Fourier Series Research Articles

An analytical model for daily‐periodic slope winds. Part 1: Surface radiation budget

AbstractThis article presents an analytical model for calculating representative daily‐periodic cycles of net surface radiation on a slope with any angle () and orientation, at any latitude and elevation (up to 2500 m), and for all seasons. Starting from the basic equations for estimating the daytime direct and diffuse components of the incoming solar radiation at the Earth's surface under clear‐sky conditions, a Fourier series expansion is derived to get an overall daily‐periodic expression, that is, covering both daytime and nighttime, which does not require data from observations, differently from most models available in the literature. Among various possible applications, this formulation is preliminary to evaluating the radiation and energy budgets driving daily‐periodic wind systems on a simple slope, as reproduced by the model presented in Part 2.

Computation of emitter-plasmon interactions using an axis-symmetric model for off-axis dipoles

The axis-symmetric modeling technique is based on expanding vector fields in cylindrical harmonics and computing the response on a two-dimensional cross-section separately for each azimuthal harmonic, significantly reducing computational costs. However, it has limitations when dealing with dipoles placed away from the symmetry axis due to challenges in the expansion of angular modes. To address this, we propose a reformulated axis-symmetric model based on the Fourier expansion of the delta function distribution concerning the azimuthal variable. This model is validated using standard Mie theory for off-axis dipoles and applied to study multiple-emitter-plasmon interactions. The emission properties of a non-cooperative ensemble near a plasmonic nanoparticle are observed to scale with the number of emitters considered, N. Notably, a Dicke effect-like superradiance (N2-dependence) is observed when a spatially disordered ensemble of dipoles oscillates collectively inside a plasmonic dimer gap. This kind of high-level cooperative quantum phenomenon is of high interest in fields such as quantum optics and light-harvesting.

Heat equation for Sturm–Liouville operator with singular propagation and potential

Abstract This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm–Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm–Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

Pointwise Convergence of Noncommutative Fourier Series

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 1 pointwise, so that the associated Fourier multipliers on noncommutative L p L_p -spaces satisfy the pointwise convergence for all p > 1 p>1 . In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

A generalized mathematical model for the damped free motion of a liquid column in a vertical U-tube

A one-dimensional mathematical model is developed to analyze the unsteady characteristics of arbitrary damped free motion, denoted as Z(τ), in a rigid, constant-diameter vertical column of a U-tube filled with an incompressible Newtonian liquid, where τ represents time. This model utilizes a simplified unsteady momentum equation derived from the Navier–Stokes equations in a circular coordinate system. Moreover, it incorporates assumptions about the periodicity of arbitrary damped free oscillations and employs the Fourier series representation to characterize the damped free motion. The combination of assumptions made for periodicity, the simplified momentum equation, and the Fourier series representation makes the current mathematical model unique and novel compared to prevailing models in the literature. In this model, the governing partial differential equation contains two dependent variables: Z(τ) is the known variable, as one can measure from experiments, and the instantaneous velocity uz is the unknown variable. Fitting the experimental data into the Fourier series provides the Fourier coefficients associated with the specific experiment. The Laplace transform method is used to determine the analytical solution for uz corresponding to the known Z(τ). The analytical expressions for instantaneous flow characteristics of practical importance, including area-averaged velocity, wall shear stress, and acceleration/deceleration, are deduced from uz. The analytical solutions presented are valid for generalized unsteady motions, including underdamped oscillations with varying amplitudes and periods, underdamped oscillations with varying amplitudes and constant period, and overdamped motion that does not exhibit a single oscillation. The findings from the present model offer insights for formulating a new friction model.

