Journal articles: 'Numerical analysis – Numerical methods in Fourier analysis – Wavelets'

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Любимова, Mariya Lyubimova, Князева, and Tatyana Knyazeva. "Processing of tomographic images by means of wavelet analysis." Journal of New Medical Technologies. eJournal 8, no. 1 (November 5, 2014): 1–4. http://dx.doi.org/10.12737/4110.

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The paper is devoted to the problem of processing of tomographic images using wavelet analysis. The features of image processing techniques, indications were analyzed. Wavelets are a signal waveform of limited duration that has an average value of zero. Wavelets are comparable to a sine wave, and they are the basis of Fourier analysis. Wavelet analysis method allows to processing of tomographic images using a large time interval, where more accurate information about the low frequency region and shorter when information is needed on high frequency. The characteristic features of the settings wavelet transforms are described. Their bad choice reduces the reliability of detection of changes in the structure of signals when changing system state. The key stages of the reconstruction tomography images in DICOM format using the method of wavelet analysis were examined; algorithm of noise reduction was investigated. Practical area of application of wavelet analysis doesn´t limited to digital signal processing; it also covers physical experiments, numerical methods and other areas of physics and mathematics. By being able to analyze the non-stationary signals, wavelet analysis has become a powerful alternative Fourier transform in medical applications.

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Podsiadlo, P., and G. W. Stachowiak. "Multi-scale representation of tribological surfaces." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 216, no. 6 (June 1, 2002): 463–79. http://dx.doi.org/10.1243/135065002762355361.

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Many numerical surface topography analysis methods exist today. However, even for the moderately complicated topography of a tribological surface these methods can provide only limited information. The reason is that tribological surfaces often exhibit a non-stationary and multi-scale nature while the numerical methods currently used work well with surface data exhibiting a stationary random process and provide surface descriptors closely related to a scale at which surface data were acquired. The suitability of different methods, including Fourier transform, windowed Fourier transform, Cohen's class distributions (especially the Wigner-Ville distribution), wavelet transform, fractal methods and a hybrid fractal-wavelet method, for the analysis of tribological surface topographies is investigated in this paper. The method best suited to this purpose has been selected.

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Boyd, John P. "Limited-Area Fourier Spectral Models and Data Analysis Schemes: Windows, Fourier Extension, Davies Relaxation, and All That." Monthly Weather Review 133, no. 7 (July 1, 2005): 2030–42. http://dx.doi.org/10.1175/mwr2960.1.

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Abstract Regional spectral models have previously periodized and blended limited-area data through ad hoc low-order schemes justified by intuition and empiricism. By using infinitely differentiable “window functions” or “bells” borrowed from wavelet theory, one can periodize with preservation of spectral accuracy. Similarly, it is shown through a mixture of theory and numerical examples that Davies relaxation for blending limited-area and global data in one-way nested forecasting can be performed using the same C∞ bells as employed for the Fourier blending.“The relative success of empirical methods . . . may be used as partial justification to allow us to make the daring approximation that the data on a limited area domain may be decomposed into a trend and a periodic perturbation, and to proceed with Fourier transformation of the latter.” Laprise (2003, p. 775)

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Sonekar, Parikshit, and Mira Mitra. "A wavelet-based model of one-dimensional periodic structure for wave-propagation analysis." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2113 (October 14, 2009): 263–81. http://dx.doi.org/10.1098/rspa.2009.0369.

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In this paper, a wavelet-based method is developed for wave-propagation analysis of a generic multi-coupled one-dimensional periodic structure (PS). The formulation is based on the periodicity condition and uses the dynamic stiffness matrix of the periodic cell obtained from finite-element (FE) or other numerical methods. Here, unlike its conventional definition, the dynamic stiffness matrix is obtained in the wavelet domain through a Daubechies wavelet transform. The proposed numerical scheme enables both time- and frequency-domain analysis of PSs under arbitrary loading conditions. This is in contrast to the existing Fourier-transform-based analysis that is restricted to frequency-domain study. Here, the dispersion characteristics of PSs, especially the band-gap features, are studied. In addition, the method is implemented to simulate time-domain wave response under impulse loading conditions. The two examples considered are periodically simply supported beam and periodic frame structures. In all cases, the responses obtained using the present periodic formulation are compared with the response simulated using the FE model without the periodicity assumption, and they show an exact match. This validates the accuracy of the periodic assumption to obtain the time- and frequency-domain wave responses up to a high-frequency range. Apart from this, the proposed method drastically reduces the computational cost and can be implemented for homogenization of PSs.

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Qian, Xu, Yaming Chen, and Songhe Song. "Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time." Communications in Computational Physics 15, no. 3 (March 2014): 692–711. http://dx.doi.org/10.4208/cicp.120313.020813a.

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AbstractIn this paper, we propose a wavelet collocation splitting (WCS) method, and a Fourier pseudospectral splitting (FPSS) method as comparison, for solving one-dimensional and two-dimensional Schrödinger equations with variable coefficients in quantum mechanics. The two methods can preserve the intrinsic properties of original problems as much as possible. The splitting technique increases the computational efficiency. Meanwhile, the error estimation and some conservative properties are investigated. It is proved to preserve the charge conservation exactly. The global energy and momentum conservation laws can be preserved under several conditions. Numerical experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

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Du, Shuangxing, Dominic A. Hudson, W. Geraint Price, Pandeli Temarel, Ruizhang Chen, and Yousheng Wu. "Wavelet Analysis of Loads on a Flexible Ship Model Traveling in Large-Amplitude Waves." Journal of Ship Research 52, no. 04 (December 1, 2008): 249–62. http://dx.doi.org/10.5957/jsr.2008.52.4.249.

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This study describes an application of a Daubechies wavelet function to analyze measured ship model data. The records of a self-propelled, flexible model of the S175 containership moving in waves are analyzed by fast Fourier transform (FFT) and wavelet methods. It is shown that the high-frequency component of the recorded rigid body motions can be omitted without substantially affecting the main features of the data set relating to dynamic loads. The decomposition of the bending moment time history into low-and high-frequency components allows the time of impact occurrence and its amplitude to be easily detected. Such quantities provide important information for the development of generic and realistic transient impact (e.g., slamming, green water) force models for ships traveling in waves.

