Fourier Series Research Articles

Investigation of the performance of oil-free linear compressor with magnetic resonance spring for pulse tube cryocooler

This study proposes a new type of oil-free linear compressor with a magnetic resonance spring, which uses a magnetic resonance spring to provide restoring force and adopts gas-lubricated bearings to provide support force for the piston. The frequency characteristics of the compressor, including magnetic spring stiffness, gas spring stiffness, and resonance frequency, are studied through experiments and theoretical analysis. The magnetic spring stiffness is 31403.31 N/m of the design compressor. Hooke’s law and Fourier decomposition are employed for the gas spring stiffness measuring the compression pressure gauged. Meanwhile, the gas spring stiffness is also obtained from the amplitude-frequency characteristic of the compressor by the back stepping technique with an electric parameter frequency scan. The experimental results demonstrate that the three measurement methods have good consistency in measuring the gas spring stiffness. The equivalent gas spring stiffness measured using the three methods and theoretical model are 72600.92 N/m, 71275.84 N/m, 71967.15 N/m, and 71929.25 N/m at 3 MPa charged pressure measured respectively. In addition, the refrigeration performance of the pulse tube cryocooler driven by the compressor is tested and the lowest temperature obtained in the cold end is 46.3 K under different charge pressure. Furthermore, an optimal frequency exists that enhances the refrigeration performance of the pulse tube cryocooler, and this optimal frequency remains constant regardless of changes in the charge pressure.

Mathematical Modeling and Applications of Differential Equations in Science and Engineering

Analytically, differential equations are a means of developing relationships between functions that define rates of changes and these rates in terms of the underlying function. They are crucial in modeling diverse, evolutive systems within various domains, to list but a few: physics, engineering, biology, and economics. The fundamentals of differential equations are categorized mainly into Two types of Differential Equations are Ordinary Differential Equations and Partial Differential Equations. Most of the differential equations are solved analytically and numerically techniques such as separation of variables, Fourier series and finite difference techniques. In situations, where exact solutions are tough to come across, computational tools like MATLAB, Maple and the Python libraries help towards approximations. Because they help to explain how systems behave, as well as predicting and improving them, differential equations are an essential tool for expanding the body of scientific and engineering knowledge.

Bayesian Inference for Estimating Heat Sources Through Temperature Assimilation

Abstract This paper utilizes a Bayesian inference framework to address the two-dimensional (2D) steady-state heat conduction problem, focusing on the estimation of unknown distributed heat sources in a thermally conducting medium with uniform conductivity. The goal is to infer the locations, strength, and shape of heaters by assimilating temperature data in Euclidean space, employing a Fourier series to represent each heater's shape. The Markov Chain Monte Carlo (MCMC) method, incorporating the random-walk Metropolis–Hasting (MH) algorithm and parallel tempering, is utilized for posterior distribution exploration in both unbounded and wall-bounded domains. It is found that multiple solutions arise in cases where the number of temperature sensors is less than the number of unknown states. Moreover, smaller heaters introduce greater uncertainty in estimated strength. To address the challenge of estimating the heater's strength and shape simultaneously due to their strong correlation, our method incorporates sharp priors on one to ensure accurate and feasible solutions of the other. The diffusive nature of heat conduction smooths out any deformations in the temperature contours, especially in the presence of multiple heaters positioned near each other, impacting convergence. In wall-bounded domains with Neumann boundary conditions, the inference of heater parameters tends to be more accurate than in unbounded domains.

One new family of smooth semi-supervised support vector classifier based on Fourier series approximation technique

One new family of smooth semi-supervised support vector classifier based on Fourier series approximation technique

A Thermal Impedance Model for IGBT Modules Considering the Nonlinear Thermal Characteristics of Chips and Ceramic Materials

The traditional method of calculating junction temperature does not consider the dependence of a material’s thermal conductivity on temperature, in which the thermal conductivity changes with temperature. However, with an increase in junction temperature, the temperature sensitivity (TS) will have a more significant impact on the actual temperature of chips. This study established an improved IGBT equivalent thermal impedance model that considers the nonlinear characteristics of the TS of chips and ceramic materials. The Fourier series analysis method was used to obtain the heat flux density curve, and then the heat diffusion angles of each layer were solved. Moreover, iterations were performed until the thermal conductivity and temperature of the chip and ceramic layers matched the nonlinear characteristics of the TS. When the power loss was less than 200 W, the maximum error of the junction temperature calculated by the proposed method considering TS was 3%, while the maximum error of the method without considering TS was 9.5%. Compared with the finite element simulation, the proposed method has a faster solving speed and high accuracy. The proposed method only requires the input material parameters, size parameters, and boundary conditions to solve the junction temperature, which has strong practicality and high accuracy.