Droplet Shape Representation Using Fourier Series and Autoencoders

The shape of liquid droplets in air plays an important role in the aerodynamic behavior and combustion dynamics of miniaturized propulsion systems such as microsatellites and small drones. Their precise manipulation can yield optimal efficiency in such systems. It is desired to have a minimal representation of droplet shapes using as few parameters as possible to automate shape manipulation using self-learning algorithms, such as reinforcement learning. In this paper, we use a neural compression algorithm to represent, with only two parameters, elliptical and bullet-shaped droplets initially represented with 200 points (400 real numbers) at the droplet boundary. The mapping of many to two points is achieved in two stages. Initially, a Fourier series is formulated to approximate the contour of the droplet. Subsequently, the coefficients of this Fourier series are condensed to lower dimensions utilizing a neural network with a bottleneck architecture. Finally, 5000 synthetically generated droplet shapes were used to train the neural network. With a two-real-number representation, the recovered droplet shapes had excellent overlap with the original ones, with a mean square error of ∼10−3. Hence, this method compresses the droplet contour to merely two numerical parameters via a fully reversible process, a crucial feature for rendering learning algorithms computationally tractable.

Study on the Accurate Magnetic Field Analytical Model of an Inertial Magnetic Levitation Actuator Considering End Effects

To address the demand for low noise and high stealthiness in ships and other vessels, this paper innovatively proposes an inertial magnetic levitation actuator based on non-uniform-sized Halbach permanent magnet arrays. To improve control accuracy, it is necessary to establish an accurate analytical model of the magnetic field and then obtain an accurate electromagnetic force model. However, the distortion of the magnetic field at the ends produces end effects, resulting in thrust fluctuations that affect the actuator’s control accuracy. Therefore, considering the end effects is necessary to establish an accurate analytical model of the magnetic field. To analyze the end leakage magnetic field of the Halbach array, the concept of a mechanical pseudo-cycle in the actuator is proposed, and the cycle of a Fourier series is redefined. A completed analytical expression of the Halbach array magnetic field distribution is derived by the new Fourier series, in which the end leakage magnetic field is contained. The accuracy of the proposed method is verified by solving the analytical model of the magnetic field, and the analytical results are compared with finite element simulations and experimental tests.

Hyperslip velocity of melting ice sliding down inclined parallel ridges

A geometric and physical model for melting ice sliding over inclined superhydrophobic (SH) surfaces with parallel ridges is presented. By analyzing the micro-shear flows of molten liquid films between the ice layer and SH surfaces, the hyperslip velocities of melting ice sliding are investigated. The stick-slip boundary condition of the SH surface is used to establish the dual-series equations analytically, and the numerical solutions are implemented by truncating Fourier series and transforming the dual-series equations into linear algebraic equations to determine the hyperslip velocities of melting ice sliding. The numerical results indicate that the non-dimensional hyperslip velocities increase nonlinearly from near 0 to approximately 1.1 for longitudinal sliding and from near 0 to approximately 0.55 for transverse sliding with an increasing air groove ratio (a). The hyperslip velocities increase with increasing δ at the beginning initially (δ < 1), after which they tend toward asymptotic solutions as δ = 1. The hyperslip velocity ratio (Wh/Uh) shows that longitudinal ridges are at least twice as effective as transverse ridges in enhancing the ice hyperslip velocity, with the velocities accounting for more than 60% of the ice sliding velocities for arbitrary θ at a = 0.95 and δ = 0.1. The relative deviations between the numerical and asymptotic solutions are less than 5% at δ = 1, with the maximum relative deviation occurring at a = 0.65 for arbitrary θ.

Bayesian Inference for Estimating Heat Sources through Temperature Assimilation

Abstract This paper utilizes a Bayesian inference framework to address the two-dimensional 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 method, incorporating the random-walk Metropolis-Hasting 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.

Laplace and Mellin transform for reconstructing the probability distribution by a limited amount of information

A method for reconstructing the Probability Density Function (PDF) of a random variable using the Laplace transform is first introduced for one-sided PDFs. This approach defines new complex quantities, referred as Shifted Characteristic Functions, which allow the PDF to be computed using a classical Fourier series expansion. The method is then extended to handle double-sided PDFs by redefining the double-sided Laplace transform. This new definition remains applicable even when the integral in the inverse Laplace transform is discretized along the imaginary axis. For comparison, a new definition of double-sided Complex Fractional Moments based on Mellin transform is also introduced, addressing the singularity at zero that arises during PDF reconstruction.

