A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.

A Novel Numerical Method for Solving Fractional Diffusion-Wave and Nonlinear Fredholm and Volterra Integral Equations with Zero Absolute Error

In this work, a new numerical method for the fractional diffusion-wave equation and nonlinear Fredholm and Volterra integro-differential equations is proposed. The method is based on Euler wavelet approximation and matrix inversion of an M×M collocation points. The proposed equations are presented based on Caputo fractional derivative where we reduce the resulting system to a system of algebraic equations by implementing the Gaussian quadrature discretization. The reduced system is generated via the truncated Euler wavelet expansion. Several examples with known exact solutions have been solved with zero absolute error. This method is also applied to the Fredholm and Volterra nonlinear integral equations and achieves the desired absolute error of 0×10−31 for all tested examples. The new numerical scheme is exceptional in terms of its novelty, efficiency and accuracy in the field of numerical approximation.

Comparison of ECG Baseline Wander Removal Techniques and Improvement Based on Moving Average of Wavelet Approximation Coefficients

The baseline wander is among the artifacts that corrupt the ECG signal. This noise can affect some signal features, in particular the ST segment, which is an important marker for the diagnosis of ischemia. This paper presents a study on the effectiveness of several methods and techniques for suppressing the baseline wonder (BW) from the ECG signals. As a result, a new technique called moving average of wavelet approximation coefficients (DWT-MAV) is proposed. The techniques concerned are the moving average, the approximation of the baseline by polynomial fitting, the Savitzky-Golay filtering, and the discrete wavelet transform (DWT). The comparison of this techniques is performed using the main criteria for assessing the BW denoising quality criteria such mean square error (MSE), percent root mean square difference (PRD) and correlation coefficient (COR). In this paper, three other criteria of comparison are proposed namely the number of samples of the ECG signal, the baseline frequency variation and the time processing. Two of these new indices are related to possible real time ECG denoising. To improve the quality of BW suppression including the new indices, a new method is proposed. This technique is a combination of the DWT and the moving average methods. This new technique performs the best compromise in terms of MSE, PRD, coefficient correlation and the time processing. The simulations were performed on ECG recording from MIT-BIH database with synthetic and real baselines.

Reconstructing Compressor Non-uniform Circumferential Flow Field from Spatially Undersampled Data - Part 1: Methodology and Sensitivity Analysis

The flow field in a compressor is circumferentially non-uniform due to the wakes from upstream stators, the potential field from both upstream and downstream stators, and blade row interactions. This non-uniform flow impacts stage performance as well as blade forced vibrations. Historically, experimental characterization of the circumferential flow variation is achieved by circumferentially traversing either a probe or the stator rows. This involves the design of complex traverse mechanisms and can be costly. To address this challenge, a novel method is proposed to reconstruct compressor nonuniform circumferential flow field using spatially under-sampled data points from a few probes at fixed circumferential locations. The paper is organized into two parts. In the present part of the paper, details of the multi-wavelet approximation for the reconstruction of circumferential flow and use of the Particle Swarm Optimization algorithm for selection of probe positions are presented. Validation of the method is performed using the total pressure field in a multi-stage compressor representative of small core compressors in aero engines. The circumferential total pressure field is reconstructed from 8 spatially distributed data points using a triple-wavelet approximation method. Results show good agreement between the reconstructed and the true total pressure fields. Also, a sensitivity analysis of the method is conducted to investigate the influence of probe spacing on the errors in the reconstructed signal.

Legendre wavelet approximation of functions having derivatives of Lipschitz class

In this paper, Legendre Wavelet approximation of functions f having first derivative f' and second derivative f'' of Lip? class, 0 < ? ? 1, have been determined. These wavelet estimators are sharper, better and best possible in Wavelet Analysis. It is observed that the LegendreWavelet estimator of f whose f'' ? Lip? is sharper than the estimator of f having f ' ?Lip? class.

Smoothness of functions versus smoothness of approximation processes

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the [Formula: see text]-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on [Formula: see text], [Formula: see text], [Formula: see text], nonlinear wavelet approximation, etc.