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We consider the inverse problem of recovering the source function with pointwise overdetermination conditions. The source is represented as a finite segment of a series with Fourier coefficients depending on time. We prove that the solution is uniquely determined and continuously depends on the data. [ABSTRACT FROM AUTHOR]

Determining the stress state in a circular ring has been a classical topic in the stress analysis literature. Based on the principle of superposition, the results may be obtained by adding known solutions to an associated ring problem, where the boundary stresses on the inner and outer walls of the ring are represented in Fourier series. In this work, the coefficients of the Fourier series are generated through an algorithm based on the fast Fourier transform (FFT). In the case of concentrated loading, the required additional fundamental solutions are derived in closed-form. The presented numerical method allows for accurate and efficient computations of the stress distributions in a circular ring in static equilibrium under arbitrary in-plane loading; and generally, the FFT-based algorithm provides a convenient and versatile tool for handing some two-dimensional problems involving circular boundaries. [ABSTRACT FROM AUTHOR]

The mathematical formulations for transverse compression of a thin elastic disc are considered, including various boundary conditions along the faces of the disc. The mixed boundary conditions corresponding to the loading by normal stresses in absence of sliding are studied in detail. These conditions support an explicit solution in a Fourier series for the boundary layers localised near the edge of the disc and also do not assume making use of the Saint-Venant principle underlying the traditional asymptotic theory for thin elastic structures. As an example, an axisymmetric problem is studied. Along with the leading order solution for a plane boundary layer, a two-term outer expansion is derived. The latter is expressed through the derivatives of the prescribed stresses. Generalisations of the developed approach are addressed. [ABSTRACT FROM AUTHOR]

We consider nonharmonic Fourier series defined in terms of arbitrarily close exponentials. Our aim is to use the positivity of quadratic polynomials in order to get new Ingham-type weighted inequalities. The proof relies on an Ingham proof technique inspired by Jaffard et al. (J Fourier Anal Appl 3:577–582, 1997). As applications, we consider families of frequencies with relevance in control approximation theory, for which we can prove the uniform (with respect to the mesh-size) controllability property of the semi-discrete model, when the spurious frequencies (the gap between them tends to zero when the mesh size goes to zero) are eliminated. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to show that in certain classes of real (or complex) functions, Bernoulli polynomials are essentially the only ones satisfying the Raabe functional equation. For the class of real 1-periodic functions which are expandable as Fourier series, we point out new solutions of the Raabe functional equation, not related to Bernoulli polynomials. Furthermore, we will give for the considered classes various proofs, making the mathematical content of the paper quite rich. [ABSTRACT FROM AUTHOR]

This paper values fixed-income (discrete- and continuous-time) European Asian and Australian options. We assume that the term structure of interest rates is modelled by the specification proposed in Moreno et al. (Econ Model 72:140–150, 2018, https://doi.org/10.1016/j.econmod.2018.01.015). We obtain closed-form expressions for the premiums of geometric average options and, for arithmetic average options, premiums are computed by numerical methods. We also perform a sensitivity analysis with respect to different parameters for both (geometric and arithmetic) options. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach for assessing the uncertainty in vibration and static responses of laminated composite beams resulting from uncertainty in material properties and distributed loads. The proposed method utilizes surrogate models developed using polynomial chaos expansion (PCE) based on a relatively small sample size. These training samples are computed using a high-order shear beam model in which the governing equations are derived using Hamilton's principle, and solved by Ritz's approach using a trigonometric series approximation. The proposed PCE model's coefficients are estimated using the spectral projection and linear regression techniques. The first four statistical moments and probability distributions of the mid-span displacement and the fundamental frequency of laminated composite beams are predicted. Global sensitivity analysis is also conducted to assess how material property variation and stochastic loads affect the beam's deflection and the fundamental frequency. The accuracy and efficiency of the proposed PCE models are compared with those from Monte Carlo simulation (MCS). A remarkable reduction in the computational cost of PCE models compared to MCS is observed without compromising the predictions' accuracy. As most real-world systems are subjected to multiple sources of uncertainty, this study provides a state-of-the-art method to quantify such uncertain parameters more efficiently and allow for a better reliability assessment in composite beam design. [ABSTRACT FROM AUTHOR]

