cost savings, operational reliability, and efficiency gains in industrial settings, revolutionizing maintenance strategies.

A numerical study on fractional order financial system with chaotic and Lyapunov stability analysis

In the last few decades, academic research has focused more on financial problems and poverty levels. These are among the two major challenges of the modern world today. To understand the challenge of financial crisis and poverty in societies. This paper explains a deterministic financial system of non linear differential equations. Which aim to provide recommendations to address the twin challenges. In this fractional chaotic model, memory and chaos are portrayed as effects on a system. Financial systems of fractional order have three compartments: interest rate x(t), investment demand rate y(t) and price indexes rate z(t). For the first time, the Legendre wavelet approximation is applied to fractional order financial systems. A major objective of this process is to convert fractional differential equations (FDEs) into algebraic equations by using Legendre wavelets and their fractional integral operators. For residual analysis, convergence and stability analysis, we have used iteration methods and Legendre wavelets. This study indicates good results between the approximate and numerical solutions and that this method is efficient and accurate. Using the iteration method, we prove that the solution exists, while the contraction theory proves its uniqueness. The findings of this study will be extremely useful for future financial management. We have demonstrated the applicability and effectiveness of these methods by analysing numerical simulations for the fractional order financial model and the numerical simulation has been calculated by MATLAB programming.

A novel stochastic ten non-polynomial cubic splines method for heat equations with noise term

In this paper, a new numerical method for solving a class of stochastic partial differential equations is presented. The proposed method is based on a non-polynomial cubic spline algorithm with an O(h4) local truncation error. The new approach has the advantage of simultaneously including ten non-polynomial cubic spline schemes. The proposed approach is also tested on three stochastic heat equations. The current results are compared to solutions obtained using the B-spline wavelet approximation method. The provided solutions demonstrate the proposed method's accuracy and applicability.

Accurate Criteria for Broken Bar Detection in Induction Motors Based on the Wavelet (Packet) Transform

Finding reliable and robust criteria for the detection of broken bars in induction motors is key for the maintenance of industrial engines, and some of the most efficient methods analyze the stator start-up current. Due to the transitory characteristics and short duration of the signal, suitable time-frequency mathematical tools are very useful for this purpose. We propose here algorithms based on the discrete wavelet and wavelet packet transform, combined with other tools in signal processing, to offer an accurate quantitative method for failure detection due to broken bars in induction motors. A good selection of the wavelet family is important for a good performance of the indicator, and the discrete approximation of the Meyer wavelet, ‘dmeyer’, consistently demonstrates the most favorable results. Our findings highlight the effectiveness of both the wavelet and wavelet packet transforms in accurately detecting broken bars in induction motors. This fact allows optimal monitoring strategies in industrial applications.

An Efficient Numerical Method Based on Bell Wavelets for Solving the Fractional Integro-Differential Equations with Weakly Singular Kernels

A novel numerical scheme based on the Bell wavelets is proposed to obtain numerical solutions of the fractional integro-differential equations with weakly singular kernels. Bell wavelets are first proposed and their properties are studied, and the fractional integration operational matrix is constructed. The convergence analysis of Bell wavelets approximation is discussed. The fractional integro-differential equations can be simplified to a system of algebraic equations by using a truncated Bell wavelets series and the fractional operational matrix. The proposed method’s efficacy is supported via various examples.

Reduced Complexity Approximation and Design of Gaussian Impulse Response Filters and Wavelets

Reduced Complexity Approximation and Design of Gaussian Impulse Response Filters and Wavelets

Implementation of Legendre wavelet method for the size dependent bending analysis of nano beam resonator under nonlocal strain gradient theory

The bending analysis due to thermoelastic loading in nanoscale materials is primarily determined by the physical aspect of the material and the modeling of structures. This paper examines the prediction of bending characteristics of the Euler-Bernoulli beam using Moore-Gibson-Thompson (MGT) thermoelasticity theory in conjunction with nonlocal strain gradient theory (NSGT). The coupled equations for dimensionless deflection and temperature change with ramp-type heating boundary conditions are formulated and solved using the Laplace transform method and the wavelet approximation method. The stiffness softening and hardening effects due to nonlocal and length-scale parameters are assessed, and their discrepancies are discussed. Findings further reveal that the impact of the thermal relaxation parameter on deflection and temperature is negligible. Moreover, the different aspect ratios of the beam's structure depict the prominently behavior of structural simulations in bending.

