Fourier Series Research Articles

Cardiac Multi-Frequency Vibration Signal Sensor Module and Feature Extraction Method Based on Vibration Modeling.

Cardiovascular diseases pose a long-term risk to human health. This study focuses on the rich-spectrum mechanical vibrations generated during cardiac activity. By combining Fourier series theory, we propose a multi-frequency vibration model for the heart, decomposing cardiac vibration into frequency bands and establishing a systematic interpretation for detecting multi-frequency cardiac vibrations. Based on this, we develop a small multi-frequency vibration sensor module based on flexible polyvinylidene fluoride (PVDF) films, which is capable of synchronously collecting ultra-low-frequency seismocardiography (ULF-SCG), seismocardiography (SCG), and phonocardiography (PCG) signals with high sensitivity. Comparative experiments validate the sensor's performance and we further develop an algorithm framework for feature extraction based on 1D-CNN models, achieving continuous recognition of multiple vibration features. Testing shows that the recognition coefficient of determination (R2), mean absolute error (MAE), and root mean square error (RMSE) of the 8 features are 0.95, 2.18 ms, and 4.89 ms, respectively, with an average prediction speed of 60.18 us/point, meeting the re-quirements for online monitoring while ensuring accuracy in extracting multiple feature points. Finally, integrating the vibration model, sensor, and feature extraction algorithm, we propose a dynamic monitoring system for multi-frequency cardiac vibration, which can be applied to portable monitoring devices for daily dynamic cardiac monitoring, providing a new approach for the early diagnosis and prevention of cardiovascular diseases.

التنبؤ باستهلاك الكهرباء باستخدام منهج التحليل الطيفي في محافظة السليمانية

In the modern world, with the extensive use of electricity in homes and businesses, studying and forecasting electrical consumption is crucial for its management and distribution. This research aims to forecast the electricity consumption in Sulaimani province using spectral analysis on time series data. For this purpose, hourly data within the time range of July to September 2022 has been utilized, and a new model has been proposed, consisting of a sinusoidal explanatory variable and a nonlinear quadratic variable. In this model, the most significant periodic cycles present in the time series under study were identified by employing the periodogram and Fourier expansion method, capable of identifying hidden cycles in the data. Subsequently, each of these periodic components was added to the model based on the principle of parsimony. The proposed composite model includes a quadratic component and two harmonic components with 24 and 12-hour periods. This model demonstrates notable efficiency and accuracy as it outperforms the classical Autoregressive Moving Average (ARMA) model based on the MSE=0.0009 rate and MAE=23.819 when compared

Optimal with Respect to Accuracy Recovery of Some Classes Functions by Fourier Series

Introduction. Function approximation (approximation or restoration) is widely used in data analysis, model building, and forecasting. The goal of function approximation is to find the function that best approximates the original function. This can be useful when the original function is too complex to analyze or when a model needs to be simplified for more efficient computation or interpretation. Function approximation is an important tool in science, engineering, economics, and other fields where data analysis and modeling are required. It allows you to simplify complex functions, identify patterns in the behavior of the object of study, and predict the value of a function beyond the available data. The purpose of the paper is consider the problems of approximation of a function, which on some interval is given by its values in some set of nodal points and belongs to some class of functions by trigonometric Fourier series with a given accuracy and at fulfillment of given constraints on its execution time. The main attention is paid to obtaining estimates of computational complexity (implementation time) and solving the problem of function approximation by Fourier series with a given or maximum possible accuracy using efficient algorithms for solving optimization problems. Results. The general formulation of the problem of approximation of functions by Fourier series in accordance with the technology of solving problems of computational and applied mathematics with specified values of quality characteristics is presented. Estimates of the error of the proposed approximation algorithms using for the computation of Fourier coefficients the optimal in accuracy and close to them quadrature formulas for the computation of integrals from rapidly oscillating functions of the classes of Helder and Lipschitz with given fixed values in the nodes of a fixed grid are given. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of the specified classes are given. Estimates of computational complexity of the given algorithms are obtained, which allow us to set real constraints on the time of algorithm implementation with a given or maximum possible accuracy. Conclusions. A comprehensive analysis of the quality of the considered algorithms for the approximation of functions by Fourier series using the accuracy-optimal (or close to them) quadrature formulas for the computation of Fourier coefficients for the computation of integrals from rapidly oscillating functions is presented. The estimates of their main characteristics – accuracy and computational complexity – are obtained. Keywords: function approximation, Fourier series, Fourier series coefficients, approximation error, computational complexity.

