Showing papers on "Fourier series published in 2021"
TL;DR: It is shown that there exist quantum models which can realise all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.
Abstract: Quantum computers can be used for supervised learning by treating parametrized quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate the practical implications of this approach, many important theoretical properties of these models remain unknown. Here, we investigate how the strategy with which data are encoded into the model influences the expressive power of parametrized quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data-encoding gates in the circuit. By repeating simple data-encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realize all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.
294 citations
TL;DR: In this article, a geometrically exact model is developed for the simply-supported beam utilizing a higher-order beam theory including thickness stretching effect, and the virtual work statement of Hamilton principle is adopted to gain the governing equation as well as boundary conditions.
68 citations
TL;DR: In this article, the authors investigated analytical and numerical solutions of fractional fuzzy hybrid systems in Hilbert space, which are devoted to model control systems that are capable of controlling complex systems with continuous time dynamics.
Abstract: The pivotal aim of this paper is to investigate analytical and numerical solutions of fractional fuzzy hybrid system in Hilbert space. Such fuzzy systems are devoted to model control systems that are capable of controlling complex systems that have discrete events with continuous time dynamics. The fractional derivative is described in Atangana-Baleanu Caputo (ABC) sense, which is distinguished by its non-local and non-singular kernel. In this orientation, the main contribution of the current numerical investigation is to generalize the characterization theory of integer fuzzy IVP to the ABC-fractional derivative under a strongly generalized differentiability, and then apply the proposed method to deal with the fuzzy hybrid system numerically. This method optimized the approximate solutions based on orthogonalization Schmidt process on Sobolev spaces, which can be straightway employed in generating Fourier expansion within a sensible convergence rate. The reproducing kernel theory is employed to construct a series solution with parametric form for the considered model in the space of direct sum W 2 2 [ a , b ] ⊕ W 2 2 [ a , b ] . Some theorems related to convergence analysis and approximation error are also proved. Moreover, we obtain the exact solution for the fuzzy model by applying Laplace transform method. So, the results obtained using the proposed method are compared with those of exact solution. To show the effect of Atangana-Baleanu fractional operator, we compare the numerical solution of fractional fuzzy hybrid system with those of integer order. Two numerical examples are carried out to illustrate that such dynamical processes noticeably depend on time instant and time history, which can be efficiently modeled by employing the fractional calculus theory. Finally, the accuracy, efficiency, and simplicity of the proposed method are evident in both classical and fractional cases.
59 citations
TL;DR: The Fourier series expansion and radial basis function neural network are incorporated into a function approximator to model time-varying-disturbed function with a known period in nonlinear systems to deal with the problems of the dead zone output and unknown control direction.
Abstract: This article considers the Nussbaum gain adaptive control issue for a type of nonlinear systems, in which some sophisticated and challenging problems, such as periodic disturbances, dead zone output, and unknown control direction are addressed. The Fourier series expansion and radial basis function neural network are incorporated into a function approximator to model time-varying-disturbed function with a known period in nonlinear systems. To deal with the problems of the dead zone output and unknown control direction, the Nussbaum-type function is recommended in the design of the control algorithm. Applying the Lyapunov stability theory and backstepping technique, the proposed control strategy ensures that the tracking error is pulled back to a small neighborhood of origin and all closed-loop signals are bounded. Finally, simulation results are presented to show the availability and validity of the analysis approach.
57 citations
TL;DR: In this paper, the analytical model is established on base of multi-segment partitioning strategy and first-order shear deformation theory, the displacement functions are made up of the Jacobi polynomials along the axial direction and Fourier series along the circumferential direction.
Abstract: In this paper, the analytical model is established on base of multi-segment partitioning strategy and first-order shear deformation theory. The displacement functions are made up of the Jacobi polynomials along the axial direction and Fourier series along the circumferential direction. In order to obtain continuity conditions and satisfy arbitrary boundary conditions, the penalty method about spring technique is adopted. The solutions about free vibration behaviors of combined composite laminated shell were obtained by approach of Rayleigh–Ritz. The convergence study and numerical verifications are carried out. Results show that the proposed method has a good stable and rapid convergence property.