In this study, the space-time fractional generalised reaction duffing model is investigated analytically, which is a generalization for a collection of prominent fractional models describing various key phenomenon in science and engineering. The governing equation is converted to a nonlinear ODE by the compatible travelling wave transform. The investigation established that for analysing nonlinear evolution equations of fractional order, the recommended approach is more effective and realistic. The findings are given extensively in rational forms of trigonometric function series or clearly in powers of specific trigonometric functions. A collection of bright, dark, periodic, and optical solitons is discovered. Mathematica is used to flourish the presence of some obtained solutions in 3D graphs with different fractional orders. The results show that the recommended methods are more practical and effective ways to illustrate the dynamics of several complex wave structures in modern science and technology. [ABSTRACT FROM AUTHOR]

This work presents novel three-dimensional solutions for the multiphysics response of magneto-electro-elastic multilayered plates with interfacial imperfections in a thermal environment. The Stroh formalism is employed, incorporating thermal coupling with the Eringen nonlocal theory to capture small-scale effects. The laminated structures are simply supported and subjected to time-harmonic temperature distributions and extended tractions represented using Fourier series expansions. The dual variable and position technique is used to address the challenges posed by non-ideal thermal and mechanical bonded contacts between constituents, ensuring the consistency and stability of the recursive field relations. A wide range of application examples are analyzed, including the influence of material arrangements, aspect ratio and nonlocal length characteristics, elastically compliant and thermally/dielectrically weakly conducting interfaces, as well as forced vibrations in combined thermo-mechanical environments. The comprehensive results shed light on the intricate multiphysics response of multilayered structures and provide valuable insights into practical engineering implications for advanced materials and structures. [ABSTRACT FROM AUTHOR]

We propose a photonics scheme to generate rectified cosinusoidal shaped (RCS) and rectified sinusoidal shaped (RSS) microwave waveforms and the duty cycle of the generated waveforms can be freely tunable. In this proposed method, two cascaded Mach-Zehnder modulators (MZMs) with polarization sensitive characteristics are employed. By setting the rotation angle of polarization controller, phase shift of the phase shifters and modulation index of the front MZM's (modulation index of the rear MZM is unfixed) properly, the obtained RCS and RSS microwave waveforms with arbitrary duty cycle by the superposition of beating signals, can be seen as an approximation of the first three terms of the Fourier series expansion of the ideal waveforms. The detailed theoretical analysis and simulations are given. In the simulations, the duty cycle of 100%, 80% and 50% of the RCS and RSS microwave waveforms are successfully obtained. Also, fitting errors are introduced to measure the similarity between the generated waveforms and the ideal waveforms. Different from previous works, the RCS and RSS microwave waveforms can be both generated by our scheme and the duty cycle is freely tunable. Moreover, the modulation index of the rear MZM is unfixed, thus increasing the flexibility of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of almost periodic response solutions for the superlinear Duffing's equation with an almost periodic external force. Assume that the system is reversible, and if the almost periodic forcing term admits a rapidly converging Fourier series, moreover the Diophantine condition for the frequencies is satisfied, the existence of response solutions will be proved. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for reversible systems. [ABSTRACT FROM AUTHOR]

We consider the corotational Maxwell model which is perhaps the simplest constitutive model with a nontrivial oscillatory shear response that can be solved analytically. The exact solution takes the form of an infinite series. Due to exponential convergence, accurate analytical approximations to the exact solution can be obtained by truncating the series after a modest number ( ≈ 10–20) of terms. We compare the speed and accuracy of this truncated analytical solution (AS) with a fast numerical method called harmonic balance (HB). HB represents the periodic steady-state solution using a Fourier series ansatz. Due to the linearity of the constitutive model, HB leads to a tridiagonal linear system of equations in the Fourier coefficients that can be solved very efficiently. Surprisingly, we find that the convergence properties of HB are superior to AS. In terms of computational cost, HB is about 200 times cheaper than AS. Thus, the answer to the question posed in the title is affirmative. [ABSTRACT FROM AUTHOR]

Based on the concept of improving the convergence of Fourier series by elimination of the Gibbs phenomenon, this work proposes a rational approach to construct hierarchic sets of shape functions of any desired degree of continuity from suitably modified (enriched) trigonometric series. Hierarchic sets of either enriched cosines or enriched sines are then used in Ritz solutions of beam and plate bending problems to illustrate the benefits provided by the enrichment in the quality of approximations. For a given degree of continuity, it is observed that a set of enriched cosines yields, in general, more accurate results and faster convergence than a set of enriched sines. To the authors' knowledge, there is no published information yet on hierarchic sets of enriched cosines. Moreover, it is shown that some hierarchic sets, which are already well-established in the literature, may lead not only to spurious oscillations but also to erroneous results. [ABSTRACT FROM AUTHOR]