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.

Elasto‐thermodiffusive interaction under void due to nonlocal stress theory

AbstractThe current investigation address a novel generalized elasto‐thermodiffusion model for a thermoelastic porous half‐space incorporating the nonlocal stress theory proposed by Eringen. Modeling of the problem is performed by adopting Moore‐Gibson‐Thompson (MGT) thermoelasticity theory defined in an integral form of a common derivative on a slipping interval, well known as the memory‐dependent derivative. The bounding plane of the medium is subjected to time‐dependent thermal and chemical shocks and there is no change in the volume fraction field. Laplace transform and the Fourier transform techniques have been adopted to represent the analytical solutions in the transformed domain. The distributions of the physical fields such as the temperature, stress, chemical potential, mass concentration and the volume fraction field were found in the real space‐time domain adopting suitable numerical scheme based on the Fourier series expansion. According to the discussion of the computational results and the respective graphical representations, the prominent role of different parameters such as the effect of nonlocality, effect of void and thermodiffusion is analyzed. Moreover, the superiority of a nonlinear kernel function compared to a linear form is also reported.

When correlations exceed system size: finite-size scaling in free boundary conditions above the upper critical dimension

We progress finite-size scaling in systems with free boundary conditions above their upper critical dimension, where in the thermodynamic limit critical scaling is described by mean-field theory. Recent works show that the correlation length is not bound by the system's physical size, a belief that long held sway. Instead, two scaling regimes can be observed — at the critical and pseudo-critical temperatures. We demonstrate that both are manifest for free boundaries. We use numerical simulations of the d = 5 Ising model to analyse the magnetization, susceptibility, magnetization Fourier modes and the partition function zeros. While some of the response functions hide the dual finite-size scaling, the precision enabled by the analysis of Lee–Yang zeros allows this be brought to the fore. In particular, finite-size scaling of leading zeros at the pseudo-critical point confirms recent predictions coming from correlations exceeding the system size. This paper is dedicated to Jaroslav Ilnytskyi on the occasion of his 60th birthday.

Nonadiabatic Conical Intersection Dynamics in the Local Diabatic Representation with Strang Splitting and Fourier Basis.

We develop and implement an exact conical intersection nonadiabatic wave packet dynamics method that combines the local diabatic representation, Strang splitting for the total molecular propagator, and discrete variable representation with uniform grids. By employing the local diabatic representation, this method captures all nonadiabatic effects, including nonadiabatic transitions, electronic coherences, and geometric phase. Moreover, it is free of singularities in the first and second derivative couplings and does not require the electronic wave function to be continuous with respect to the nuclear coordinates. We further show that in contrast to the adiabatic representation, the split-operator method can be directly applied to the full molecular propagator with the locally diabatic ansatz. The Fourier series, employed as the primitive nuclear basis functions, is universal and can be applied to all types of reactive coordinates. The combination of local diabatic representation, Strang splitting, and Fourier basis allows numerically exact modeling of conical intersection quantum dynamics directly with adiabatic electronic states that can be obtained from standard electronic structure computations.

A novel time-varying grey Fourier model for variable amplitude seasonal fluctuation sequences

PurposeA time-varying grey Fourier model (TVGFM(1,1,N)) is proposed for the simulation of variable amplitude seasonal fluctuation time series, as the performance of traditional grey models can't catch the time-varying trend well.Design/methodology/approachThe proposed model couples Fourier series and linear time-varying terms as the grey action, to describe the characteristics of variable amplitude and seasonality. The truncated Fourier order N is preselected from the alternative order set by Nyquist-Shannon sampling theorem and the principle of simplicity, then the optimal Fourier order is determined by hold-out method to improve the robustness of the proposed model. Initial value correction and the multiple transformation are also studied to improve the precision.FindingsThe new model has a broader applicability range as a result of the new grey action, attaining higher fitting and forecasting accuracy. The numerical experiment of a generated monthly time series indicates the proposed model can accurately fit the variable amplitude seasonal sequence, in which the mean absolute percentage error (MAPE) is only 0.01%, and the complex simulations based on Monte-Carlo method testify the validity of the proposed model. The results of monthly electricity consumption in China's primary industry, demonstrate the proposed model catches the time-varying trend and has good performances, where MAPEF and MAPET are below 5%. Moreover, the proposed TVGFM(1,1,N) model is superior to the benchmark models, grey polynomial model (GMP(1,1,N)), grey Fourier model (GFM(1,1,N)), seasonal grey model (SGM(1,1)), seasonal ARIMA model seasonal autoregressive integrated moving average model (SARIMA) and support vector regression (SVR).Originality/valueThe parameter estimates and forecasting of the new proposed TVGFM are studied, and the good fitting and forecasting accuracy of time-varying amplitude seasonal fluctuation series are testified by numerical simulations and a case study.