Results from the further development of CGI inversion on simulation and field data systems

AbstractA new series expansion-based method, the Combined Geoelectric Weighted Inversion (CGWI) procedure is presented and tested by using synthetic and in-field measured datasets. The method is an improved version of the Combined Geoelectric Inversion (CGI) robustified by involving Cauchy-Steiner weights in an Iteratively Reweighted Least Squares technique. The new procedure is compared to the Fourier series expansion-based 1.5D and the CGI methods as well as to the broadly applied RES2DINV inversion procedure. The field measurements are performed during stone exploration in an active quarry on the south-western slopes of the Mátra mountains, in northern Hungary. It is shown that the CGWI method gives stable and robust parameter estimation with acceptable accuracy. The comparison with other inversion methods is based on data distances, estimation errors and correlation parameters calculated on the base of the parameter correlation matrix.

Nonlinear Finite Element Formulation for Thin-Walled Conical Shells

This study presents a novel finite element formulation to predict the geometrically nonlinear response of conical shells under a wide range of practical loading conditions. The formulation expresses the discretized equilibrium equations in terms of the first Piola-Kirchhoff stress tensor and its conjugate gradient of the virtual displacements, is based on the kinematics of Love-Kirchhoff thin shell theory and the Saint-Venant-Kirchhoff constitutive model, and captures the follower effect of pressure loading. The formulation takes advantage of the axisymmetric nature of the shell geometries by adopting a Fourier series to characterize the displacement distributions along the circumferential direction while using Hermitian interpolation along the meridional direction. Comparisons with general shell models show the accuracy of the formulation under various loading conditions with a minimal number of degrees of freedom, resulting in a significant computational efficiency compared to conventional general-purpose shell solutions.

Application of the Cubic Exponential B-spline Collocation Operator Splitting Method for Numerical solutions of Burgers Equation

In this study, the cubic exponential B-spline collocation method has been proposed for the numerical solutions of the Burgers equation with the operator splitting. To apply the operator splitting method, the Burgers' equation has decomposed into two sub-equations based on the time term: the linear part (diffusion) and the nonlinear part (convection). Subsequently, for each sub-equation, Crank-Nicolson finite difference schemes in the temporal direction and cubic exponential B-spline functions and their derivatives, have applied at the x_m nodal points in the spatial direction. The algebraic equation systems obtained have been solved numerically using the Lie-Trotter and Strang splitting schemes to get the solutions of the main equation. Some advantages of the splitting methods include preserving the physical characteristics of the solution, yielding more convergent results over long time intervals, enabling simpler algorithms, and facilitating the storage of solution vectors on computer. To assess the accuracy of the computed numerical results the L_2 and L_∞ error norms have been used. Additionally, the obtained results have been compared with some studies in the literature. The stability analysis of the applied method has been investigated using the von Neumann Fourier series method.

Effect of step bottom and waterway on flexural gravity wave scattering

Flexural gravity wave scattering by two semi-infinite non-identical ice sheets, which are separated by a clean water surface, is investigated in the presence of a step bottom topography. The problem is investigated in the cases of (i) intermediate water depth and (ii) shallow water. The effect of two edge conditions, (i) simply supported edge and (ii) free edge, on wave scattering is also analyzed. Employing linear velocity potential theory, the problem is studied in the frequency domain. The physical phenomenon is modeled as a boundary value problem having the Laplace equation as the governing equation, and it is solved using the eigenfunction expansion method. Considering the bottom topography and upper boundary, the fluid domain is divided into three regions, and in each region, velocity potential is expressed in terms of infinite Fourier series. Velocity and pressure are matched at the intermediate surface of two regions, and a system of algebraic equations with unknown coefficients is obtained. The complete solution of the present problem is recognized by solving the system of equations numerically. The energy relation is derived using Green's theorem. The reflection and transmission coefficients are computed and compared with the energy relation to check the accuracy of the present method. For different parameters (depth ratio, clean waterway, and different ice properties), reflection coefficient and transmission coefficient are computed and presented as a function of angular frequency. The value of the reflection coefficient ranges from zero to unity, whereas at certain frequencies, the transmission coefficient attends value more than unity.