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Chen, Jian Hui. "An Improved EMD Method and its Application in Nonstationary Signals Analysis." Advanced Materials Research 429 (January 2012): 313–17. http://dx.doi.org/10.4028/www.scientific.net/amr.429.313.

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Empirical mode decomposition (EMD) method based on HHT has exhibited unique advantages such as adaptability and highly efficiency in many nonlinear, nonstationary signals processing applications. It breaks the uncertainty principle limit, but the traditional EMD still has its deficiencies. In this article, we construct a new wavelet which has excellent decomposing-frequency performance and energy concentration, and then an improved EMD method based on this wavelet is presented. Results of numerical simulation show the validity and efficiency of the method proposed in paper are better than traditional one. Furthermore, some foreseeable trends of time-frequency distribution technologies are described. The systems in reality, strictly speaking, tend to non-linear, so most practical signals are non-stationary random signals. Nonlinear, nonstationary signals analysis is a very significant and difficult problem in almost all technical fields such as automation, communication, aerospace- engineering, biomedicine, structural fault diagnosis and so on. Owed to the rapid development of large scale integrated circuit technology and artificial intelligence, the exploration of signal processing theories have got a sharply impetus. A series of new modern signal processing theories and methods have appeared to meet the need of time-frequency joint analysis of nonlinear, non-Gaussian and non-stationary signals, including discrete short-time Fourier transform, wavelet transform, Hilbert-Huang transform and so on. Time-frequency joint analysis can observe the evolution of the signal in the time domain and the frequency domain simultaneously, provide local time-frequency characteristics of the signal.

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Babajanian Bisheh, Hossein, Gholamreza Ghodrati Amiri, and Ehsan Darvishan. "Ensemble Classifiers and Feature-Based Methods for Structural Damage Assessment." Shock and Vibration 2020 (December 19, 2020): 1–14. http://dx.doi.org/10.1155/2020/8899487.

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In this paper, a new structural damage detection framework is proposed based on vibration analysis and pattern recognition, which consists of two stages: (1) signal processing and feature extraction and (2) damage detection by combining the classification result. In the first stage, discriminative features were extracted as a set of proposed descriptors related to the statistical moment of the spectrum and spectral shape properties using five competitive time-frequency techniques including fast S-transform, synchrosqueezed wavelet transform, empirical wavelet transform, wavelet transform, and short-time Fourier transform. Then, forward feature selection was employed to remove the redundant information and select damage features from vibration signals. By applying different classifiers, the capability of the feature sets for damage identification was investigated. In the second stage, ensemble-based classifiers were used to improve the overall performance of damage detection based on individual classifiers and increase the number of detectable damages. The proposed framework was verified by a suite of numerical and full-scale studies (a bridge health monitoring benchmark problem, IASC-ASCE SHM benchmark structure, and a cable-stayed bridge in China). The results showed that the proposed framework was superior to the existing single classifier and could assess the damage with reduced false alarms.

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Chen, Binqiang, Qixin Lan, Yang Li, Shiqiang Zhuang, and Xincheng Cao. "Enhancement of Fault Feature Extraction from Displacement Signals by Suppressing Severe End Distortions via Sinusoidal Wave Reduction." Energies 12, no. 18 (September 15, 2019): 3536. http://dx.doi.org/10.3390/en12183536.

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Displacement signals, acquired by eddy current sensors, are extensively used in condition monitoring and health prognosis of electromechanical equipment. Owing to its sensitivity to low frequency components, the displacement signal often contains sinusoidal waves of high amplitudes. If the digitization of the sinusoidal wave does not satisfy the condition of full period sampling, an effect of severe end distortion (SED), in the form of impulsive features, is likely to occur because of boundary extensions in discrete wavelet decompositions. The SED effect will complicate the extraction of weak fault features if it is left untreated. In this paper, we investigate the mechanism of the SED effect using theories based on Fourier analysis and wavelet analysis. To enhance feature extraction performance from displacement signals in the presence of strong sinusoidal waves, a novel method, based on the Fourier basis and a compound wavelet dictionary, is proposed. In the procedure, ratio-based spectrum correction methods, using the rectangle window as well as the Hanning window, are employed to obtain an optimized reduction of strong sinusoidal waves. The residual signal is further decomposed by the compound wavelet dictionary which consists of dyadic wavelet packets and implicit wavelet packets. It was verified through numerical simulations that the reconstructed signal in each wavelet subspace can avoid severe end distortions. The proposed method was applied to case studies of an experimental test with rub impact fault and an engineering test with blade crack fault. The analysis results demonstrate the proposed method can effectively suppress the SED effect in displacement signal analysis, and therefore enhance the performance of wavelet analysis in extracting weak fault features.

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Stanković, Ljubiša, Jonatan Lerga, Danilo Mandic, Miloš Brajović, Cédric Richard, and Miloš Daković. "From Time–Frequency to Vertex–Frequency and Back." Mathematics 9, no. 12 (June 17, 2021): 1407. http://dx.doi.org/10.3390/math9121407.

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The paper presents an analysis and overview of vertex–frequency analysis, an emerging area in graph signal processing. A strong formal link of this area to classical time–frequency analysis is provided. Vertex–frequency localization-based approaches to analyzing signals on the graph emerged as a response to challenges of analysis of big data on irregular domains. Graph signals are either localized in the vertex domain before the spectral analysis is performed or are localized in the spectral domain prior to the inverse graph Fourier transform is applied. The latter approach is the spectral form of the vertex–frequency analysis, and it will be considered in this paper since the spectral domain for signal localization is well ordered and thus simpler for application to the graph signals. The localized graph Fourier transform is defined based on its counterpart, the short-time Fourier transform, in classical signal analysis. We consider various spectral window forms based on which these transforms can tackle the localized signal behavior. Conditions for the signal reconstruction, known as the overlap-and-add (OLA) and weighted overlap-and-add (WOLA) methods, are also considered. Since the graphs can be very large, the realizations of vertex–frequency representations using polynomial form localization have a particular significance. These forms use only very localized vertex domains, and do not require either the graph Fourier transform or the inverse graph Fourier transform, are computationally efficient. These kinds of implementations are then applied to classical time–frequency analysis since their simplicity can be very attractive for the implementation in the case of large time-domain signals. Spectral varying forms of the localization functions are presented as well. These spectral varying forms are related to the wavelet transform. For completeness, the inversion and signal reconstruction are discussed as well. The presented theory is illustrated and demonstrated on numerical examples.

