Smooth Periodic Function Research Articles

Fourier–Gegenbauer Pseudospectral Method for Solving Periodic Higher-Order Fractional Optimal Control Problems

Optimal control (OC) problems involve determining the best control strategy for a system to achieve the desired outcome. Fractional-order OC models have emerged in recent decades to represent complex systems that exhibit memory effects and nonlocal interactions. These models offer advantages over the traditional integer-order models. However, a key challenge with existing fractional calculus methods to define fractional derivatives, such as Riemann–Liouville, Caputo, and Grünwald–Letnikov derivatives, is that they often fail to maintain the periodic nature of dynamical systems. This limitation hinders their application to real-world scenarios, in which periodicity is crucial. This study proposes a novel model for periodic higher-order fractional OC problems (PHFOCPs). We achieve this by employing recently developed periodic fractional derivatives that specifically preserve the periodicity within the model. This study outlines also a robust numerical solution method for this new class of problems. The numerical method is applicable for any positive fractional order [Formula: see text] and leverages several key techniques, including a useful change of variables, Fourier collocation, and Fourier and Gegenbauer quadratures, to retain numerical stability and high accuracy. We theoretically prove the exponential convergence of the proposed numerical method for sufficiently smooth periodic functions. The effectiveness of the proposed approach is further validated through various numerical simulations. In essence, this work presents a novel framework for the OC of periodic systems using fractional calculus, overcoming the limitations of classical methods and offering a powerful tool for various scientific and engineering applications.

Enhanced trapezoidal rule for discontinuous functions

In many applications, data to be integrated is available only in the form of function values at predetermined equispaced points. For smooth periodic functions, the trapezoidal rule gives very accurate approximations to the integral, but is only second order accurate in more commonly arising non-periodic cases. The Gregory approach for end corrections can improve the accuracy order to 9 before the onset of negative weights, and of rapidly increasing ill-conditioning. In recent work, this has been extended to accuracy orders around 20.This present study focuses on the more general case when one or both end points of the integration interval do not coincide with any of the equispaced grid points. We show that accuracy orders up to around 10 can also then be achieved, with all quadrature weights still remaining non-negative. This method can be utilized for example when functions to be integrated feature discontinuities at locations that can be separately determined (at or in between the equispaced grid points).

THE ESTIMATES OF ALEXANDROV’S <i>n</i>-WIDTH OF A COMPACT SET FOR SOME INFINITELY DIFFERENTIABLE PERIODIC FUNCTIONS

In this paper, we obtain two-way estimates of the Alexandrov’s n-width of a compact set of infinitely smooth periodic functions that are bounded embedded in the space of continuous functions on the unit circle.

Boundedness in Asymmetric Oscillations at Resonance in a Critical Situation

In this article, by using Moser's twist theorem, we prove that all solutions of the equation $x'' + ax^{+} - bx^{-} + \varphi(x) = p(t)$ are bounded in the critical situation, where $p$ is a smooth periodic function, and $\varphi$ is bounded one.

Liouville-type theorem of positive periodic solutions for the periodic parabolic system

ABSTRACT In this paper, we are concerned with the problem for a periodic parabolic system where p, q>0, Ω is a bounded and smooth domain in , and are properly smooth, bounded and positive periodic functions with periodicity ω. It is known that for the elliptic system in , there exists a Sobolev hyperbola , where is defined by (3), which characterizes the existence and nonexistence of nontrivial nonnegative solutions. By using the classical blowing-up method, Liouville-type theorem, Leray–Schauder fixed point theory and Pohozaev identity, we will prove that the Sobolev hyperbola is also a critical curve for the existence and non-existence of the positive periodic solutions for the periodic parabolic system.

Hermite-Wavelet Transforms of Multivariate Functions on $[0,1]^{d}$

For a $d$ -dimensional smooth target function $f$ on the cube $[0,1]^{d}$ , we propose the Hermite-wavelet transform to overcome boundary effects. In details, we first give a decomposition of $f$ based on its even-order Hermite interpolation on sections of the cube $[0,1]^{d}$ : $f=G+r$ , where $G$ is a combination of polynomials and the restriction of derivative functions on some part of the boundary of the cube, and $r$ can be extended to a smooth periodic function on $\mathbb{R}^{d}$ after odd extension. Noticing that the restriction of derivative functions on some part of the boundary of the cube has less number of free variables, using similar decomposition again and again, finally the multivariate smooth function $f$ on the cube $[0,1]^{d}$ can be decomposed into a combination of smooth periodic functions and polynomials whose coefficients are completely determined by partial derivatives of $f$ at vertices of the cube $[0,1]^{d}$ . After that, we expand all smooth periodic functions into periodic wavelet series. Since these periodic functions have the same smoothness as the target function $f$ , the corresponding periodic wavelet coefficients decay fast. Hence the $d$ -dimensional smooth target function $f$ on the cube $[0,1]^{d}$ can be reconstructed by values of partial derivatives of $f$ at all vertices of $[0,1]^{d}$ and few periodic wavelet coefficients with small error.