Advanced unconditional signal processing model for cross-section contour reconstruction using multi-channel measurements

In mechanical engineering the standard method to assess the geometric tolerances of rotational parts is by analysing parts’ rotational motion in relation to the measuring system. Traditional tolerance compliance measurement models are conditional signal processing models that require pre-defined contour parameters of measured sections. However, due to part diversity and complexity, pre-determining and pre-setting the section parameters before each measurement process is very time-consuming and, in some cases, unachievable. To address this challenge, this paper proposes a universal unconditional signal processing model, which uses multi-channel measurements for cross-section contour reconstruction that operates without predefined section parameters. This model can handle multi-point measurement signals and accurately estimate the contour shape and engineering center coordinates of measured sections through circumferential Fourier expansion and an approximated signal-contour transform matrix. An efficient iterative algorithm then reconstructs the contour and precisely locates the engineering center. The computational accuracy and robustness of the proposed model have been confirmed through rigorous theoretical analysis and comprehensive experimental validation. Due to its inherent unconditional advantage, the proposed signal processing model can achieve intelligent monitoring of rotational parts throughout their entire life, which not only adapts and remains stable across a wide variety of cross-section types but also significantly improves measurement efficiency, ensuring precise and accurate results.

Innovative Multi-Temporal Evapotranspiration Forecasting Using Empirical Fourier Decomposition and Bidirectional Long Short-Term Memory

Reference evapotranspiration (ETo) is an essential environmental variable that is intimately significant to agriculture. Managing water and crop planning relies heavily on precise forecasting of ETo. This research used a novel time series decomposition technique, Empirical Fourier Decomposition (EFD), to forecast ETo accurately. Four machine learning techniques were used to forecast ETo using decomposed lagged ETo values. The input data source was from Prince Edward Island (PEI) weather stations (Harrington and St Peters Stations). First, autocorrelation analysis was performed to determine effective lags. Then, ETo data were decomposed using EFD, and lagged data was created based on EFD results. The Kbest feature selection algorithm was used to choose effective inputs, reducing the training time. The accuracy of models was evaluated using different statistical metrics such as correlation coefficient (R) and root mean square error (RMSE). The results showed that using EFD decomposition can significantly improve forecast accuracy. The comparison between different machine learning models showed that the deep learning-based model (Bidirectional LSTM (Long Short Term Memory)) (R=0.956, RMSE= 0.451 mm/day for Harrington station and R=0.956, RMSE= 0.451 mm/day for St Peters station) performed better than the Generalized Regression Neural Network (GRNN), K-nearest neighbor (KNN), and Random Forest (RF) models. Finally, the best model (EFD-Bidirectional LSTM) was used to forecast multitemporal ETo at both stations. Results showed that the developed model can forecast ETo for up to 28 days with reasonable accuracy. However, the accuracy of multi-step ahead forecasting decreases when evapotranspiration values are high, as the models tend to underestimate these values. The findings of this study can assist in accurately calculating crop water requirements and help farmers optimize their irrigation schedules.

Review In Fourier series

It is known that Fourier series are one of linear transformations and have many uses in mathematics, physics, engineering…etc. It is used to influence on periodic functions that repeats in a certain period to transform it into a sum of sine and cosine functions also used for non-periodic functions, as we will notice, and it has many other uses. In this article, I study, with examples, the important uses of Fourier series in physics, engineering, and mathematics, with proofs and examples where it necessary. One of results that we have reached in this article is that use Fourier series to facilitated a solution of differential equations in particular partial differential equations by the sinusoidal part of it, also obtaining the best approximation element for a particular function by means of its mathematical mean which called Feger operators. Moreover, there are more than one auxiliary conclusion for, example if a function has two different periods, then the sum of these periods, its subtraction, and its multiplying by constant, will be a new period for this function. Here, I refer to another conclusion that, Fourier series itself do not give the best approximation element for a particular function.