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Koziol, Piotr. "Analytical approximation of rail bending stress." MATEC Web of Conferences 148 (2018): 05002. http://dx.doi.org/10.1051/matecconf/201814805002.

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Phenomena associated with railway dynamics are usually analysed by using numerical approaches due to high computational complexity of such systems. However, classical methods based on analytical modelling are still highly valued and desirable by researchers and railway industry. This paper presents analytical solution representing dynamic response of railway track due to moving train in the case of nonlinear foundation. In published papers, one can find analyses of various characteristics such as velocity and acceleration of vibrations of track layers or bending moments of rails. The approach applied in this paper uses the Fourier transform combined with wavelet based approximation applied to the systems of infinitely long beams. The system of Euler-Bernoulli beams resting on viscoelastic foundation represents two-layer model (or one-layer model) of railway track, commonly used in engineering studies. It is shown that although both methods give good results for displacements, analysis of other characteristics, involving derivatives of higher orders, might lead to wrong results, even in the linear case. Possible reasons of this problem are pointed out. Some modifications of the known dynamic railway track models are proposed for further work.

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Strawderman, Robert L., C. Gasquet, P. Witomski, and Robert Ryan. "Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets." Journal of the American Statistical Association 95, no. 449 (March 2000): 349. http://dx.doi.org/10.2307/2669586.

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Yu, Zhou, Ray Abma, John Etgen, and Claire Sullivan. "Attenuation of noise and simultaneous source interference using wavelet denoising." GEOPHYSICS 82, no. 3 (May 1, 2017): V179—V190. http://dx.doi.org/10.1190/geo2016-0240.1.

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High-resolution seismic imaging requires noise attenuation to achieve signal-to-noise ratio (S/N) improvements without compromising data bandwidth. Amplitude versus offset analysis requires good amplitude fidelity in premigration processes. Any nonreflected wavefield energy in the data will degrade the seismic image quality. Despite significant progress over the years, preserving low-frequency signals without compromising the S/N and avoiding the smearing of aliased signal are still a challenge for conventional methods. This problem is compounded when additional interference noise is added with simultaneous source acquisition. Because noise characteristics vary from shot to shot and receiver to receiver, we need a method that is robust and effective. In addition, we also want the method to be efficient and easy to use from a practical perspective. We have recently developed an approach using a wavelet transform to deterministically separate the primary signal from the noise, including simultaneous source interference. The goals are (1) improving the S/N without compromising bandwidth, (2) preserving the low-frequency and near-offset primaries without compromising the S/N, and (3) preserving the local primary wavefield while attenuating noise. For distance-separated simultaneous source acquisition, the goal is preserving long-offset primaries while removing interference. This wavelet denoising flow consists of a linear transformation and filtering using the complex wavelet transform (CWT). For reflection signals, normal moveout (NMO) is used. NMO transforms the low-velocity surface waves and the interference noise to where it is easily identified and rejected with a dip filter in the multidimensional CWT domain. Land field data examples have demonstrated significantly improved S/Ns and low-frequency signal preservation in migrated images after wavelet denoising. Since the numerical implementation of the CWT is as fast as a fast Fourier transform, this flow is able to suppress noise and interference simultaneously on the 3D land data much faster than the other inversion methods.

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Chiang, C. H., and C. C. Cheng. "Detecting Rebars and Tubes Inside Concrete Slabs Using Continuous Wavelet Transform of Elastic Waves." Journal of Mechanics 20, no. 4 (December 2004): 297–302. http://dx.doi.org/10.1017/s172771910000352x.

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AbstractA typical problem of elastic wave methods, such as the impact echo method, is due to peak detection based solely on amplitude spectrum. Current study aims to improve the feature identification of impact-echo signals obtained from buried objects in concrete slabs. Steel rebar, steel tubes, and PVC tubes embedded in a concrete slab are tested. Numerical simulations are carried out based on models constructed using the finite element method. The received signals, both experimental and simulated, are analyzed using both fast Fourier transform and continuous wavelet transform (CWT). The amplitude spectra can only provide global information and lose some important local effects of frequency components. This can be resolved by continuous wavelet transform for preserving the transient effects in the frequency domain. Localized spectral contents are analyzed and thus better understanding is achieved for the impulse responses due to different objects below the surface of the concrete slab. Features related to steel rebar, PVC and steel tubes are readily identified in the coefficient plot of wavelet coefficients. Multiple reflections and vibration modes related to various characteristics of wave propagation in the concrete slab can now be decomposed into distinctive frequency bands with different time durations. The result of CWT provides more information and is easier to interpret than that of the spectral analysis. The same peak frequency found in the amplitude spectrum is now distinguishable between PVC and steel tubes at a resolution of 0.1kHz or better. Such findings provide a more effective way to pick up true rebar signals using the impact-echo method.

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AWREJCEWICZ, J., A. V. KRYSKO, I. E. KUTEPOV, N. A. ZAGNIBORODA, M. V. ZHIGALOV, and V. A. KRYSKO. "ANALYSIS OF CHAOTIC VIBRATIONS OF FLEXIBLE PLATES USING FAST FOURIER TRANSFORMS AND WAVELETS." International Journal of Structural Stability and Dynamics 13, no. 07 (August 23, 2013): 1340005. http://dx.doi.org/10.1142/s0219455413400051.

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In this paper chaotic vibrations of flexible plates of infinite length are studied. The Kirchhoff–Love hypotheses are used to derive the nondimensional partial differential equations governing the plate dynamics. The finite difference method (FDM) and finite element method (FEM) are applied to validate the numerical results. The numerical analysis includes both standard (time histories, fast Fourier Transform, phase portraits, Poincaré sections, Lyapunov exponents) as well as wavelet-based approaches. The latter one includes the so called Gauss 1, Gauss 8, Mexican Hat and Morlet wavelets. In particular, various plate dynamical regimes including the periodic, quasi-periodic, sub-harmonic, chaotic vibrations as well as bifurcations of the plate are illustrated and studied. In addition, the convergence of numerical results obtained via different wavelets is analyzed.