Couple-stress elasticity of intrinsic and extrinsic dislocations in three-dimensional multilayered materials

This work investigates the influence of couple stresses on intrinsic and extrinsic dislocations within three-dimensional heterogeneous multilayered structures. The materials consist of orthotropic and dissimilar layers with different elastic properties, incorporating the microstructural lengths from a special version of the anisotropic couple-stress elasticity. Double Fourier series expansions are used to describe extended displacements and tractions, considering rotation and couple-stress components and accounting for arbitrary displacement and rotation discontinuities associated with dislocation and disclination loops as well as periodic dislocation and disclination networks at semi-coherent interfaces. The dual-variable and position technique is used to recursively determine the field responses at any point in the multilayered laminate. The application examples focus on dislocation problems with couple stresses, revealing the effects of the core-spreading dislocation technique, free surfaces, anisotropic elasticity, and couple stresses on the field solutions. The analysis also uncovers pressure and von Mises stress amplitudes, non-zero rotation, and non-symmetric stress profiles due to couple stresses in copper/niobium bimaterials. Changes in the sign and magnitude of the Peach-Koehler forces are observed when couple stresses are introduced, which deviate from the classical theory of dislocations. The investigation finds a significant 29% reduction in energy per unit area at a semi-coherent interface between copper and niobium influenced by couple stresses. These results deepen our understanding of how couple stresses affect dislocation behavior in multilayered materials, and provide prospects for future discrete dislocation and disclination dynamics simulations with engineering applications in various nano- and micro-structured materials.

Delay-adaptive compensation for 3-D formation control of leader-actuated multi-agent systems

This paper focuses on the control of collective dynamics in large-scale multi-agent systems (MAS) operating in a 3-D space, with a specific emphasis on compensating for the influence of an unknown delay affecting the actuated leaders. The communication graph of the agents is defined on a mesh-grid 2-D cylindrical surface. We model the agents’ collective dynamics by a complex- and a real-valued reaction–advection–diffusion 2-D partial differential equations (PDEs) whose states represent the 3-D position coordinates of the agents. The leader agents on the boundary suffer unknown actuator delay due to the cumulative computation and information transmission time. We design a delay-adaptive controller for the 2-D PDE by using PDE backstepping combined with a Lyapunov functional method, where the latter is employed to design an update law that generates real-time estimates of the unknown delay. Capitalizing on our recent result on the control of 1-D parabolic PDEs with unknown input delay, we use Fourier series expansion to bridge the control of 1-D PDEs to that of 2-D PDEs. To design the update law for the 2-D system, a new target system is defined to establish the closed-loop local boundedness of the system trajectories in H2 norm and the regulation of the states to zero assuming a measurement of the spatially distributed plant’s state. We illustrate the performance of the delay-adaptive controller by numerical simulations.

MIXED PROBLEM FOR A NONLINEAR IMPULSIVE DIFFERENTIAL EQUATION OF PARABOLIC TYPE

In this paper, we consider a nonlinear impulsive parabolic type partial differential equation with nonlinear impulsive conditions. Dirichlet type boundary value conditions with respect to spatial variable is used, and eigenvalues and eigenfunctions of the spectral problem are founded. The Fourier method of the separation of variables is applied. A countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of the unknown function. A theorem on a unique solvability of the countable system of nonlinear functional equations is proved by the method of successive approximations. A criteria of uniqueness and existence of a solution for the nonlinear impulsive mixed problem is obtained. A solution of the mixed problem is derived in the form of the Fourier series. The absolute and uniform convergence of the Fourier series is proved.