Waves of Islamic Radicalism

Islamic radicalism has become a constant disturbing issue in world politics. Since the early 1980s, the dormant madness of Islamic radicalism has exploded in vicious waves of terrorist attacks around the world. Previous studies, though they contributed enormously to understanding this issue, fell short in analyzing the process that led to the dominance of Islamic radicalism and the damping of Islamic moderateness. This article (a) models the rise and fall of Islamic radical/ moderate waves utilizing Fourier series and the damping notion of waves (2) Identify and test empirically the factors that advance/hamper radicalism using time-series and panel data from 1980 to 2015 across six Islamic countries: Algeria, Bangladesh, Egypt, Jordan, Morocco, and Turkey.

APPLIED MIXED KERNEL AND FOURIER SERIES MODELLING IN NONPARAMETRIC REGRESSION

There are three nonparametric regression approaches, namely, parametric, nonparametric and semi-parametric regression. Nonparametric regression allows the response variable to follow a different curve from one predictor variable to another. In paired data, the components of predictor variable and the response variable are assumed to follow unknown data patterns, so that they can be approximated by a kernel based regression model and Fourier series. The base components are approximated by kernel functions and Fourier series functions. Error is assumed to be normally distributed with zero mean and constant variance. The aim of this research is to make a kernel-based nonparametric regression model and Fourier series on poverty data in the Province of Bali. The research phase method begins with the introduction of the kernel based nonparametric regression model and the Fourier series. The next step is to study the estimation of the regression curve. The function estimation results are highly dependent on the bandwidth, smoothing and oscillation parameters. The resulting model gives a value of R2=0.6278, means that the used variables can explain the model by 62.78 percent. From the obtained modeling re-sults, the Open Unemployment Rate has a positive effect on the percentage of poor people in Bali. Keywords: Nonparametric Regression; Modeling; Kernel; Fourier Series.

Rate of convergence of certain Fourier series of functions of generalized bounded variation

In this paper, we discuss the rate of convergence of the rational Fourier series and conjugate rational Fourier series of functions of generalized bounded variation. In particular, well- known Wiener’s and Siddiqi’s theorems for functions of p-bounded variation are proved in more general complete rational orthogonal system. Also some results are obtained for a class of functions wider than the class of functions of bounded variation and of {n\alpha }-bounded variation.

An analytical method for the vibration characteristics of double panel structures with uniform elastic boundary supports

ABSTRACT In this paper, the methods for solving the free vibration and forced vibration of double panel structures under arbitrary boundaries are studied. Based on the Mindlin plate theory, the natural boundary between plates and plates is constrained by artificial springs, and the theoretical model of the structure is derived. The displacement function of in-plane vibration and bending vibration is expressed by a modified Fourier series expansion. The effectiveness and reliability of this method are verified by a series of numerical calculations and comparison with those of the finite element method. Through the parameter analysis, it can be seen that the vibration isolation effect can be increased by properly increasing the coupling height or making the connecting plate closer to the boundary of the bottom plate. The method presented in this paper can be used in vibration absorption and structural design of such structures.

Solving Partial Differential Equations Using Fourier Series

This research paper aims to study the methods of solving partial differential equations using Fourier series. Partial differential equations are a powerful and practical tool in mathematics and physics to describe many phenomena and processes that involve continuous change at the level of equations and functions. The basic concepts related to the Fourier series and how to represent functions using it in the field of mathematics will be reviewed. Application examples of partial differential equations, such as waves, heat and diffusion, and how to use the Fourier series to solve them will also be presented. The steps of solving the partial differential equation using the Fourier series will be explained. The importance of Fourier series in the theory of partial differential equations is that periodic functions f(x) defined on (-∞,∞) or functions defined on a finite interval can be represented by an infinite interval of sines and cosines. In this research, the solution of the partial differential equation using the Fourier series was studied with the solution of an example.

The weak-type Carleson theorem via wave packet estimates

We prove that the weak- L p L^{p} norms, and in fact the sparse ( p , 1 ) (p,1) -norms, of the Carleson maximal partial Fourier sum operator are ≲ ( p − 1 ) − 1 \lesssim (p-1)^{-1} as p → 1 + p\to 1^+ . This is an improvement on the Carleson-Hunt theorem, where the same upper bound on the growth order is obtained for the restricted weak- L p L^p type norm, and which was the strongest quantitative bound prior to our result. Furthermore, our sparse ( p , 1 ) (p,1) -norms bound imply new and stronger results at the endpoint p = 1 p=1 . In particular, we obtain that the Fourier series of functions from the weighted Arias de Reyna space Q A ∞ ( w ) \mathrm {QA}_{\infty }(w) , which contains the weighted Antonov space L log ⁡ L log ⁡ log ⁡ log ⁡ L ( T ; w ) L\log L\log \log \log L(\mathbb T; w) , converge almost everywhere whenever w ∈ A 1 w\in A_1 . This is an extension of the results of Antonov [Proceedings of the XXWorkshop on Function Theory (Moscow, 1995), 1996, pp. 187–196] and Arias De Reyna, where w w must be Lebesgue measure. The backbone of our treatment is a new, sharply quantified near- L 1 L^1 Carleson embedding theorem for the modulation-invariant wave packet transform. The proof of the Carleson embedding relies on a newly developed smooth multi-frequency decomposition which, near the endpoint p = 1 p=1 , outperforms the abstract Hilbert space approach of past works, including the seminal one by Nazarov, Oberlin and Thiele [Math. Res. Lett. 17 (2010), pp. 529–545]. As a further example of application, we obtain a quantified version of the family of sparse bounds for the bilinear Hilbert transforms due to Culiuc, Ou and the first author.