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Candela, D. "Numerical Methods for Nonlinear Fourier Analysis, Prediction, and Filtering." Journal of Computational Physics 117, no. 2 (March 1995): 205–14. http://dx.doi.org/10.1006/jcph.1995.1059.

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Yakovleva, T. V., V. G. Bazhenov, V. S. Kruzhilin, and V. A. Krysko. "MATHEMATICAL MODELING OF NONLINEAR VIBRATIONS OF A PLATE WITH EXPOSURE TO COLOR NOISE TAKING INTO ACCOUNT OF CONTACT INTERACTION WITH THE BEAM." Problems of strenght and plasticity 81, no. 3 (2019): 324–32. http://dx.doi.org/10.32326/1814-9146-2019-81-3-324-332.

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A theory of contact interaction of a plate locally supported by a beam, under the influence of external lateral load and external additive color noise (pink, red, white) was constructed. Also described design is in a stationary temperature field. For the plate, the Kirchhoff kinematic model was adopted; for the beam, Euler - Bernoulli, the physical nonlinearity is taken into account according to the theory of small elastic-plastic deformations. The temperature field is taken into account according to the Duhamel - Neumann theory, and there are no restrictions on the temperature distribution over the plate thickness and the height of the beam. The temperature field is determined from the solution of the three-dimensional (plate) and two-dimensional (beam) heat conduction equations. The theory of B.Ya. Cantor. The heat conduction equations are solved by the finite difference method of the second and fourth order of accuracy. The system of differential equations is reduced to the Cauchy problem by the Bubnov - Galerkin methods in higher approximations and finite differences in spatial variables. Next, the Cauchy problem is solved by the fourth-order Runge - Kutta method and the Newmark method. At each time step, the iterative procedure of I. Birger was applied. The results of a numerical experiment are given. To analyze the results, the methods of nonlinear dynamics were used (construction of signals, phase portraits, Poincare sections, Fourier power spectra and Morlet wavelet spectra, analysis of the sign of Lyapunov indices by three methods: Wolf, Kantz, Rosenstein). The effect of color noise on the contact interaction between the plate and the beam has been studied. It has been established that red additive noise has the most significant effect on the oscillation pattern of the lamellar-beam structure compared to pink and white noise.

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Quarteroni, Alfio. "Fourier Spectral Methods for Pseudoparabolic Equations." SIAM Journal on Numerical Analysis 24, no. 2 (April 1987): 323–35. http://dx.doi.org/10.1137/0724024.

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Skopina, M. A. "Local Convergence of Fourier Series with Respect to Periodized Wavelets." Journal of Approximation Theory 94, no. 2 (August 1998): 191–202. http://dx.doi.org/10.1006/jath.1998.3191.

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Tadmor, Eitan. "The Exponential Accuracy of Fourier and Chebyshev Differencing Methods." SIAM Journal on Numerical Analysis 23, no. 1 (February 1986): 1–10. http://dx.doi.org/10.1137/0723001.

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E, Weinan. "Convergence of Fourier Methods for the Navier–Stokes Equations." SIAM Journal on Numerical Analysis 30, no. 3 (June 1993): 650–74. http://dx.doi.org/10.1137/0730032.

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Akimov, Pavel A., Alexander M. Belostotsky, Taymuraz B. Kaytukov, Marina L. Mozgaleva, and Mojtaba Aslami. "ABOUT SEVERAL NUMERICAL AND SEMIANALYTICAL METHODS OF LOCAL STRUCTURAL ANALYSIS." International Journal for Computational Civil and Structural Engineering 14, no. 4 (December 21, 2018): 59–69. http://dx.doi.org/10.22337/2587-9618-2018-14-4-59-69.

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Numerical or semianalytical solution of problems of structural mechanics with immense number of unknowns is time-consuming process. High-accuracy solution at all points of the model is not required normally, it is necessary to find only the most accurate solution in some pre-known domains. The choice of these domains is a priori data with respect to the structure being modelled. Designers usually choose domains with the so-called edge effect (with the risk of significant stresses that could lead to destruction of structures) and regions which are subject to specific operational requirements. Stress-strain state in such domains is important. Wavelets provide effective and popular tool for local structural analysis. Operational and variational formulations of problems of structural mechanics with the use of method of extended domain are presented. After discretization and obtaining of governing equations, problems are transformed to a multilevel space by multilevel wavelet transform. Discrete wavelet basis is used and corresponding direct and inverse algorithms of transformations are performed. Due to special algorithms of averaging, reduction of the problems is provided. Wavelet-based methods allows reducing the size of the problems and obtaining accurate results in selected domains simultaneously. These are rather efficient methods for evaluation of local phenomenon in structures.

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Osborne, Alfred R. "Nonlinear Fourier Analysis: Rogue Waves in Numerical Modeling and Data Analysis." Journal of Marine Science and Engineering 8, no. 12 (December 9, 2020): 1005. http://dx.doi.org/10.3390/jmse8121005.

