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This paper aims to explore a new wavelet adaptive threshold de-noising method to resolve the shortcomings of wavelet hard-threshold method and wavelet soft-threshold method, which are usually used in gear fault diagnosis.

A new threshold function and a new determined method of threshold for each layer are proposed. The principle and the implementation of the algorithm are given. The simulated signal and the measured gear fault signal are analyzed, and the obtained results are compared with those from wavelet soft-threshold method, wavelet hard-threshold method and wavelet modulus maximum method.

The presented wavelet adaptive threshold method overcomes the defects of the traditional wavelet threshold method, and it can effectively eliminate the noise hidden in the gear fault signal at different decomposition scales. It provides more accurate information for the further fault diagnosis.

A new threshold function is adopted and the multi-resolution unbiased risk estimation is used to determine the adaptive threshold, which overcomes the defect of the traditional wavelet method.

In the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose a wavelet collocation method based on Haar wavelets to identify a parameter in parabolic partial differential equations (PDEs). As Haar wavelet is defined in a very simple way, implementation of the Haar wavelet method becomes easier than the other numerical methods such as finite element method and spectral method. The computational time taken by this method is very less because Haar matrices and Haar integral matrices are stored once and used for each iteration. In the case of Haar wavelet method, Dirichlet boundary conditions are incorporated automatically. Apart from this property, Haar wavelets are compactly supported orthonormal functions. These properties lead to a huge reduction in the computational cost of the method.

The aim of this paper is to reconstruct the source control parameter arises in quasilinear parabolic partial differential equation using Haar wavelet-based numerical method. Haar wavelets possess various properties, for example, compact support, orthonormality and closed form expression. The main difficulty with the Haar wavelet is its discontinuity. Therefore, this paper cannot directly use the Haar wavelet to solve partial differential equations. To handle this difficulty, this paper represents the highest-order derivative in terms of Haar wavelet series and using successive integration this study obtains the required term appearing in the problem. Taylor series expansion is used to obtain the second-order partial derivatives at collocation points.

An efficient and accurate numerical method based on Haar wavelet has been proposed for parameter identification in quasilinear parabolic partial differential equations. Numerical results are obtained from the proposed method and compared with the existing results obtained from various finite difference methods including Saulyev method. It is shown that the proposed method is superior than the conventional finite difference methods including Saulyev method in terms of accuracy and CPU time. Convergence analysis is presented to show the accuracy of the proposed method. An efficient algorithm is proposed to find the wavelet coefficients at target time.

The outcome of the paper would have a valuable role in the scientific community for several reasons. In the current scenario, the parabolic inverse problem has emerged as very important problem because of its application in many diverse fields such as tomography, chemical diffusion, thermoelectricity and control theory. In this paper, higher-order derivative is represented in terms of Haar wavelet series. In other words, we represent the solution in multiscale framework. This would enable us to understand the solution at various resolution levels. In the case of Haar wavelet, this paper can achieve a very good accuracy at very less resolution levels, which ultimately leads to huge reduction in the computational cost.

In this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.

The authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.

The purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.

Many engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

Fabric defects detection is vital in the automation of textile industry. The purpose of this paper is to develop and implement a new fabric defects detection method based on adaptive wavelet.

Fabric defects can be regarded as the abrupt features of textile images with uniform background textures. Wavelets have compact support and can represent these textures. When there is an abrupt feature existed, the response is totally different with the response of the background textures, so wavelets can detect these abrupt features. This method designs the appropriate wavelet bases for different fabric images adaptively. The defects can be detected accurately.

The proposed method achieves accurate detection of fabric defects. The experimental results suggest that the approach is effective.

This paper develops an appropriate method to design wavelet filter coefficients for detecting fabric defects, which is called adaptive wavelet. And it is helpful to realize the automation of textile industry.

The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane–Emden equations:

To deal with singularity, Haar wavelets are used, and to deal with the nonlinear system of equations that arise during computation, the Newton-Raphson method is used. The convergence of these methods is also established and the results are compared with existing techniques.

The authors propose three methods based on uniform Haar wavelets approximation coupled with the Newton-Raphson method. The authors obtain quadratic convergence for the Haar wavelets collocation method. Test problems are solved to validate various computational aspects of the Haar wavelets approach. The authors observe that with only a few spatial divisions the authors can obtain highly accurate solutions for both initial value problems and boundary value problems.

The results presented in this paper do not exist in the literature. The system of nonlinear singular differential equations is not easy to handle as they are singular, as well as nonlinear. To the best of the knowledge, these are the first results for a system of nonlinear singular differential equations, by using the Haar wavelets collocation approach coupled with the Newton-Raphson method. The results developed in this paper can be used to solve problems arising in different branches of science and engineering.

The technique for hiding confidential data in specific digital media by enhancing the graphical contents is known as watermarking. The dissemination of information over a secure channel is essential for multimedia applications. The purpose of this study is to develop a secure communication approach for OFDM system.

This paper exploits a secure communication in the orthogonal frequency division multiplexing (OFDM) system using wavelet-based video watermarking technique. In this work, the Chronological-MS algorithm is used for securing the data communication in the OFDM system. Here, the secret message is embedded in video frames using wavelet transform for hiding sensitive information and the hidden information is transmitted over the OFDM system. The Chronological-MS algorithm is used for selecting the optimal regions in the video for embedding secret message. In embedding phase, wavelet coefficients are obtained by applying wavelet transform on the frame for embedding the secret message. Meanwhile, in extraction phase, the inverse wavelet transform is applied to extract the secret message.