A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed. The displacement and bending angle functions are represented through an improved Fourier series, which ensures the continuity of the derivative at the boundary and enhances the convergence. The exact characteristic equations of the multi-span spinning beams with elastic constraints under an axial compressive force are derived by the Lagrange equation. The efficiency and accuracy of the present method are validated in comparison with the finite element method (FEM) and other methods. The effects of the boundary spring stiffness, the number of spans, the spinning velocity, and the axial compressive force on the dynamic characteristics of the multi-span spinning beams are studied. The results show that the present method can freely simulate any boundary constraints without modifying the solution process. The elastic range of linear springs is larger than that of torsion springs, and it is not affected by the number of spans. With an increase in the axial compressive force, the attenuation rate of the natural frequency of a spinning beam with a large number of spans becomes larger, while the attenuation rate with an elastic boundary is lower than that under a classic simply supported boundary. [ABSTRACT FROM AUTHOR]

Recently, Devaiya and Srivastava [3] studied the T r -strong convergence of numerical sequences and Fourier series using a lower triangular matrix T = (b m , n) , and generalized the results of Kórus [8]. The main objective of this paper is to introduce [ T r , G , u , q ] -strongly convergent sequence spaces for r ∈ N , and defined by a sequence of modulus functions. We also provide a relationship between [ T , G , u , q ] and [ T r , G , u , q ] -strongly convergent sequence spaces. Further, we investigate some geometrical and topological characteristics and establish some inclusion relationships between these sequence spaces. In the last, we derive some results on characterizations for T r -strong convergent sequences, statistical convergence and Fourier series using the idea of [ T r , G , u , q ] -strongly convergent sequence spaces. [ABSTRACT FROM AUTHOR]

We obtain an estimate for the convergence rate of the Fourier series of a continuous periodic function in terms of the modulus of continuity of the function and the value of its -variation. We prove that the leading term of the estimate is sharp. [ABSTRACT FROM AUTHOR]

It is well known that Luzin's conjecture has a positive solution for one-dimensional trigonometric Fourier series, but in the multidimensional case it has not yet found its confirmation for spherical partial sums of multiple Fourier series. Historically, progress in solving Luzin's conjecture has been achieved by considering simpler problems. In this paper, we consider three of these problems for spherical partial sums: the principle of generalized localization, summability almost everywhere, and convergence almost everywhere of multiple Fourier series of smooth functions. A brief overview of the work in these areas is given and unsolved problems are mentioned and new problems are formulated. Moreover, at the end of the work, a new result on the convergence of spherical sums for functions from Sobolev classes is proved. [ABSTRACT FROM AUTHOR]

Considering the approximation of a function from a Sobolev space by the partial sums of Fourier series in a system of Sobolev orthogonal polynomials generated by classical Laguerre polynomials, we obtain an estimate for the convergence rate of the partial sums to . [ABSTRACT FROM AUTHOR]

A new flow fluctuation abatement method based on noncircular gear variable speed drive is proposed and the mechanism principle of variable speed lobe pump is expounded. Considering the instantaneous flow rate of lobe pump and the closed constraint condition of noncircular gear, the design method of noncircular gear for flow pulsation suppression based on Fourier series is established. Based on the transient numerical simulation, the effects of design parameter of the noncircular gear, angular deviation, rotor-rotor gap and input velocity on fluid pulsation were investigated. The result shows that the outlet flow rate fluctuation of lobe pump decreases by 84.1 % after installing a Fourier noncircular gear, while the increase of angular deviation deteriorates the flow characteristics. With the increase of the number of terms in the Fourier series, the intensity of the fluid pulsation decreases gradually. Decreasing of the rotor-rotor gap from 0.4 mm down to 0.2 mm produces about 54.1 % reduction of flow rate fluctuation. As the input velocity is increased from 200 (r/min) to 600 (r/min), the flow pulsation is reduced by more than 60.0 %. A pump prototype was made to verify that the flow pulsation of the lobe pump can be optimized effectively by the external Fourier noncircular gears drive, which provides a beneficial support for the design of lobe pump with low flow pulsation. [ABSTRACT FROM AUTHOR]

Boundary-value problems are considered for harmonic and biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on a plane. The problems are reduced to solving linear integral equations at boundary contours, which are assumed to be planar. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which the integral equations are expressed. The algorithm uses the periodicity of functions defined for closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed power. Applications to solving problems of hydrodynamics, filtration, and other problems of theoretical physics are considered. [ABSTRACT FROM AUTHOR]