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Nonlinear Fourier Analysis (NLFA) as developed herein begins with the nonlinear Schrödinger equation in two-space and one-time dimensions (the 2+1 NLS equation). The integrability of the simpler nonlinear Schrödinger equation in one-space and one-time dimensions (1+1 NLS) is an important tool in this analysis. We demonstrate that small-time asymptotic spectral solutions of the 2+1 NLS equation can be constructed as the nonlinear superposition of many 1+1 NLS equations, each corresponding to a particular radial direction in the directional spectrum of the waves. The radial 1+1 NLS equations interact nonlinearly with one another. We determine practical asymptotic spectral solutions of the 2+1 NLS equation that are formed from the ratio of two phase-lagged Riemann theta functions: Surprisingly this construction can be written in terms of generalizations of periodic Fourier series called (1) quasiperiodic Fourier (QPF) series and (2) almost periodic Fourier (APF) series (with appropriate limits in space and time). To simplify the discourse with regard to QPF and APF Fourier series, we call them NLF series herein. The NLF series are the solutions or approximate solutions of the nonlinear dynamics of water waves. These series are indistinguishable in many ways from the linear superposition of sine waves introduced theoretically by Paley and Weiner, and exploited experimentally and theoretically by Barber and Longuet-Higgins assuming random phases. Generally speaking NLF series do not have random phases, but instead employ phase locking. We construct the asymptotic NLF series spectral solutions of 2+1 NLS as a linear superposition of sine waves, with particular amplitudes, frequencies and phases. Because of the phase locking the NLF basis functions consist not only of sine waves, but also of Stokes waves, breather trains, and superbreathers, all of which undergo complex pair-wise nonlinear interactions. Breather trains are known to be associated with rogue waves in solutions of nonlinear wave equations. It is remarkable that complex nonlinear dynamics can be represented as a generalized, linear superposition of sine waves. NLF series that solve nonlinear wave equations offer a significant advantage over traditional periodic Fourier series. We show how NLFA can be applied to numerically model nonlinear wave motions and to analyze experimentally measured wave data. Applications to the analysis of SINTEF wave tank data, measurements from Currituck Sound, North Carolina and to shipboard radar data taken by the U. S. Navy are discussed. The ubiquitous presence of coherent breather packets in many data sets, as analyzed by NLFA methods, has recently led to the discovery of breather turbulence in the ocean: In this case, nonlinear Fourier components occur as strongly interacting, phase locked, densely packed breather modes, in contrast to the previously held incorrect belief that ocean waves are weakly interacting sine waves.

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Sathar, Mohammad Hasan Abdul, Ahmad Fadly Nurullah Rasedee, Anvarjon A. Ahmedov, and Norfifah Bachok. "Numerical Solution of Nonlinear Fredholm and Volterra Integrals by Newton–Kantorovich and Haar Wavelets Methods." Symmetry 12, no. 12 (December 9, 2020): 2034. http://dx.doi.org/10.3390/sym12122034.

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The current study proposes a numerical method which solves nonlinear Fredholm and Volterra integral of the second kind using a combination of a Newton–Kantorovich and Haar wavelet. Error analysis for the Holder classes was established to ensure convergence of the Haar wavelets. Numerical examples will illustrate the accuracy and simplicity of Newton–Kantorovich and Haar wavelets. Numerical results of the current method were then compared with previous well-established methods.

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Le Roux, Daniel Y., Christopher Eldred, and Mark A. Taylor. "Fourier Analyses of High-Order Continuous and Discontinuous Galerkin Methods." SIAM Journal on Numerical Analysis 58, no. 3 (January 2020): 1845–66. http://dx.doi.org/10.1137/19m1289595.

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Abd-Elhameed, W. M., and Y. H. Youssri. "New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/626275.

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We introduce two new spectral wavelets algorithms for solving linear and nonlinear fractional-order Riccati differential equation. The suggested algorithms are basically based on employing the ultraspherical wavelets together with the tau and collocation spectral methods. The main idea for obtaining spectral numerical solutions depends on converting the differential equation with its initial condition into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. For the sake of illustrating the efficiency and the applicability of our algorithms, some numerical examples including comparisons with some algorithms in the literature are presented.

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Chan, Tony F., and Tom Kerkhoven. "Fourier Methods with Extended Stability Intervals for the Korteweg–de Vries Equation." SIAM Journal on Numerical Analysis 22, no. 3 (June 1985): 441–54. http://dx.doi.org/10.1137/0722026.

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Awrejcewicz, J., I. V. Papkova, E. U. Krylova, and V. A. Krysko. "Wavelet-Based Analysis of the Regular and Chaotic Dynamics of Rectangular Flexible Plates Subjected to Shear-Harmonic Loading." Shock and Vibration 19, no. 5 (2012): 979–94. http://dx.doi.org/10.1155/2012/658298.

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We investigate non-linear dynamics of flexible rectangular plates subjected to external shear harmonic load action. We show that an application of the classical and widely used Fourier analysis does not allow to obtain real picture of the frequency vibration characteristics in each time instant. On the other hand, we show that application of the wavelets approach allows to follow frequency time evolutions. Our numerical results indicate that vibrations in different plate points occur with the same frequencies set although their power is different. Hence, the vibration characteristics can be represented by one arbitrary taken plate point. Furthermore, using wavelets scenarios of transitions from regular to chaotic dynamics are illustrated and discussed including two novel scenarios not reported so far in the existing literature.

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Hamiga, Władysław Marek, and Wojciech Bronisław Ciesielka. "Aeroaocustic Numerical Analysis of the Vehicle Model." Applied Sciences 10, no. 24 (December 18, 2020): 9066. http://dx.doi.org/10.3390/app10249066.

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Understanding local phenomena connected with airflow around road vehicles allows to reduce the negative impact of transportation on the environment. This paper presents using numerical tools for Computational Fluid Dynamics (CFD) and Computational AeroAcoustic (CAA) calculation. As a model for simulation, simplified car geometry is used, which is known in the research community as an Ahmed body. The study is divided into two main parts: a validation process and a CAA analysis using the Ffowcs Williams–Hawkings (FW-H) analogy. Research is performed using k−ω Shear Stress Transport (SST) and the Large Eddy Simulation (LES) turbulence model. To compare results with other authors’ studies, three different comparison criteria are introduced: a drag coefficient for different velocities, characteristic flow structure, and velocity profiles. The CAA analysis is presented using colormaps and Fast Fourier Transformation (FFT). The methods used in this work allow visualizing the acoustic field around reference geometry and determining the frequency range for which the A-weighted sound pressure level is the highest.

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Carles, Rémi. "On Fourier Time-Splitting Methods for Nonlinear Schrödinger Equations in the Semiclassical Limit." SIAM Journal on Numerical Analysis 51, no. 6 (January 2013): 3232–58. http://dx.doi.org/10.1137/120892416.

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Vigdergauz, S. "Plane elastostatic stress analysis in complex variables: A wavelet processing perspective." Mathematics and Mechanics of Solids 22, no. 2 (August 5, 2016): 176–90. http://dx.doi.org/10.1177/1081286515577689.