Approximation of Functions by n-Euler Product Means of Fourier Series and its Conjugate

Approximation of Functions by n-Euler Product Means of Fourier Series and its Conjugate

Martingale Hardy Orlicz–Lorentz–Karamata spaces and applications in Fourier analysis

Abstract We summarize some results as well as we prove some new results about the Orlicz–Lorentz–Karamata spaces and martingale Hardy Orlicz–Lorentz–Karamata spaces. More precisely, Doob's maximal inequality for submartingales and Burkholder–Davis–Gundy inequality are presented. We also show some fundamental martingale inequalities and modular inequalities. Additionally, based on atomic decompositions, duality theorems and fractional integral operators are discussed. As applications in Fourier analysis, we consider the Walsh–Fourier series on Orlicz–Lorentz–Karamata spaces. The dyadic maximal operators on martingale Hardy Orlicz–Lorentz–Karamata spaces are presented. The boundedness of maximal Fejér operator is proved, which further implies some convergence results of the Fejér means.

On an Exact Convergence of Quasi-Periodic Interpolations for the Polyharmonic–Neumann Eigenfunctions

Fourier expansions employing polyharmonic–Neumann eigenfunctions have demonstrated improved convergence over those using the classical trigonometric system, due to the rapid decay of their Fourier coefficients. Building on this insight, we investigate interpolations on a finite interval that are exact for polyharmonic–Neumann eigenfunctions and exhibit similar benefits. Furthermore, we enhance the convergence of these interpolations by incorporating the concept of quasi-periodicity, wherein the basis functions are periodic over a slightly extended interval. We demonstrate that those interpolations achieve significantly better convergence rates away from the endpoints of the approximation interval and offer increased accuracy over the entire interval. We establish these properties for a specific case of polyharmonic–Neumann eigenfunctions known as the modified Fourier system. For other basis functions, we provide supporting evidence through numerical experiments. While the latter methods display superior convergence rates, we demonstrate that interpolations using the modified Fourier basis offer distinct advantages. Firstly, they permit explicit representations via the inverses of certain Vandermonde matrices, whereas other interpolation methods require approximate computations of the eigenvalues and eigenfunctions involved. Secondly, these matrix inverses can be efficiently computed for numerical applications. Thirdly, the introduction of quasi-periodicity improves the convergence rates, making them comparable to those of other interpolation techniques.

Generating functions of Bernstein polynomials‎: ‎Fourier series expansion and applications

Generating functions of Bernstein polynomials‎: ‎Fourier series expansion and applications

Analytical prediction of electromagnetic performance for surface-embedded permanent magnet in-wheel machines considering iron’s nonlinearity

Accurate magnetic field calculation is the premise of electromagnetic performance prediction. Conventional subdomain (SD) techniques assume that the iron’s relative permeability is infinite, leading to falsely overestimated flux density. We propose an accurate magnetic field analytical model for permanent magnet (PM) in-wheel machines considering iron’s magnetization nonlinearity and saturation. Specifically, according to the excitation source and topology, the entire solution domain of the machine is divided into sub-regions such as stator slots/teeth, stator slot-openings/tooth-tips, air-gap, and rotor slots/teeth, etc. Poisson’s or Laplace’s magnetic vector potential (MVP) equations are solved using Maxwell’s electromagnetic theory and complex Fourier series methods in each sub-region. Specifically, in our approach, The Cauchy product theorem addresses the discontinuous magnetic permeability change at the slot and tooth interface. The machine’s magnetic saturation effect is considered by combining the actual magnetization characteristics of iron with an iterative algorithm. The general solution for the MVP is solved using the boundary conditions between adjacent subregions. Subsequently, electromagnetic properties such as air-gap flux density, back electromotive force (EMF), and electromagnetic torque are obtained. The accuracy of the analytical model is verified by finite element analysis (FEA) and prototype tests, which proved that the proposed analytical model can consider the iron’s nonlinearity and the magnetic saturation. In addition, the inaccurate overestimation of electromagnetic torque and air-gap magnetic flux density by the conventional SD techniques has also been proven.