The problem of high-frequency diffraction of a dipole field by a perfectly conducting strongly elongated spheroid is considered in parabolic equation approximation. The leading order term is represented in the form of Fourier series with each harmonics expressed by an integral involving the Whittaker functions. The amplitudes under the sign of integration are obtained as solutions of integral equations and are expressed explicitly in terms of the Whittaker functions. [ABSTRACT FROM AUTHOR]

The subject of the research is the control algorithms for a seven-phase converter that implement space-vector voltage modulation of a seven-phase motor as an alternative to a three-phase engine in modern electric traction. The study used elements of set theory, combinatorics, Fourier series expansion and vector analysis. Checking research results was implemented on a special stand for experimental studies of spatial vector voltage modulation of a seven-phase motor. [ABSTRACT FROM AUTHOR]

Boundary-value problems are considered for harmonic and biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on a plane. The problems are reduced to solving linear integral equations at boundary contours, which are assumed to be planar. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which the integral equations are expressed. The algorithm uses the periodicity of functions defined for closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed power. Applications to solving problems of hydrodynamics, filtration, and other problems of theoretical physics are considered. [ABSTRACT FROM AUTHOR]

In this study, ship turning in regular waves is investigated after sufficient time has passed after steering. At that time, the ship reaches the so-called periodic steady turning condition, under which the ship turns periodically with a constant drift in waves. The analytical formulas for the turning trajectory in waves are derived using the Fourier series expansion of the surge velocity, lateral velocity, and yaw rate under periodic steady turning conditions. From the obtained formulas, it is found that the ship turning in waves has the following characteristics. The turning trajectory in waves is deformed by ship drift in proportion to time. The drift velocity and the drift direction during turning are then determined by a combination of the average velocity components and first-order velocity components with the turning frequency. Furthermore, the turning trajectory in waves is composed of periodic circular motions with various frequencies in addition to the drift motion, and it is generally not a perfect circle. As a concrete example, using the derived analytical formulas, the turning trajectory for the KCS model in regular waves is analyzed. For the analysis, two turning trajectories obtained using a simulation method based on a two-time scale concept and a free-running model test are used under the same wave conditions. The reasons for the difference between the two are then clarified analytically. [ABSTRACT FROM AUTHOR]

Let b → = (b 1 , b 2 , ... , b m) be a collection of locally integrable functions and T Σ b → the commutator of multilinear singular integral operator T. Denote by L (δ) and L (δ (⋅)) the Lipschitz spaces and the variable Lipschitz spaces, respectively. The main purpose of this paper is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of multilinear commutator T Σ b → in the context of the variable exponent Lebesgue spaces, that is, the authors give the necessary and sufficient conditions for bj (j = 1, 2, ..., m) to be L (δ) or L (δ (⋅)) via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. The authors do so by applying the Fourier series technique and some pointwise estimate for the commutators. The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator. [ABSTRACT FROM AUTHOR]

Let be a smooth bounded domain in , and let be a linear elliptic differential operator of order with singular coefficients acting in . Under some assumptions of singularity for the coefficients of , we obtain Green's function estimates that hold up to the boundary of the domain and study the absolute convergence of the corresponding Fourier series. [ABSTRACT FROM AUTHOR]

The novel fault diagnosis method of gearbox based on Fourier Bessel series expansion-based empirical wavelet transform (FBEWT) and manifold regularization extreme learning machine (MRELM) is proposed to obtain excellent fault diagnosis results of gearbox in this paper. A new feature extraction strategy based on Fourier Bessel series expansion-based empirical wavelet transform is used to capture the key non-stationary features of the vibrational signal of gearbox, and significantly improve the diagnosis ability of gearbox. The ELM with manifold regularization is proposed for fault diagnosis of gearbox. In order to outstand the superiority and stability of the proposed FBEWT and manifold regularization ELM, the balanced dataset and unbalanced dataset, respectively, are used. The experimental results testify that FBEWT-MRELM are more superior and stable than FBEWT-ELM, EWT-MRELM, and EWT-ELM regardless of balanced dataset and unbalanced dataset. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear dynamic model for rotating beams with elastic boundary is established. The model considers the influence factors such as pre-twisted, setting angle, thermal gradient and geometric nonlinearity. Firstly, according to the Euler–Bernoulli beam theory, the Lagrange function of the rotating beam with elastic constraints is formulated, and a modified Fourier series method is used to solve the linear part to determine the modal function with elastic boundary. Secondly, the modal expansion of the displacements is carried out, and the nonlinear dynamic equations of a rotating pre-twisted beam with elastic boundary are obtained by Lagrange equation. Finally, the multi-scale method is used to solve the nonlinear problem to study the vibration response of the rotating beam with elastic constraints. The accuracy and stability of this method are verified by convergence analysis and comparison with other literatures. After determining the possibility of 2:1 internal resonance of the system, the influence of key system parameters such as rotational speed, spring stiffness and thermal gradient on vibration characteristics under elastic boundary is analyzed. The results show that the system parameters have a significant influence on the nonlinear phenomena of the system and the influence of elastic boundary cannot be ignored. [ABSTRACT FROM AUTHOR]