Investigation of FGM membrane wrinkling using reduced models based on a multi-scale method

In this article, we show that the reduced model proposed in our recent article to study the wrinkling of homogeneous elastic membranes does not reproduce the wrinkling observed in the case of an FGM-type composite membrane. New reduced models are then proposed to investigate the wrinkling of membranes made from Functionally Graded Materials (FGM). We examine, by this way, the influence of material functional parameters on determining the critical load and post-critical behavior under different loads. These reduced models, created using a multi-scale method based on Fourier series with slowly varying coefficients, enrich the reference model. In addition, these reduced models also allow us to significantly decrease the computation time and the number of finite elements compared to the complete model. The robustness of these reduced models is demonstrated in the case of wrinkling of membranes modeled by very thin shells, which are discretized using DKT18-type finite elements as described in literature. The resulting nonlinear problems are solved using a High-Order Continuation Method (HOCM).

SIMULATION OF FRICTION IN HINGED SUPPORTS DURING NON-STATIONARY VIBRATIONS OF A TIMOSHENKO BEAM

When carrying out research and technological processes in modern science and technology, the suppression or reduction of vibrations is often required for better operation of sensitive equipment and precision instruments. This paper presents a study on the damping of vibrations of a beam caused by the action of a dynamic load. The motion of an isotropic elastic beam is described using the model of S. P. Timoshenko. The beam has hinged supports at the edges. Damping moments located and acting at the anchoring points of the beam are used to damp the vibrations. As the beam moves in the hinges, a frictional moment with linear viscous damping occurs, which is proportional to the damping coefficient and the angular velocity of the beam in the hinge. In order to estimate the vibration damping, the solution of the direct problem of modeling the motion of the Timoshenko beam at zero ini- tial conditions is considered. The beam motion is modeled by a system of differential equations according to the model of S. P. Timoshenko. The re- quired functions are set in the form of Fourier series. Laplace integral transformation is used. The peculiarity of solution of a direct problem is that at this stage the friction moments in joints are unknown and are defined by solving corresponding inverse problem using Volterra integral equation the- ory. An analytical and numerical solution of the practical problem is obtained. Numerical results have been obtained in the form of graphs of beam point displacements and friction moments for different damping coefficients. A comparative evaluation of vibration reduction for different damping parameters has been carried out. The research results compare well with the results obtained by other authors.

Estimation of Multiresponse Multipredictor Nonparametric Regression Model Using Mixed Estimator

In data analysis using a nonparametric regression approach, we are often faced with the problem of analyzing a set of data that has mixed patterns, namely, some of the data have a certain pattern and the rest of the data have a different pattern. To handle this kind of datum, we propose the use of a mixed estimator. In this study, we theoretically discuss a developed estimation method for a nonparametric regression model with two or more response variables and predictor variables, and there is a correlation between the response variables using a mixed estimator. The model is called the multiresponse multipredictor nonparametric regression (MMNR) model. The mixed estimator used for estimating the MMNR model is a mixed estimator of smoothing spline and Fourier series that is suitable for analyzing data with patterns that partly change at certain subintervals, and some others that follow a recurring pattern in a certain trend. Since in the MMNR model there is a correlation between responses, a symmetric weight matrix is involved in the estimation process of the MMNR model. To estimate the MMNR model, we apply the reproducing kernel Hilbert space (RKHS) method to penalized weighted least square (PWLS) optimization for estimating the regression function of the MMNR model, which consists of a smoothing spline component and a Fourier series component. A simulation study to show the performance of proposed method is also given. The obtained results are estimations of the smoothing spline component, Fourier series component, MMNR model, weight matrix, and consistency of estimated regression function. In conclusion, the estimation of the MMNR model using the mixed estimator is a combination of smoothing spline component and Fourier series component estimators. It depends on smoothing and oscillation parameters, and it has linear in observation and consistent properties.

A Stochastically Correlated Bivariate Square-Root Model

We introduce a novel stochastically correlated two-factor (i.e., bivariate) diffusion process under the square-root format, for which we analytically obtain the corresponding solutions for the conditional moment-generating functions and conditional characteristic functions. Such solutions recover verbatim those of the uncorrelated case which encompasses a range of processes similar to those produced by a bivariate square-root process in which entries are correlated in the standard way, that is, via a constant correlation coefficient. Note that closed-form solutions for the conditional characteristic and moment-generating functions are not available for the latter. We focus on the financial scenario of obtaining closed-form expressions for the exact price of a zero-coupon bond and Asian option prices using a Fourier cosine series method.