Noncontace Monitoring of Respiration and Heartbeat Based on Two-Wave Model Using a Millimeter-Wave MIMO FM-CW Radar

This paper deals with the non-contact measurement of heartbeat and respiration using a millimeter-wave multiple-input–multiple-output (MIMO) frequency-modulated continuous-wave (FM-CW) radar. Monitoring heartbeat and respiration is useful for detecting cardiac diseases and understanding stress levels. Contact sensors are not suitable for these sorts of long-term measurements due to the discomfort and skin irritation they cause. Therefore, the use of non-contact sensors, such as radars, is desirable. In this study, we obtained heartbeat and respiration information from phase data measured using a millimeter-wave MIMO FM-CW radar. We propose a two-wave model based on a Fourier series expansion and extract respiration and heartbeat information as a minimization problem. This model makes it possible to produce respiration and heartbeat waveforms. The produced heartbeat waveform can be used for estimating the interbeat interval (IBI). Experiments were conducted to confirm the usefulness of the proposed method. Moreover, the estimated results were compared with the contact sensor’s results. The results for both types of sensors were in good agreement.

Finite Tube Method for buckling analysis of tubular members using Fourier-approximation for the displacements

In this paper an efficient numerical method for the static analysis of cylindrical tubes is introduced. The method is designed for the linear buckling analysis of wind turbine support towers which are, most typically, built up from conical and/or cylindrical cans. Accordingly, the developed method uses cylindrical tube segments as elementary building blocks, along with specialized shape functions, and is named the Finite Tube Method. Within a tube segment the displacements are approximated by two-dimensional Fourier series. The curved nature of the surface is directly considered in the kinematic equations. The segments are joined and/or supported to the ground by constraint equations or by elastic links. In the current implementation internal stresses are determined in a simplified way: the circumferential stress distributions are calculated from the internal forces/moment by classic strength of material formulae, while the longitudinal distribution within each segment is quadratic. The considered internal forces/moments are: normal force, shear force, bending moment, and torsional moment. The internal forces can be arbitrarily combined. In the paper the underlying derivations are briefly summarized, then the method is demonstrated and validated by numerical examples, comparing the results to analytical and alternative numerical solutions. The authors are actively developing the method and will provide future work on utilization of the method for buckling mode identification and decomposition, as well as practical advancements to make the method a useful tool in the engineering design and analysis of wind turbine support towers.

The spectral representation method: A framework for simulation of stochastic processes, fields, and waves

The Spectral Representation Method (SRM) was developed in the 1970s to simulate Gaussian stochastic processes and fields from a Fourier series expansion according to the Spectral Representation Theorem. Since those early developments, the SRM has continuously evolved into a comprehensive framework for the simulation of stochastic processes, fields, and waves with a rigorous theoretical foundation. Its major advantages are conceptual simplicity and computational efficiency. In the 1990s, much of the theory for simulation of Gaussian stochastic processes, fields, and waves was firmly established and early methods for simulation of non-Gaussian processes, fields, and waves were introduced. In the 2000s and 2010s, methods that coupled the SRM with Translation Process Theory were improved to enable efficient and accurate simulations of stochastic processes, fields, and waves with strongly non-Gaussian marginal probability distributions. More recently, the SRM was extended for higher-order non-Gaussian processes, fields, and waves by extending the Fourier stochastic expansion to include non-linear wave interactions derived from higher-order spectra. This paper reviews the key theoretical developments related with the SRM and provides the relevant algorithms necessary for its practical implementation for the simulation of stochastic processes, fields, and waves that can be either stationary or non-stationary, homogeneous or non-homogeneous, one-dimensional or multi-dimensional, scalar or multi-variate, Gaussian or non-Gaussian, or any combination thereof. The paper concludes with some brief remarks addressing the open research challenges in SRM-based theory and simulations.

Hygrothermoelastic analysis of the nano-circular plate with memory effect

In hygrothermal environments, the coupling effects of temperature and moisture substantially impact deflection and stresses play a significant role. This study presents a coupled hygrothermoelastic model with non-Fourier and non-Fick effects established by introducing relaxation times or phase lags of heat and moisture flux accompanied by memory-dependent derivatives. The boundary value problem is formulated by considering a thin circular plate as an exemplary example, where the perimetric edge is clamped. The upper and lower edges of the plate is subjected to zero temperature, whereas the curved surface is exposed to hygrothermal shock. The closed-form solution of temperature and moisture distribution is obtained via the integral transform approach. The Fourier series expansion approach is used to calculate the numerical Laplace inversion. The effects of both heat and moisture flux relaxation times on the thermal deflection/stresses of the plate are analyzed and illustrated graphically.