55 citations
TL;DR: In this paper, a power-full higher-order shear-deformation theory in curvilinear coordinate is developed to model the doubly-curved nano-size shell.
49 citations
TL;DR: In this article, the authors derived a nonasymptotic lower bound for the minimum singular value of a Vandermonde matrix whose nodes are determined by the point sources, given as a weighted l 2 sum, where each term only depends on the configuration of each individual clump.
43 citations
TL;DR: In this article, an iterative reproducing-kernel (IRK) algorithm was proposed to solve the Riccati and Bernoulli differential equations in the Caputo-Fabrizio sense.
41 citations
TL;DR: In this paper, it was shown that the Benjamin-Ono equation admits global Birkhoff coordinates on the space of the torus, which allow to integrate it by quadrature.
Abstract: In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are coordinates which allow to integrate it by quadrature and hence are also referred to as nonlinear Fourier coefficients. As a consequence, all the $L^2(\T)$ solutions of the Benjamin--Ono equation are almost periodic functions of the time variable. The construction of such coordinates relies on the spectral study of the Lax operator in the Lax pair formulation of the Benjamin--Ono equation and on the use of a generating functional, which encodes the entire Benjamin--Ono hierarchy.
40 citations
TL;DR: In this article, the resonance phenomenon in anisotropic and functionally graded nano-size structure is investigated in a doubly curved shell which is modeled exploiting a quasi-three-dimensional model and nonlocal strain gradient theory in order to predict the small-size effects.
32 citations
TL;DR: In this paper, a strategy based on modeling poroelastic doubly curved composite shells on a Pasternak-type Elastic Foundation (PEF) was proposed to analyze wave propagation through a porous core.
TL;DR: In this article, the Fourier series coefficients that describe the spatial distribution of the plate properties are used as optimization variables to obtain solutions that maximize an objective function capable of yielding low-frequency band gaps.
TL;DR: The proposed method is based on the Fourier series expansion along the spatial coordinate which transforms the original time fractional partial differential equations into a sequence of multi-term fractional ordinary differential equations.
TL;DR: In this article, an adaptive waveform design method using multitone sinusoidal frequency modulation (MTSFM) is presented, where the MTSFM waveform's modulation function is represented as a finite Fourier series expansion.
Abstract: This article presents an adaptive waveform design method using multitone sinusoidal frequency modulation (MTSFM). The MTSFM waveform's modulation function is represented as a finite Fourier series expansion. The Fourier coefficients are utilized as a discrete set of design parameters. These design parameters can be modified to shape the spectrum, auto-correlation function (ACF), and ambiguity function (AF) shapes of the waveform. The MTSFM waveform model naturally possesses the constant envelope and spectral compactness properties that make it well suited for transmission on practical radar/sonar transmitters which utilize high power amplifiers. The MTSFM has an exact mathematical definition for its time-series using generalized Bessel functions which allow for deriving closed-form analytical expressions for its spectrum, AF, and ACF. These expressions allow for establishing well-defined optimization problems that finely tune the MTSFM's properties. This adaptive waveform design model is demonstrated by optimizing MTSFM waveforms that initially possess a “thumbtack-like” AF shape. The resulting optimized designs possess substantially improved sidelobe levels over specified regions in the range-Doppler plane without increasing their time-bandwidth product. Simulations additionally demonstrate that the optimized thumbtacklike MTSFM waveforms are competitive with thumbtacklike phase-coded waveforms derived from design algorithms available in the published literature.
TL;DR: By running several tests with experimentally collected backscattering data, it is found that the numerical convexification algorithm can accurately image both the dielectric constants and shapes of targets of interests including a challenging case of targets with voids.
Abstract: This report extends our recent progress in tackling a challenging 3D inverse scattering problem governed by the Helmholtz equation. Our target application is to reconstruct dielectric constants, el...
TL;DR: Experimental results show that the fault diagnosis accuracy rate of the proposed FDM-based method using the acoustic signals reaches up to 96.32% under the limited sample data conditions, which achieved better fault diagnosis effect than vibration signals in the experimental conditions.