Detection of Unauthorized Actions in Networks Using Wavelet Analysis

Signal processing techniques are used to analyze and detect network anomalies because of their ability to detect new and unknown intrusions. The paper proposes a method of modeling network signals for the detection of network anomalies, which combines wavelet approximation and the theory of system identification. To characterize the behavior of network traffic, fifteen functions are provided, which are used as input signals within the system. At the same time, it is assumed that security violations within the network can be detected by checking abnormal patterns of system functioning according to audit data.
 Despite the fact that machine learning methods have achieved significant results in detecting network anomalies, they still face the difficulty of using the implemented algorithms, in the presence of differences in the behavior of the training data and test data, which in turn leads to inefficient performance of the algorithms. This effect is exacerbated by the limitation of algorithms to detect previously unknown types of attacks due to the large number of false positives.
 The paper develops a new method of modeling network signals for detecting anomalies in networks using wavelet analysis. In particular, the general architecture of the approach consists of three components: feature analysis, modeling of normal network traffic based on wavelet approximation and prediction using ARX model, intrusion or non-intrusion decision making
 The result is evaluated using the DARPA intrusion detection dataset, which performs a comprehensive analysis of the intrusions in the dataset. Evaluation results show that this approach provides a high level of detection of both instances and types of attacks.

Bernoulli polynomial based wavelets method for solving chaotic behaviour of financial model

This paper presents an algorithm for solving systems of non integer financial chaotic model. The Bernoulli wavelets function approximation applies to fractional order financial systems for the first time. The main objective of this process is to convert fractional differential equation systems into algebraic equation systems using wavelets approximation based on Bernoulli wavelets and their fractional integral operators. This fractional financial chaotic model that represents the effects of memory and chaos in a system. There are 3 compartments in this fractional order financial system: price indexes, interest rates, and investment demand. Analyses are depends on iteration method and Bernoulli wavelets method to determine the stability analysis, residual error and convergence analysis of the solutions. This study indicates that the presented approximate solutions fit exactly with the numerical solution and that the technique is accurate and effective. Our proposed model has been discussed for its uniqueness, boundedness, and non negativity. Its findings are extremely important for dealing with financial issues in the future.

A wavelet analysis method to retrieve thermal conductivity and radiative properties of intense extinction and low conductive medium

The accurate acquisition of the physical properties of strongly extinction materials limits their application development. In order to solve the problem of large errors in the high temperature measurement of these materials, this paper constructs an inversion model for the radiative and conductive coupled heat transfer through wavelet decomposition coefficients, and compares it with traditional methods. The radiative and conductive coupled heat transfer model is solved by finite volume method and Monte Carlo method. Daubechies nine wavelet is used for signal wavelet analysis. It has been discovered that the wavelet decomposition parameters of the temperature field are mainly concentrated in the 4–8 wavelet frequency band of wavelet, it can separate from the random error that is mainly concentrated in the 1–3 frequency band of wavelet. Most of physical properties like the heating power, convective heat transfer coefficient, material thermal conductivity, extinction coefficient has a great impact on the waveform and amplitude of wavelet detail coefficient and approximation coefficient, and the change trend of approximation coefficient and detail coefficient amplitude is consistent with the trend of temperature field. However, the effect of scattering albedo on wavelet parameters is tiny. Through inversion, it is found that the inversion model based on wavelet decomposition parameters of 5–7 frequency band has higher accuracy when random error exists.

A Numerical Technique Based on Bernoulli Wavelet Operational Matrices for Solving a Class of Fractional Order Differential Equations

In this paper, we present an efficient, new, and simple programmable method for finding approximate solutions to fractional differential equations based on Bernoulli wavelet approximations. Bernoulli Wavelet functions involve advantages such as orthogonality, simplicity, and ease of usage, in addition to the fact that fractional Bernoulli wavelets have exact operational matrices that improve the accuracy of the applied approach. A fractional differential equation was simplified to a system of algebraic equations using the fractional order integration operational matrices of Bernoulli wavelets. Examples are used to demonstrate the technique’s precision.