We study the evolution Navier–Stokes equations in a cube under Navier boundary conditions. For the related stationary Stokes problem, we determine explicitly all the eigenvectors, eigenvalues and the corresponding Weyl asymptotic. We introduce the notion of rarefaction, namely families of eigenvectors that weakly interact with each other through the nonlinearity. By combining the spectral analysis with rarefaction, we expand the solutions in Fourier series, making explicit some of their properties. We then suggest several new points of view in order to explain the striking difference in uniqueness results between 2D and 3D. First, we construct examples of solutions for which the nonlinearity plays a minor role, both in 2D and 3D. Second, we show that, if a solution is rarefied, then its energy is decreasing: hence, rarefaction may be seen as an almost two dimensional assumption. Finally, by exploiting the explicit form of the eigenvectors we provide a numerical explanation of the difficulty in using energy methods for general solutions of the 3D equations. [ABSTRACT FROM AUTHOR]

We study how the well-known criterion for the uniform convergence of a sine series with monotone coefficients changes if, instead of monotonicity, one imposes the condition of -monotonicity with . Moreover, we obtain an addition to the well-known Kolmogorov theorem on the integrability of the sum of a cosine series with convex coefficients tending to zero. [ABSTRACT FROM AUTHOR]

The novel fault diagnosis method of gearbox based on Fourier Bessel series expansion-based empirical wavelet transform (FBEWT) and manifold regularization extreme learning machine (MRELM) is proposed to obtain excellent fault diagnosis results of gearbox in this paper. A new feature extraction strategy based on Fourier Bessel series expansion-based empirical wavelet transform is used to capture the key non-stationary features of the vibrational signal of gearbox, and significantly improve the diagnosis ability of gearbox. The ELM with manifold regularization is proposed for fault diagnosis of gearbox. In order to outstand the superiority and stability of the proposed FBEWT and manifold regularization ELM, the balanced dataset and unbalanced dataset, respectively, are used. The experimental results testify that FBEWT-MRELM are more superior and stable than FBEWT-ELM, EWT-MRELM, and EWT-ELM regardless of balanced dataset and unbalanced dataset. [ABSTRACT FROM AUTHOR]

B ± → D K ± transitions are known to provide theoretically clean information about the CKM angle γ , with the most precise available methods exploiting the cascade decay of the neutral D into CP self-conjugate states. Such analyses currently require binning in the D decay Dalitz plot, while a recently proposed method replaces this binning with the truncation of a Fourier series expansion. In this paper, we present a proof of principle of a novel alternative to these two methods, in which no approximations at the level of the data representation are required. In particular, our new strategy makes no assumptions about the amplitude and strong phase variation over the Dalitz plot. This comes at the cost of a degree of ambiguity in the choice of test statistic quantifying the compatibility of the data with a given value of γ , with improved choices of test statistic yielding higher sensitivity. While our current proof-of-principle implementation does not demonstrate optimal sensitivity to γ , its conceptually novel approach opens the door to new strategies for γ extraction. More studies are required to see if these can be competitive with the existing methods. [ABSTRACT FROM AUTHOR]

A modified Fourier method and six-parameter constrained model are employed to investigate the static bending characteristics of functionally graded sandwich beams under classical and non-classical boundary conditions based on the first-order shear deformation theory. Three types of sandwich beams including isotropic hardcore, functionally graded core, and isotropic softcore are considered. The effective material properties of functionally graded materials are assumed to vary according to power law distribution of volume fraction of constituents by Voigt model. The governing equations and boundary conditions are derived from the principle of minimum potential energy and are solved using the modified Fourier series method which includes the standard Fourier cosine series together with two auxiliary polynomials terms. The high convergence rate, availability and accuracy of the formulation are verified by comparisons with results of other methods. Moreover, numerous new bending results for functionally graded sandwich beams with general boundary conditions are presented. The significant effects of various boundary conditions, different types of sandwich beams, power-law index, span-to-height ratio and skin–core-skin thickness ratio on the displacements, axial stresses, and shear stresses of the sandwich beams with symmetrical and unsymmetrical forms are also investigated. [ABSTRACT FROM AUTHOR]