Notes on the Overconvergence of Fourier Series and Hadamard–Ostrowski Gaps

This paper examines the relationship between the overconvergence of Fourier series and the existence of Hadamard–Ostrowski gaps. Ostrowski’s result on the overconvergence of power series serves as a motivating factor for obtaining a natural generalization: the overconvergence of Fourier series. The connection between Hadamard–Ostrowski gaps and the overconvergence of Fourier series is clarified by applying the Hadamard three-circle theorem and the theory of orthogonal polynomials. Our main result is obtained by applying the Hadamard three-circle theorem.

Fourier Transform Thermoreflectance Method Under Front-Heat Front-Detect Configuration

The thermoreflectance method, which can measure thermal diffusivity in the cross-plane direction of thin films, mainly has two possible configurations; rear-heat front-detect (RF) and front-heat front-detect (FF) configuration. FF configuration is applicable to a wide variety of thin films including thin films deposited on opaque substrates, but this configuration has some problems in determination of the thermal diffusivity. One of the main problems is the effect of the penetration of pump beam and probe beam in thin film, which affects the initial temperature distribution near the sample’s surface after pulse heating. Several studies have tried to analyze the effect but there have been no practical analytical solutions which can solve this problem in FF configuration. In this paper, we propose a new analytical solution which considers the penetration of pump beam and probe beam into thin film, and by applying Fourier expansion analysis which we developed in a previous study to thermoreflectance signals, we have determined the thermal diffusivity of thin film in the thermoreflectance method under FF configuration. We measured platinum thin films with different thickness under both FF and RF configuration and obtained consistent thermal diffusivity values from both configurations.

Two-sided estimates of Lebesgue constants for spaces of functions of generalized bounded variation and Hölder spaces

A proof of two-sided estimates of Lebesgue constants for spaces Hω⋂BV(X), where Hω is the Hölder space and BV(X) is the space of functions of bounded variation constructed from the symmetric space of sequences X. Choosing different spaces of sequences as X, we obtained new criteria for the convergence of classical Fourier series.

Generalized Approach to Optimal Polylinearization for Smart Sensors and Internet of Things Devices

This study introduces an innovative numerical approach for polylinear approximation (polylinearization) of non-self-intersecting compact sensor characteristics (transfer functions) specified either pointwise or analytically. The goal is to partition the sensor characteristic optimally, i.e., to select the vertices of the approximating polyline (approximant) along with their positions, on the sensor characteristics so that the distance (i.e., the separation) between the approximant and the characteristic is rendered below a certain problem-specific tolerance. To achieve this goal, two alternative nonlinear optimization problems are solved, which differ in the adopted quantitative measure of the separation between the transfer function and the approximant. In the first problem, which relates to absolutely integrable sensor characteristics (their energy is not necessarily finite, but they can be represented in terms of convergent Fourier series), the polylinearization is constructed by the numerical minimization of the L1-metric (a distance-based separation measure), concerning the number of polyline vertices and their locations. In the second problem, which covers the quadratically integrable sensor characteristics (whose energy is finite, but they do not necessarily admit a representation in terms of convergent Fourier series), the polylinearization is constructed by numerically minimizing the L2-metric (area- or energy-based separation measure) for the same set of optimization variables—the locations and the number of polyline vertices.

Thermoelastic response of multilayer cylinders: The direct integration and single-solid approach

An approach for solving the plane nonaxisymmetric thermoelasticity problems in multilayer cylinders is suggested from the single-solid standpoint, which implies considering a multilayer structure as a single unity with a step-wise variation of material properties. The direct integration method was used to the advantage of our approach by applying toward the compatibility equation in terms of strains instead of the one in terms of stresses. As a result, the governing equation is derived without involving generalized derivatives by the radial coordinate and then reduced to an integral equation of the second kind. Using the resolvent-kernel for solving the latter equation, the thermal stresses are computed as the Fourier series whose constituents are explicitly determined by the corresponding constituents of the thermal loading.