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The well-known Kolosov–Muskhelishvili (KM) representation of the Airy function for 2D stress analysis in complex variable terms is enhanced by combining it with Walsh wavelets decomposition. It allows us to perform general analytical derivations up to the maximum extent possible which, in turn, provides a basis for developing a new stress computation algorithm readily incorporated into the routine single scale KM scheme. The mathematical treatment of the wavelet application is supported by a number of examples where non-trivial closed-form solutions are known and serve as a benchmark for numerical simulations. The comparison shows that the proposed framework has better performance than the conventional Fourier transform, especially when it comes to non-smooth stress distributions.

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Pandit, Sapna, Manoj Kumar, R. N. Mohapatra, and Ali Saleh Alshomrani. "Shock waves analysis of planar and non planar nonlinear Burgers’ equation using Scale-2 Haar wavelets." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 8 (August 7, 2017): 1814–50. http://dx.doi.org/10.1108/hff-05-2016-0188.

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Purpose This paper aims to find the numerical solution of planar and non-planar Burgers’ equation and analysis of the shock behave. Design/methodology/approach First, the authors discritize the time-dependent term using Crank–Nicholson finite difference approximation and use quasilinearization to linearize the nonlinear term then apply Scale-2 Haar wavelets for space integration. After applying this scheme on partial differential, the equation transforms into a system of algebraic equation. Then, the system of equation is solved using Gauss elimination method. Findings Present method is the extension of the method (Jiwari, 2012). The numerical solutions using Scale-2 Haar wavelets prove that the proposed method is reliable for planar and non-planar nonlinear Burgers’ equation and yields results better than other methods and compatible with the exact solutions. Originality/value The numerical results for non-planar Burgers’ equation are very sparse. In the present paper, the authors identify where the shock wave and discontinuity occur in planar and non-planar Burgers’' equation.

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Santos, Diogo, Tiago Abreu, Paulo A. Silva, and Paulo Baptista. "Estimation of Coastal Bathymetry Using Wavelets." Journal of Marine Science and Engineering 8, no. 10 (October 1, 2020): 772. http://dx.doi.org/10.3390/jmse8100772.

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When waves propagate in coastal areas at depths lower than one half the wavelength, they exhibit a different signature at the sea surface and the observed wavelength pattern enables inferring bathymetries. Commonly, a spectral analysis using the fast Fourier transform (FFT) is employed to derive wavelength and wave direction of swell waves, in nearshore regions. Nevertheless, it is recognized that this method presents limitations, particularly regarding depth inversion limits that do not allow obtaining bathymetric data close to the shoreline. This work explores a wavelet spectral analysis to obtain bathymetric data. This new imaging methodology is applied over different seafloors with 2D and 3D features such as longshore bars or headlands. The synthetic images of the water surface are generated from a numerical Boussinesq-type model that simulates the propagation of both regular and irregular waves. The spectral analysis is carried to estimate the water depths, which are validated with the model’s bathymetry. Wavelet image processing methodology shows very positive results, revealing the capabilities of this new methodology to map shallow marine environments.

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Toutounian, F., Emran Tohidi, and A. Kilicman. "Fourier Operational Matrices of Differentiation and Transmission: Introduction and Applications." Abstract and Applied Analysis 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/198926.

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This paper introduces Fourier operational matrices of differentiation and transmission for solving high-order linear differential and difference equations with constant coefficients. Moreover, we extend our methods for generalized Pantograph equations with variable coefficients by using Legendre Gauss collocation nodes. In the case of numerical solution of Pantograph equation, an error problem is constructed by means of the residual function and this error problem is solved by using the mentioned collocation scheme. When the exact solution of the problem is not known, the absolute errors can be computed approximately by the numerical solution of the error problem. The reliability and efficiency of the presented approaches are demonstrated by several numerical examples, and also the results are compared with different methods.

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Sheng, Changtao, Jie Shen, Tao Tang, Li-Lian Wang, and Huifang Yuan. "Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains." SIAM Journal on Numerical Analysis 58, no. 5 (January 2020): 2435–64. http://dx.doi.org/10.1137/19m128377x.

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Andersson, Robert, Léon F. van Heijkamp, Ignatz M. de Schepper, and Wim G. Bouwman. "Analysis of spin-echo small-angle neutron scattering measurements." Journal of Applied Crystallography 41, no. 5 (September 13, 2008): 868–85. http://dx.doi.org/10.1107/s0021889808026770.

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Spin-echo small-angle neutron scattering (SESANS) is, in contrast to conventional small-angle neutron scattering (SANS), a real-space technique. SESANS measures the projection of the density–density correlation function of a sample, rather than, as in SANS, its Fourier transform. This paper introduces a toolkit for interpretion and analysis of a SESANS measurement. Models that are used in SANS are discussed and translated into a SESANS formalism. These models can be used to analyse and fit the data obtained by SESANS. Dilute, concentrated, random, fractal and anisotropic density distributions are considered. Numerical methods used to calculate the projection from numerical data are presented, either by using Fourier transformation orviathe real-space pair correlation function.

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Artioli, E., and E. Viola. "Free Vibration Analysis of Spherical Caps Using a G.D.Q. Numerical Solution." Journal of Pressure Vessel Technology 128, no. 3 (July 6, 2005): 370–78. http://dx.doi.org/10.1115/1.2217970.

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In this paper we present the frequency evaluation of spherical shells by means of the generalized differential quadrature method (G.D.Q.M.), an effective numerical procedure which pertains to the class of generalized collocation methods. The shell theory used in this study is a first-order shear deformation theory with transverse shearing deformations and rotatory inertia included. The shell governing equations in terms of mid-surface displacements are obtained and, after expansion in partial Fourier series of the circumferential coordinate, solved with the G.D.Q.M. Several comparisons are made with available results, showing the reliability and modeling capability of the numerical scheme in argument.

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Zou, Guang-an, Bo Wang, and Mu Mu. "Stability Analysis of Numerical Methods for a 1.5-Layer Shallow-Water Ocean Model." Journal of Applied Mathematics 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/478054.