For the first time, a modified two-dimensional Fourier series approach is proposed for new thermal buckling analysis of rectangular thin plates under various edge conditions. The solution form of plate deflection is considered to be in terms of a double Fourier Sine series (Navier-form solution) whose derivatives are determined via utilizing the Stoke's transform technic. The present study exhibits the following significant merits: (a) the method adopted allows one to consider any possible combination of boundary conditions with no necessity to be satisfied in the Fourier series; (b) the given solution procedure is simple and straightforward since the complicated boundary value problem (BVP) of partial differential equation (PDE) can be changed into solving sets of linear algebra equations, which heavily decreases the complicated mathematical manipulations of plate thermal buckling problem; (c) all the results acquired converge rapidly because of using the sum function of series. Greeting agreements between the present analytical solutions with the numerical results provided by FEM testifies the accuracy of the approach proposed. The present results are believed to be severe as new benchmarks for validating other methods and providing better design for plate structures. The influences of the aspect ratio and boundary condition on the thermal buckling behaviors of plates are also investigated and discussed. Furthermore, it is capable to extend the present solution procedure to deal with problems of plates under more complex edge conditions by ways of utilizing other Fourier series. [ABSTRACT FROM AUTHOR]

In this paper, we study the exact boundary controllability of the linear fourth-order Schrödinger equation, with variable physical parameters and clamped boundary conditions on a bounded interval. The control acts on the first spatial derivative at the right endpoint. We prove that this control system is exactly controllable at any time T > 0 . The proofs are based on a detailed spectral analysis and the use of nonharmonic Fourier series. [ABSTRACT FROM AUTHOR]

The problem of determining a periodic external force acting on a non-linear self-oscillating system of general type (the Liénard oscillator) is considered. A procedure for the asymptotic estimation of the unknown coefficients in the partial sum of the Fourier series is proposed, which approximates the periodic disturbance of the oscillator on the basis of information about the motion of the original system. The procedure is based on the synthesis of invariant relationships; the method allows finding dependences between the variables on the trajectories of a specially constructed extended dynamic system and deter-mining the sought unknown quantities as functions of known ones. The results of numerical simulation of the asymptotic estimation of external force parameters for the Duffing oscillator model are reported. [ABSTRACT FROM AUTHOR]

We consider the inverse problem of recovering the heat transfer coefficient represented as a finite segment of the Fourier series with coefficients depending on time. The overdetermination data are taken in the form of integrals of a solution with weights over a space domain. We prove that a solution to the problem is uniquely determined and continuously depends on the data. [ABSTRACT FROM AUTHOR]

This paper presents a steady-state Fourier series-based analysis (FSA) of class-D converters driven by general asymmetrical switching waveforms and its application to the design of inductive power transfer (IPT) systems. This formulation of the FSA extends an earlier analysis that was strictly limited to the symmetrically driven case. For analyzing asymmetrical waveforms, the presented FSA provides an advantage over other multi-harmonic approaches that either use less approximate, simplified circuits to model the nonlinear rectifier or involve a larger number of unknowns that scale with the number of harmonics. While the presented method is restricted to continuous conduction mode (CCM) operation, it only requires solving for two unknowns (the time delay and duty cycle of the rectangular switching waveform at the rectifier's input), independent of the number of harmonics considered. IPT systems are commonly operated in CCM, and the FSA is applied to the analysis and design of a series–series compensated IPT system with asymmetrical duty cycle control, in which the optimal switching frequency and input voltage are selected to maximize efficiency subject to design constraints for ensuring low switching losses. Finally, a web-based IPT calculator application is presented to showcase the practicality and usefulness of the FSA. [ABSTRACT FROM AUTHOR]