Free Vibration of Graphene Nanoplatelet-Reinforced Porous Double-Curved Shells of Revolution with a General Radius of Curvature Based on a Semi-Analytical Method

Based on domain decomposition, a semi-analytical method (SAM) is applied to analyze the free vibration of double-curved shells of revolution with a general curvature radius made from graphene nanoplatelet (GPL)-reinforced porous composites. The mechanical properties of the GPL-reinforced composition are assessed with the Halpin–Tsai model. The double-curvature shell of revolution is broken down into segments along its axis in accordance with first-order shear deformation theory (FSDT) and the multi-segment partitioning technique, to derive the shell’s functional energy. At the same time, interfacial potential is used to ensure the continuity of the contact surface between neighboring segments. By applying the least-squares weighted residual method (LWRM) and modified variational principle (MVP) to relax and achieve interface compatibility conditions, a theoretical framework for analyzing vibrations is developed. The displacements and rotations are described through Fourier series and Chebyshev polynomials, accordingly, converting a two-dimensional issue into a suite of decoupled one-dimensional problems. The obtained solutions are contrasted with those achieved using the finite element method (FEM) and other existing results, and the current formulation’s validity and precision are confirmed. Example cases are presented to demonstrate the free vibration of GPL-reinforced porous composite double-curved paraboloidal, elliptical, and hyperbolical shells of revolution. The findings demonstrate that the natural frequency of the shell is related to pore coefficients, porosity, the mass fraction of the GPLs, and the distribution patterns of the GPLs.

A displacement prediction model Based on IQGA-LSSVM for Mogu toppling rock slope in Lianghekou reservoir, China

Predicting landslide displacement is essential for developing early warning systems, assessing disaster risks, and understanding spatiotemporal evolution patterns. To address the limitations of insufficient accuracy in traditional landslide displacement prediction models, this study proposes a landslide displacement prediction model that combines an improved quantum genetic algorithm (IQGA) with a least squares support vector machine (LSSVM). The Mogu toppling rock slope is employed as a case study to demonstrate the application of the empirical mode decomposition (EMD) method for the decomposition of landslide displacement into trend and periodic components. Subsequently, these components are accurately predicted using Fourier series fitting and the IQGA-LSSVM model, respectively. The prediction results are subsequently evaluated and analyzed. The findings indicate that this model outperforms traditional QGA-LSSVM models, BP neural networks, and Support Vector Machine models in terms of prediction accuracy. Furthermore, the study illustrates how high-accuracy landslide displacement prediction models can effectively prevent and control disasters in critical areas such as hydraulic and hydropower engineering, urban planning, and geological disaster management.

Two-dimensional hyperchaos-based encryption and compression algorithm for agricultural UAV-captured planar images.

This study presents an approach that integrates compressed sensing technology with two-dimensional hyperchaotic coupled Fourier oscillator systems (2D-HCFOS) to address the challenge of slow encryption speeds in agricultural unmanned aerial vehicles (UAVs). The primary challenge in enhancing encryption speed lies in the limited capacity inherent in traditional chaotic-based systems and the computational complexity of their processes. The 2D-HCFOS utilizes a complex two-dimensional hybrid chaotic system, which significantly enhances the security of agricultural UAV image data. Notably, the image encryption process is performed on a personal computer connected to the drone, ensuring efficient processing. By integrating advanced Fourier series and nonlinear coupled oscillators, the model surpasses existing chaotic-based methods, improving both the pseudo-randomness and robustness of encryption. Additionally, incorporating Bonouille functions into the discrete cosine transform (DCT) domain results in a sparser measurement matrix, which is essential for efficient encryption on personal computers. The effectiveness of 2D-HCFOS in securely encrypting agricultural drone images has been rigorously validated through simulations and analytical evaluations using sophisticated row, rotation, and matrix encryption techniques. The improved security performance is further verified by comparative analysis. Compared with other models, the Lyapunov index of 2D-HCFOS is 15.1039, and the sample entropy is 2.4987, indicating that it possesses superior chaotic performance and encryption reliability.