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A 1.5-layer reduced-gravity shallow-water ocean model in spherical coordinates is described and discretized in a staggered grid (standard Arakawa C-grid) with the forward-time central-space (FTCS) method and the Leap-frog finite difference scheme. The discrete Fourier analysis method combined with the Gershgorin circle theorem is used to study the stability of these two finite difference numerical models. A series of necessary conditions of selection criteria for the time-space step sizes and model parameters are obtained. It is showed that these stability conditions are more accurate than the Courant-Friedrichs-Lewy (CFL) condition and other two criterions (Blumberg and Mellor, 1987; Casulli, 1990, 1992). Numerical experiments are proposed to test our stability results, and numerical model that is designed is also used to simulate the ocean current.

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Kargar, Zeynab, and Habibollah Saeedi. "B-spline wavelet operational method for numerical solution of time-space fractional partial differential equations." International Journal of Wavelets, Multiresolution and Information Processing 15, no. 04 (April 19, 2017): 1750034. http://dx.doi.org/10.1142/s0219691317500345.

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In this paper, the linear B-spline scaling functions and wavelets operational matrix of fractional integration are derived. A new approach implementing the linear B-spline scaling functions and wavelets operational matrices combining with the spectral tau method is introduced for approximating the numerical solutions of time-space fractional partial differential equations with initial-boundary conditions. They are utilized to reduce the main problem to a system of algebraic equations. The uniform convergence analysis for the linear B-spline scaling functions and wavelets expansion and an efficient error estimation of the presented method are also introduced. Illustrative examples are given and numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Special attention is given to a comparison between the numerical results obtained by our new technique and those found by other known methods.

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Brannick, James, Xiaozhe Hu, Carmen Rodrigo, and Ludmil Zikatanov. "Local Fourier Analysis of Multigrid Methods with Polynomial Smoothers and Aggressive Coarsening." Numerical Mathematics: Theory, Methods and Applications 8, no. 1 (February 2015): 1–21. http://dx.doi.org/10.4208/nmtma.2015.w01si.

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We focus on the study of multigrid methods with aggressive coarsening and polynomial smoothers for the solution of the linear systems corresponding to finite difference/element discretizations of the Laplace equation. Using local Fourier analysis we determineautomaticallythe optimal values for the parameters involved in defining the polynomial smoothers and achieve fast convergence of cycles with aggressive coarsening. We also present numerical tests supporting the theoretical results and the heuristic ideas. The methods we introduce are highly parallelizable and efficient multigrid algorithms on structured and semi-structured grids in two and three spatial dimensions.

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SERIANI, GÉZA, and SAULO POMPONET OLIVEIRA. "DFT MODAL ANALYSIS OF SPECTRAL ELEMENT METHODS FOR ACOUSTIC WAVE PROPAGATION." Journal of Computational Acoustics 16, no. 04 (December 2008): 531–61. http://dx.doi.org/10.1142/s0218396x08003774.

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Spectral element methods are now widely used for wave propagation simulations. They are appreciated for their high order of accuracy, but are still used on a heuristic basis. In this work we present the numerical dispersion of spectral elements, which allows us to assess their error limits and to devise efficient numerical simulations, particularly for 2D and 3D problems. We propose a novel approach based on a discrete Fourier transform of both the probing plane waves and the discrete wave operators. The underlying dispersion relation is estimated by the Rayleigh quotients of the plane waves with respect to the discrete operator. Together with the Kronecker product properties, this approach delivers numerical dispersion estimates for 1D operators as well as for 2D and 3D operators, and is well suited for spectral element methods, which use nonequidistant collocation points such as Gauss–Lobatto–Chebyshev and Gauss–Lobatto–Legendre points. We illustrate this methodology with dispersion and anisotropy graphs for spectral elements with polynomial degrees from 4 to 12. These graphs confirm the rule of thumb that spectral element methods reach a safe level of accuracy at about four grid points per wavelength.

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AWREJCEWICZ, J., A. V. KRYSKO, and V. SOLDATOV. "ON THE WAVELET TRANSFORM APPLICATION TO A STUDY OF CHAOTIC VIBRATIONS OF THE INFINITE LENGTH FLEXIBLE PANELS DRIVEN LONGITUDINALLY." International Journal of Bifurcation and Chaos 19, no. 10 (October 2009): 3347–71. http://dx.doi.org/10.1142/s0218127409024803.

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Both classical Fourier analysis and continuous wavelets transformation are applied to study non-linear vibrations of infinitely long flexible panels subject to longitudinal sign-changeable external load actions. First the governing PDEs are derived and then the Bubnov–Galerkin method is applied to yield 2N first order ODEs. The further used Lyapunov exponent computation is described. Transition scenarios from regular to chaotic dynamics of the being investigated plate strip are analyzed using different wavelets, and their suitability and advantages/disadvantages to nonlinear dynamics monitoring and quantifying are illustrated and discussed. A few novel results devoted to the beam nonlinear dynamics behavior are reported. In addition, links between the largest Lyapunov exponent computation and the wavelet spectrum numerical estimation are also illustrated and discussed.

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Yang, Zhibo, Xuefeng Chen, Yumin He, Zhengjia He, and Jie Zhang. "The Analysis of Curved Beam Using B-Spline Wavelet on Interval Finite Element Method." Shock and Vibration 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/738162.

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A B-spline wavelet on interval (BSWI) finite element is developed for curved beams, and the static and free vibration behaviors of curved beam (arch) are investigated in this paper. Instead of the traditional polynomial interpolation, scaling functions at a certain scale have been adopted to form the shape functions and construct wavelet-based elements. Different from the process of the direct wavelet addition in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space by aid of the corresponding transformation matrix. Furthermore, compared with the commonly used Daubechies wavelet, BSWI has explicit expressions and excellent approximation properties, which guarantee satisfactory results. Numerical examples are performed to demonstrate the accuracy and efficiency with respect to previously published formulations for curved beams.

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Ucar, Yusuf, Nuri Murat Yagmurlu, and Orkun Tasbozan. "Numerical Solutions of the Modified Burgers’ Equation by Finite Difference Methods." Journal of Applied Mathematics, Statistics and Informatics 13, no. 1 (May 24, 2017): 19–30. http://dx.doi.org/10.1515/jamsi-2017-0002.