Considering number of frames, the maximum Peak signal-to-noise ratio (PSNR) value is attained by proposed Wavelet + Chronological MS method for Video 2 with value 73.643 dB, respectively. Meanwhile, the minimum mean squared error (MSE) attained by the proposed Wavelet + Chronological MS method is when considering number of frames with MSE values as 0.001 for both Videos 1 and 2. The minimum bit error rate (BER) value is attained by the proposed method with value 0.00009 considering random noise with Video 1. Thus, the proposed Wavelet + Chronological MS have shown better results than the existing techniques.

This work proposes a wavelet-based watermarking method using Chronological-MS, for initiating secured communication over an OFDM. One of the main advantages of wavelets is that they offer a simultaneous localization in time and frequency domain. Hence, the proposed method offers the highly secured data transmission over the OFDM.

Fueled by the rapid growth of internet, steganography has emerged as one of the promising techniques in the communication system to obscure the data. Steganography is defined as the process of concealing the data or message within media files without affecting the perception of the image. Media files, like audio, video, image, etc., are utilized to embed the message. Nowadays, steganography is also used to transmit the medical information or diagnostic reports. The paper aims to discuss these issues.

In this paper, the novel wavelet transform-based steganographic method is proposed for secure data communication using OFDM system. The embedding and extraction process in the proposed steganography method exploits the wavelet transform. Initially, the cost matrix is estimated by the following three aspects: pixel intensity, edge transformation and wavelet transform. The cost estimation matrix provides the location of the cover image where the message is to be entrenched. Then, the wavelet transform is utilized to embed the message into the cover image according to the cost value. Subsequently, in the extraction process, the wavelet transform is applied to the embedded image to retrieve the message efficiently. Finally, in order to transfer the secret information over the channel, the newly developed wavelet-based steganographic method is employed for the OFDM system.

The experimental results are evaluated and performance is analyzed using PSNR and MSE parameters and then compared with existing systems. Thus, the outcome of our wavelet transform steganographic method achieves the PSNR of 71.5 dB which ensures the high imperceptibility of the image. Then, the outcome of the OFDM-based proposed steganographic method attains the higher PSNR of 71.07 dB that proves the confidentiality of the message.

In the authors’ previous work, the embedding and extraction process was done based on the cost estimation matrix. To enhance the security throughout the communication system, the novel wavelet-based embedding and extraction process is applied to the OFDM system in this paper. The idea behind this method is to attain a higher imperceptibility and robustness of the image.

We describe how wavelets may be used to solve partial differential equations. These problems are currently solved by techniques such as finite differences, finite elements and multigrid. The wavelet method, however, offers several advantages over traditional methods. Wavelets have the ability to represent functions at different levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies) are localized in space, which means that the solution can be refined in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem. To demonstrate the wavelet technique, we consider Poisson's equation in two dimensions. By comparison with a simple finite difference solution to this problem with periodic boundary conditions we show how a wavelet technique may be efficiently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method described by Proskurowski and Widlund and others. The convergence of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method.

The purpose of this paper is to examine contagion among the major world markets during the last 25 years and propose a new way to analyze contagion with wavelet methods.

The analysis uses a novel way to study contagion using wavelet methods. The comparison is made between co‐movements at different time scales. Co‐movement methods of the discrete wavelet transform and the continuous wavelet transform are applied.

Clear signs of contagion among the major markets are found. Short time scale co‐movements increase during the major crisis while long time scale co‐movements remain approximately at the same level. In addition, gradually increasing interdependence between markets is found.

Because of the chosen method, the approach is limited to large data sets.

The research has practical implications to portfolio managers etc. who wish to have better view of the dynamics of the international equity markets.

The research uses novel wavelet methods to analyze world equity markets. These methods allow the markets to be analyzed in the whole state space.

The real challenges in online crack detection testing based on guided waves are random noise as well as narrow-band coherent noise; and to achieve efficient structural health assessment methodology, magnificent extraction of noise and analysis of the signals are essential. The purpose of this paper is to provide optimal noise filtering technique for Lamb waves in the diagnosis of structural singularities.

Filtration of time-frequency information of guided elastic waves through the noisy signal is investigated in the present analysis using matched filtering technique which “sniffs” the signal buried in noise and most favorable mother wavelet based denoising methods. The optimal wavelet function is selected using Shannon’s entropy criterion and verified by the analysis of root mean square error of the filtered signal.

Wavelet matched filter method, a newly developed filtering technique in this work and which is a combination of the wavelet transform and matched filtering method, significantly improves the accuracy of the filtered signal and identifies relatively small damage, especially in enormously noisy data. A comparative study is also performed using the statistical tool to know acceptability and practicability of filtered signals for guided wave application.

The proposed filtering techniques can be utilized in online monitoring of civil and mechanical structures. The algorithm of the method is easy to implement and found to be successful in accurately detecting damage.

Although many techniques have been developed over the past several years to suppress random noise in Lamb wave signal but filtration of interferences of wave modes and boundary reflection is not in a much matured stage and thus needs further investigation. The present study contains detailed information about various noise filtering methods, newly developed filtration technique and their efficacy in handling the above mentioned issues.