Wavelets approach for the solution of nonlinear variable delay differential equations

Abstract In this study, the Laguerre wavelet-oriented numerical scheme for nonlinear first and second-order delay differential equations (DDEs) is offered. The proposed technique is dependent on the truncated series of the Laguerre wavelets approximation of an unknown function. Here, we transform the different ordered DDEs into a system of non-linear algebraic equations with the help of limit points of a sequence of collocation points. Four nonlinear illustrations are involved to prove the efficiency of the planned technique. the Obtained results are equated with the current results, indicating the proposed technique’s accuracy and efficiency.

A Virtual Impactor-Based Portable Optical Sensor With Probabilistic Neural Network: For Particle Property Identification

With the increasing problem of atmospheric particulate matter pollution, an increasing number of scholars have conducted studies related to particulate matter. Most of the research on particulate matter has focused on information measurement such as particle size, where light scattering method is widely used in the field of particulate matter measurement. Numerous studies have shown that the particle properties have a great influence on the measurement accuracy of light scattering method. However, there are relatively few studies related to the identification of particle properties. Therefore, according to the difference of particle signals with different properties, this paper proposes a new method of particle property identification based on probabilistic neural network. Also, a new portable light scattering sensor integrated with virtual impactor is designed for particle signal acquisition. In the particle signal feature extraction session, we innovatively select Image entropy, kurtosis and sample entropy of discrete wavelet approximation signal as eigenvalues of the particle signal. The experimental results show that the virtual impactor has a good separation with a cutting particle size of 2.67 μm. The designed new portable light scattering sensor has good performance which can realize the task of particle light scattering signal acquisition. Meanwhile, the proposed new method of particle property identification has a 99% correct classification rate. In future work, we will further optimize the design of portable sensors and thus reduce the specifications of the measurement system.

LINEAR AND NON-LINEAR WAVELET APPROXIMATIONS OF FUNCTIONS OF LIPSCHITZ CLASS AND RELATED CLASSES USING THE HAAR WAVELET SERIES

In this paper, linear wavelet approximation of a functions f belonging to Lipξ, and Lip(ξ, p), 1 ≤ p < ∞, have been determined by using Haar scaling function and Haar wavelet series. A non-linear wavelet approximation of function belonging L2(R) and Lipα class has been obtained. The comparison of the order of linear and non-linear approximations have been studied. It is observed that a non-linear approximation error is better and sharper than linear approximation error.

The Least Squares Time Element Method Based on Wavelet Approximation for Structural Dynamic Load Identification

Dynamic load identification is a commonly used and quite important approach to obtain the excitation loads of structures in engineering practice. In this paper, a novel dynamic load identification method combining the least squares time element method (LSTEM), wavelet scaling function and regularization method is proposed, which performs a better accuracy and a stronger anti-noise ability. It decomposes the time history of dynamic load into a series of time elements, and approximates the load profile at each time element using wavelet scaling functions. In order to balance the accuracy and efficiency for load identification, an optimal wavelet resolution is then determined. Simultaneously, the least squares time element model is derived which establishes the forward model for computing the wavelet coefficient. Finally, the wavelet coefficients for dynamic load identification are accurately and stably solved by implementing regularization. By this method, on the one hand, the wavelet scaling function and LSTEM improve the identification accuracy, and on the other hand, the integral process in the least squares operation gains the anti-noise ability for the load identification. A numerical example of a roof structure and an experiment of a composite laminate are studied and verify the effectiveness of the proposed method.

Fusion of CT and MRI Medical Images Using Dual Tree Complex Wavelet Transform

Abstract: Combining various medical pictures will improve illness diagnosis accuracy and illustrate the complex link between them for medical study. Existing approaches take a long time and require a large number of data to train the models. In this model, we will use multi-stage fusion networks to extract sophisticated information from medical pictures such as Magnetic Resonance Imaging (MRI) and Computed Tomography (CT). In the suggested approach, we will extract the difficult and correlated information from each picture using the Dual Tree Complex Wavelet Transform (DTCWT), and then segment the fused image to obtain the segmented image. The proposed approach entails, the fusion of multimodal medical pictures may be accomplished using the Dual Tree Complex Wavelet Transform, which converts the original medical image to grayscale and decomposes it before extracting the wavelet coefficients using DTCWT. Following that, wavelet approximation is utilized to produce the fused coefficients. To obtain a final fused picture, the Inverse Dual Tree Complicated Wavelet Transform is performed. Additionally, segmentation is carried out in order to provide a segmented image for better visual representation. The quality of the final fused picture can be increased using the proposed strategy.