TL;DR: The main concern of as mentioned in this paper is to deal with the thermoelastic interaction in a thick plate subjected to a moving heat source and being enlightened by memory-dependent derivative (MDD).
Abstract: The main concern of this article is to deal with the thermoelastic interaction in a thick plate subjected to a moving heat source and being enlightened by memory-dependent derivative (MDD). Due to ...
21 Jun 2021
TL;DR: In this paper, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov-Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0 <α≤1 order.
Abstract: In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0<α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven.
TL;DR: In this paper, the existence and uniqueness of the inverse boundary value problem for a mixed type partial differential equation with Hilfer operator with spectral parameter in a positive rectangular domain and a negative rectangular domain was studied.
Abstract: In this paper, we consider an inverse boundary value problem for a
mixed type partial differential equation with Hilfer operator of
fractional integro-differentiation in a positive rectangular
domain and with spectral parameter in a negative rectangular
domain. The differential equation depends from another positive
parameter in mixed derivatives. With respect to first variable
this equation is a fractional-order nonhomogeneous differential
equation in the positive part of the considering segment, and with
respect to second variable is a second-order differential equation
with spectral parameter in the negative part of this segment.
Using the Fourier series method, the solutions of direct and
inverse boundary value problems are constructed in the form of a
Fourier series. Theorems on the existence and uniqueness of the
problem are proved for regular values of the spectral parameter.
It is proved the stability of the solution with respect to
redefinition functions, and with respect to parameter given in
mixed derivatives. For irregular values of the spectral parameter,
an infinite number of solutions in the form of a Fourier series
are constructed.
TL;DR: By using fractional order accumulation to assign appropriate weights to samples, a fractional grey prediction model with Fourier series that offers high prediction accuracy is proposed that performs well compared with other considered prediction models.
Abstract: Tourism demand forecasting has played an important role in supporting governments to devise development policies for travel and tourism. However, time series related to tourism often do not conform to statistical assumptions and feature significant temporal fluctuations. Because a Fourier series is often applied to oscillating sequences to remove noise, it is reasonable to develop a grey prediction model in conjunction with a Fourier series to forecast tourism demand. However, grey prediction models traditionally use one-order accumulation, treating each sample with equal weight, to identify regularities concealed in data sequences. Furthermore, when generating residuals from Fourier series, the prediction accuracy of the newly generated predicted values is not taken into account. In this study, by using fractional order accumulation to assign appropriate weights to samples, we propose a fractional grey prediction model with Fourier series that offers high prediction accuracy. Experimental results demonstrate that the proposed grey prediction model performs well compared with other considered prediction models.
TL;DR: In this article, it was shown that the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.
Abstract: Consider the cubic nonlinear Schrodinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.
TL;DR: This approach optimizes approximate solutions based on the Gram-Schmidt process on Sobolev spaces that execute to generate Fourier expansion within a fast convergence rate, whereby the constructed kernel function fulfills homogeneous integral boundary conditions.
Abstract: In this paper, a coupled system of fractional differential equations along with integral boundary conditions is discussed by means of the iterative reproducing kernel algorithm Towards this end, a recently advanced analytical approach is proposed to obtain approximate solutions of nonclassical-types boundary value problems of fractional derivatives in Caputo sense This approach optimizes approximate solutions based on the Gram-Schmidt process on Sobolev spaces that execute to generate Fourier expansion within a fast convergence rate, whereby the constructed kernel function fulfills homogeneous integral boundary conditions Moreover, the solution is presented in the form of a fractional series over the entire Hilbert spaces without unwarranted assumptions on the considered models The validity of the present algorithm is illustrated by expounding and testing two numerical examples The achieved results indicate that the proposed algorithm is systematic, feasibility, stability, and convenient for dealing with other fractional systems emerging in the physical, technology and engineering
TL;DR: This work presents a procedure to generate an approximate optimal trajectory through a finite Fourier series, using a nonlinear programming solver, in which the optimization parameters are the coefficients of the Fourierseries and the positions of the spacecraft along the initial and target orbits.
TL;DR: In this paper, a mathematical model for rotating thermal nanobeams is presented, which is based on Eringen's nonlocal elasticity theory, Euler-Bernoulli's assumptions, and generalized thermoelasticity with two different phase lags.