Conventional impedance inversion from post-stack zero-offset seismic data is usually based on the convolution model, and wave-equation based inversion is rarely used, although it is capable to precisely describe seismic wave propagation and invert impedance model with higher resolution. That is because there are more than one physical parameters involved in the conventional wave equation, making impedance inversion complicated. In this study, a one-dimensional (1D) wave equation, containing only the impedance parameter, is adopted and applied for the inversion of 1D impedance model by seismic waveform tomography. However, for a three-dimensional (3D) model, direct application of the 1D waveform tomography may lead to lateral discontinuities. Therefore, we propose to utilize a truncated Fourier series to parameterize the 3D impedance model, and then invert for the Fourier coefficients. With this parameterization scheme, the large- and small-scale components of the impedance model can be reconstructed stepwise by gradually increasing the number of Fourier coefficients. To efficiently and effectively invert the coefficients for the 3D model with salt structure, we propose a joint strategy, in which the low-frequency seismic data is used to invert for the Fourier coefficients representing the large-scale components of the model, while the high-frequency seismic data is applied to invert for the Fourier coefficients representing the small-scale components of the model. Tests on a 3D impedance model with salt structure result in models with high resolution and good spatial continuity, proving the feasibility and stability of the impedance inversion procedure. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problems on the solvability and constructing solutions of one nonlocal boundary-value problem for the multidimensional, fourth-order, integro-differential Benney–Luke equation with degenerate kernel and spectral parameters. For various values of spectral parameters, necessary and sufficient conditions of the existence of a solution are obtained. The Fourier series for solutions of the problem corresponding to various sets of spectral parameters are obtained. For regular values of spectral parameters, the absolute and uniform convergence of the series and the possibility of their termwise differentiation with respect to all variables are proved. The problem is also examined studied for cases of irregular values of spectral parameters. [ABSTRACT FROM AUTHOR]

Olive pomace is generated during the production of olive oil. The disposal of olive pomace presents a serious environmental issue to the agricultural community in Morocco. Among all actions devoted to the reduction of olive pomace waste, their revaluation seems to be the most efficient and advantageous from both environmental and economic points of view. Due to its high moisture content, solar drying constitutes an important and promising solution for a possible bio-fuel conversion of olive pomace. For this purpose, a modified uneven-span greenhouse dryer was designed and installed at the faculty of sciences Semlalia, Marrakech, Morocco. An experimental study of solar drying kinetics of the olive pomace was conducted to test the performances of the modified greenhouse dryer under a hybrid (solar/hot air) forced convection mode. The hot-air blower was supplied by electricity from a PV array installed near the greenhouse. The mean temperatures of air within the greenhouse and of the olive waste during the drying hours reached, respectively, 63 °C and 55 °C (from August 3rd to August 5th). The duration of the drying operation of 300 kg of waste from a moisture content of 50 mass% to 19 mass% was 27 h, which was low compared to the open sun drying. Therefore, hybrid forced mode enabled a considerable reduction in drying time, being an aspect to take into account for its use during low solar radiation. The drying kinetics of thin layer two-phase olive pomace was also investigated. The Two-term Gaussian model was found to be the most suitable model to describe the thin layer drying behavior of the two-phase olive pomace dried under hybrid forced convection greenhouse drying mode. In addition, the Fourier series was used to solve the diffusivity equation inside the two-phase olive pomace. The effective diffusivity was found to be 2.0 10–09 m2 s−1 and 2.3 10–09 m2 s−1 using the linear fit and the slope from two successive points methods, respectively. [ABSTRACT FROM AUTHOR]

We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices A = (a i , j) i , j = 1 , 2 , ... of the form A = T (a) + E , where E represents a compact operator, and T(a) is a semi-infinite Toeplitz matrix associated with the function a, with Fourier series ∑ k = - ∞ ∞ a k e i k t , in the sense that (T (a)) i , j = a j - i . If a is real valued and essentially bounded, then these matrices represent bounded self-adjoint operators on ℓ 2 . We prove that if a 1 , ... , a p are continuous and positive functions, or are in the Wiener algebra with some further conditions, then matrix geometric means, such as the ALM, the NBMP and the weighted mean of quasi-Toeplitz positive definite matrices associated with a 1 , ... , a p , are quasi-Toeplitz matrices associated with the function (a 1 ⋯ a p) 1 p , which differ only by the compact correction. We introduce numerical algorithms for their computation and show by numerical tests that these operator means can be effectively approximated numerically. [ABSTRACT FROM AUTHOR]

The distribution of Riemann zeros may represent a spectrum of a certain point distribution. Potentially suitable distributions were synthesized by summing complex exponentials over Riemann zeros. If the magnitudes of all harmonics are equal to each other, the resulting spectrum has peaks at the logarithms of primes and prime powers in accordance with the von Mangoldt function. If the magnitudes are set inversely proportional to the derivative of the Riemann zeta function at zeros, the spectrum has peaks at the logarithms of primes and products of distinct primes following the Möbius function. Combining trigonometric series over Riemann zeros with the Möbius function, we obtained the spectrum that consists of equal intensity peaks at the logarithms of primes only. [ABSTRACT FROM AUTHOR]