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Abstract In this study, a numerical solution of the modified Burgers’ equation is obtained by the finite difference methods. For the solution process, two linearization techniques have been applied to get over the non-linear term existing in the equation. Then, some comparisons have been made between the obtained results and those available in the literature. Furthermore, the error norms L2 and L∞ are computed and found to be sufficiently small and compatible with others in the literature. The stability analysis of the linearized finite difference equations obtained by two different linearization techniques has been separately conducted via Fourier stability analysis method.

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Pretzier, Georg. "A New Method for Numerical Abel-Inversion." Zeitschrift für Naturforschung A 46, no. 7 (July 1, 1991): 639–41. http://dx.doi.org/10.1515/zna-1991-0715.

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A new numerical solution of Abel-type integral equations is presented which is based on the principle of Fourier-analysis. The unknown radial distribution is expanded in a series of cosine-functions. the amplitudes of which are calculated by least-squares-fitting of the Abel-transformed series to the measured data. Advantages and applications of the new method are discussed, followed by a short comparison with methods, commonly used for Abel-inversion.

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Meng, Fanming, and Yuanpei Chen. "Analysis of elasto-hydrodynamic lubrication of journal bearing based on different numerical methods." Industrial Lubrication and Tribology 67, no. 5 (August 10, 2015): 486–97. http://dx.doi.org/10.1108/ilt-03-2015-0026.

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Purpose – The purpose of this study is to analyze and compare the tribological performances of journal bearings at different conditions through four numerical methods, which are based on the Boussinesq elastic half-space hypothesis or finite element theory. Design/methodology/approach – An elasto-hydrodynamic lubrication (EHL) model of journal bearings is established, with the oil film pressure obtained by the finite difference method, and the deformation of bearing calculated by four different numerical methods, i.e. the direct finite element method (DFEM), influence coefficient method (ICM), fast-Fourier transform method (FFTM) and direct Boussinesq method (DBM). The tribological performances of the journal bearings obtained with the four methods along with the computation efficiency of the methods are discussed. Findings – Under different operation conditions, the tribological performances with the finite element method-based methods (DFEM and ICM) agree with each other, and so do those with the Boussinesq-based methods (FFTM and DBM). Compared with the former two methods, the latter two overestimate the friction coefficient, film thickness and bearing deformation, but underestimate the film pressure, load-carrying capacity and friction force. The above discrepancies depend on the lubricant viscosity, the eccentricity ratio and rotational speed of the shaft and the length–diameter ratio of the bearing. Among the four methods, the FFTM has the best computation efficiency, followed by the DBM and the FEM-based methods. Originality/value – This study conducts detailed discussions of the numerical methods used in the EHL calculation of journal bearings and gives a helpful reference to analyses and designs of journal bearings.

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Ding, Hengfei, and Changpin Li. "Numerical Algorithms for the Fractional Diffusion-Wave Equation with Reaction Term." Abstract and Applied Analysis 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/493406.

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Two numerical algorithms are derived to compute the fractional diffusion-wave equation with a reaction term. Firstly, using the relations between Caputo and Riemann-Liouville derivatives, we get two equivalent forms of the original equation, where we approximate Riemann-Liouville derivative by a second-order difference scheme. Secondly, for second-order derivative in space dimension, we construct a fourth-order difference scheme in terms of the hyperbolic-trigonometric spline function. The stability analysis of the derived numerical methods is given by means of the fractional Fourier method. Finally, an illustrative example is presented to show that the numerical results are in good agreement with the theoretical analysis.

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Malmir, Iman. "A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems." Fractal and Fractional 3, no. 3 (September 3, 2019): 46. http://dx.doi.org/10.3390/fractalfract3030046.

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Fractional integration operational matrix of Chebyshev wavelets based on the Riemann–Liouville fractional integral operator is derived directly from Chebyshev wavelets for the first time. The formulation is accurate and can be applied for fractional orders or an integer order. Using the fractional integration operational matrix, new Chebyshev wavelet methods for finding solutions of linear-quadratic optimal control problems and analysis of linear fractional time-delay systems are presented. Different numerical examples are solved to show the accuracy and applicability of the new Chebyshev wavelet methods.

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Orel, Bojan, and Andrej Perne. "Chebyshev-Fourier Spectral Methods for Nonperiodic Boundary Value Problems." Journal of Applied Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/572694.

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A new class of spectral methods for solving two-point boundary value problems for linear ordinary differential equations is presented in the paper. Although these methods are based on trigonometric functions, they can be used for solving periodic as well as nonperiodic problems. Instead of using basis functions periodic on a given interval−1,1, we use functions periodic on a wider interval. The numerical solution of the given problem is sought in terms of the half-range Chebyshev-Fourier (HCF) series, a reorganization of the classical Fourier series using half-range Chebyshev polynomials of the first and second kind which were first introduced by Huybrechs (2010) and further analyzed by Orel and Perne (2012). The numerical solution is constructed as a HCF series via differentiation and multiplication matrices. Moreover, the construction of the method, error analysis, convergence results, and some numerical examples are presented in the paper. The decay of the maximal absolute error according to the truncation numberNfor the new class of Chebyshev-Fourier-collocation (CFC) methods is compared to the decay of the error for the standard class of Chebyshev-collocation (CC) methods.

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Polonsky, I. A., and L. M. Keer. "A Fast and Accurate Method for Numerical Analysis of Elastic Layered Contacts." Journal of Tribology 122, no. 1 (March 17, 1999): 30–35. http://dx.doi.org/10.1115/1.555323.

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Solution of contact problems for layered elastic solids generally requires the use of numerical methods. Recently, the fast Fourier transform (FFT) technique has been applied to such contacts. While very fast, FFT is strictly applicable only to periodic contact problems. When it is applied to essentially non-periodic contacts, an error is introduced in the numerical solution. A new method that overcomes the limitation of the “straightforward” FFT approach for solving non-periodic layered contact problems is introduced in the present article. A special correction procedure based on the multi-level multi-summation technique is used to compensate the FFT results for the periodicity error. The use of a robust iteration scheme based on the conjugate gradient method ensures that the new method is applicable to contact problems involving real rough surfaces. Numerical examples demonstrate that the new method is both accurate and fast. [S0742-4787(00)00501-4]

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Journal articles on the topic 'Numerical analysis – Numerical methods in Fourier analysis – Wavelets'

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