Abstract: In this study, a mathematical model for rotating thermal nanobeams is presented. A system of equations is derived that describes the thermoelastic behaviour of rotating nanoscale beams. The proposed model is based on Eringen’s nonlocal elasticity theory, Euler–Bernoulli's assumptions, and generalized thermoelasticity with two different phase lags. The nanoscale beam material is completely surrounded by an axial magnetic field and exposed to a time-dependent variable temperature field. The Laplace transform in the state-space approach is employed to solve the problem studied. Because of the difficulty in finding the inversion of the Laplace transforms, it was obtained numerically using one of the techniques based on the technique of the Fourier series expansion. The significance of different parameters such as the rotational angular velocity, nonlocal parameter, temperature change, and magnetic field on the nanobeam response has been investigated. Moreover, the results obtained are verified with the corresponding results from the literature.
TL;DR: In this paper, a direct Lyapunov approach to the full-order closed-loop system was proposed, where the finite-dimensional state is coupled with the infinite-dimensional tail of the state Fourier expansion, and provided Linear Matrix Inequalities (LMIs) for finding the controller dimension and resulting exponential decay rate.
TL;DR: In this paper, an efficient approach to obtain accurate solutions for free vibration of functionally graded beams (FGB) with variable cross-sections and resting on Pasternak elastic foundations is presented.
TL;DR: In this paper, the elasticity-based locally exact asymptotic homogenization theory is extended to accommodate non-ageing linearly viscoelastic composites under antiplane shear loading.
TL;DR: In this article, a quasi-reversibility method was proposed to solve the inverse source problem of computing the initial condition of hyperbolic equations from the measurements of Cauchy data.
Abstract: We propose a numerical method to solve an inverse source problem of computing the initial condition of hyperbolic equations from the measurements of Cauchy data. This problem arises in thermo- and photo-acoustic tomography in a bounded cavity, in which the reflection of the wave makes the widely-used approaches, such as the time reversal method, not applicable. In order to solve this inverse source problem, we approximate the solution to the hyperbolic equation by its Fourier series with respect to a special orthonormal basis of $$L^2$$
. Then, we derive a coupled system of elliptic equations for the corresponding Fourier coefficients. We solve it by the quasi-reversibility method. The desired initial condition follows. We rigorously prove the convergence of the quasi-reversibility method as the noise level tends to 0. Some numerical examples are provided. In addition, we numerically prove that the use of the special basic above is significant.
TL;DR: The analysis results of the measured bearing data indicate that the mono-components obtained by APSFDM contain more accurate fault feature information that can be used for an effective failure diagnosis of bearing.
TL;DR: In this article, the free vibration characteristics of a joined shell structure in which three shells of revolution made of functionally graded material are elastically bonded are reported, and the structure of the joined shell is formed by elastically bonding double-curved shells to both sides of the cylindrical shell in the middle.
Abstract: In this paper, for the first time, the free vibration characteristics of a joined shell structure in which three shells of revolution made of functionally graded material are elastically bonded are reported. The structure of the joined shell is formed by elastically bonding double-curved shells to both sides of the cylindrical shell in the middle, and the hyperbolic shells have elliptic, parabolic, and hyperbolic shapes. The Haar wavelet discretization method (HWDM), one of the effective, convenient and accurate numerical solution methods, is applied to investigate the free vibration behavior of various elastically joined functionally graded shells (EJFGSs). The theoretical model of the EJFGSs is formulated based on first-order shear deformation theory (FSDT). The displacement and rotation of arbitrary point of the EJFGS are expended as Haar wavelet series in the meridian direction and as a Fourier series in the circumferential direction. The boundary condition at both ends of the EJFGS and the continuous condition between its shells are modeled by the spring stiffness technique. By solving the governing equation of the whole system, the vibration characteristics of the EJFGS such as the natural frequency and the corresponding mode shape can be obtained. Through the comparisons with the previous literature and finite element method (FEM) results, the accuracy and reliability of our method are verified. Finally, new free vibration analysis results of various EJFGSs, which can be used as benchmark data for researchers in this field, are reported with parameter study.