This work aims to theoretically analyze the nonuniform heat transfer through a micropolar miniature half-space by investigating the magneto-thermo-viscoelastic interactions. To examine the micromechanical coupling and the influence of thermo-mechanical relaxation, a higher-order two-phase-lag thermoelastic concept and a viscoelastic model of Kelvin–Voigt type are considered. The theoretical framework has been extended to incorporate the Eringen's nonlocal model to include the small-scale effects. The so-called Hall effect occurs in a conductive material when it is subjected to a high magnetic field orthogonal to the direction of the current traveling through it. Without nonlocality and viscoelastic effects, it is possible to achieve different types of generalized theories of thermoelasticity. The governing equations are developed and numerically solved using Laplace transforms. The Laplace transform is then inverted numerically using a method based on Fourier series expansion, and then the numerical values of physical fields are presented. The consequences of variations in nonlocality, viscoelasticity and the Hall effect are finally demonstrated and a comparison between the findings of this study and those for the generalized thermoelastic models are reported. [ABSTRACT FROM AUTHOR]

On the basis of the linear three-dimensional theory of elasticity, we consider the problem of equilibrium of nonthin isotropic cylindrical shells with dents under certain boundary conditions imposed on the end faces. To describe the cross section of the reference surface, we use the Pascal snail equation in polar coordinates. By the method of separation of variables, with the help of the approximation of functions by discrete Fourier series, we reduce the three-dimensional boundary-value problem to a one-dimensional problem, which is solved by the stable numerical method of discrete orthogonalization. We estimate the accuracy of the obtained solutions. The results of solving the problems are presented in the form of plots and tables. [ABSTRACT FROM AUTHOR]

The dynamic response of nanobeams has attracted noticeable attention in the scientific community. Boundary conditions and other effects on the element are very important in the dynamic behavior of these elements. To the authors' knowledge, there is no paper that provides a general solution for the vibration of a nanobeam with deformable boundary conditions and subjected to a point load according to the hardening nonlocal approach. The present study reports an efficient solution method based on the Stokes' transformation which can investigate the impacts of deformable boundary conditions and axial point load on the transverse vibration of a nanobeam restrained with lateral springs. In this study, an eigenvalue problem obtained by using Fourier sine series and Stokes' transform can be used to easily analyze the frequencies of nanobeam applications subjected to vibration and axial force at both rigid and non-rigid boundaries. It is seen from the presented problem that axial load intensity, nanoscale parameter, boundary condition and length are important variables in the vibration of nanobeams. Also, it should be noted here that the present analytical method can be applicable to a variety of nanotechnology structures and machines, especially micro-electromechanical systems and nano-electromechanical systems. [ABSTRACT FROM AUTHOR]

The stress state problem of non-thin cylindrical shells with a perturbed cross-sectional shape is solved using spatial problem statement, analytical methods of separation of variables in two coordinate directions, approximation of functions by discrete Fourier series, and numerical solution in the third coordinate direction with the method of discrete orthogonalization. The shells with certain boundary conditions at the ends are under an external normal load. Typical features of the stress state of the shells are established depending on the length and thickness for two variants of the perturbed surface. The reference cross-sectional surface is described by Pascal's limacon parametric equation. The results obtained are presented in the form of graphs of stresses in the shells. [ABSTRACT FROM AUTHOR]

The development of a variational method for solving the problem of quasi-geostrophic dynamics in a two-layer periodic channel is considered. The development of the method is as follows. First, the formulation of the variational problem is generalized: the turbulent exchange coefficient of a quasi-geostrophic potential vorticity (QGPV) is included in the control vector. Second, the solution area more accurately describes the size of the Antarctic Circumpolar Current (ACC). Using the selection of linear meridional transport and the expansion of the solution in a Fourier series, the problem is reduced to a nonlinear system of ordinary differential equations (ODEs) in time. The doubly connected domain leads to the fact that the solution of the ODE must satisfy an additional stationary relation that determines the transport of the ACC. The variational algorithm is reduced to solving a system of forward and adjoint equations minimizing the mean squared error of the stationary relation. The QGPV turbulent exchange coefficient is determined in the process of solving the optimal problem. The numerical runs are carried out for a periodic channel simulating the water area of the ACC in the Southern Ocean. The characteristics of stationary current regimes are studied for different values of the model parameters. Sinusoidal circulation in both layers with a linear transfer with the wind, depending on the bottom topography, is typical. In some cases, under the sinusoidal, in the lower layer, a cellular circulation is formed, and sometimes an undercurrent occurs. In this case, the solution of the optimal problem is characterized by a low value of the turbulent viscosity coefficient and a low transport in the lower layer. [ABSTRACT FROM AUTHOR]