A periodic map for linear barycentric rational trigonometric interpolation

Consider the set of equidistant nodes in [0, 2π),θk:=k·2πn,k=0,⋯,n−1.For an arbitrary 2π–periodic function f(θ), the barycentric formula for the corresponding trigonometric interpolant between the θk’s isT[f](θ)=∑k=0n−1(−1)kcst(θ−θk2)f(θk)∑k=0n−1(−1)kcst(θ−θk2),where cst(·):=ctg(·) if the number of nodes n is even, and cst(·):=csc(·) if n is odd. Baltensperger [3] has shown that the corresponding barycentric rational trigonometric interpolant given by the right-hand side of the above equation for arbitrary nodes introduced in [9] converges exponentially toward f when the nodes are the images of the θk’s under a periodic conformal map. In the present work, we introduce a simple periodic conformal map which accumulates nodes in the neighborhood of an arbitrarily located front, as well as its extension to several fronts. Despite its simplicity, this map allows for a very accurate approximation of smooth periodic functions with steep gradients.

Uniform Approximation of the Curvature of Smooth Plane Curves with the Use of Partial Fourier Sums

An error bound for the approximation of the curvature of graphs of periodic functions from the class Wr for r ≥ 3 in the uniform metric is obtained with the use of the simplest approximation technique for smooth periodic functions, which is approximation by partial sums of their trigonometric Fourier series. From the mathematical point of view, the interest in this problem is connected with the specific nonlinearity of the graph curvature operator on the class of smooth functions Wr on a period or a closed interval for r ≥ 2. There are several papers on curvature approximation for plane curves in the mean-square and Chebyshev norms. In previous works, the approximation was performed by partial sums of trigonometric series (in the L2 norm), interpolation splines with uniform knots, Fejer means of partial sums of trigonometric series, and orthogonal interpolating wavelets based on Meyer wavelets (in the C∞ norm). The technique of this paper, based on the lemma, can possibly be generalized to the Lp metric and other approximation methods.

The coexistence of quasi-periodic and blow-up solutions in a superlinear Duffing equation

In this paper, we construct a continuous positive periodic function $p(t)$ such that the corresponding superlinear Duffing equation possesses a solution which escapes to infinity in some finite time, and also has infinitely many subharmonic and quasi-periodic solutions, where the coefficient $a(x)$ is an arbitrary positive smooth periodic function defined in the whole real axis.

Strong approximation of lacunary series with random gaps

We investigate the asymptotic behavior of sums $$\sum _{k=1}^N f(n_kx)$$ , where f is a mean zero, smooth periodic function on $$\mathbb {R}$$ and $$(n_k)_{k\ge 1}$$ is a random sequence such that the gaps $$n_{k+1}-n_k$$ are i.i.d. Our result shows that, in contrast to the classical Salem–Zygmund theory, the almost sure behavior of lacunary series with random gaps can be described very precisely without any assumption on the size of the gaps.

On families of differential equations on two-torus with all phase-lock areas

We consider two-parametric families of non-autonomous ordinary differential equations on the two-torus with coordinates (x, t) of the type . We study its rotation number as a function of the parameters (A, B). The phase-lock areas are those level sets of the rotation number function that have non-empty interiors. Buchstaber, Karpov and Tertychnyi studied the case when in their joint paper. They observed the quantization effect: for every smooth periodic function f(t) the family of equations may have phase-lock areas only for integer rotation numbers. Another proof of this quantization statement was later obtained in a joint paper by Ilyashenko, Filimonov and Ryzhov. This implies a similar quantization effect for every and rotation numbers that are multiples of . We show that for every other analytic vector field v(x) (i.e. having at least two Fourier harmonics with non-zero non-opposite degrees and nonzero coefficients) there exists an analytic periodic function f(t) such that the corresponding family of equations has phase-lock areas for all the rational values of the rotation number.

The spatial Hill lunar problem: periodic solutions emerging from equilibria

ABSTRACTIn this paper, we provide sufficient conditions for the existence of periodic solutions emerging of the equilibrium points of the spatial Hill lunar problem having the following equations of motion:ϵ is a small parameter and Fi, i ∈ {1, 2, 3}, are smooth periodic functions in t which define a perturbation in resonance p:q with some of the periodic solutions of the spatial Hill lunar problem being p and q positive relatively prime integers.

Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface

We study two-dimensional equilibrium configurations of nematic liquid crystal (NLC) director in a cell confined between two parallel surfaces: a planar surface and a spatially modulated one. The relief of the modulated surface is described by a smooth periodic sine-like function. The director easy axis orientation is homeotropic at one of the bounding surfaces and is planar at the other one. Strong NLC anchoring with both surfaces is assumed. We consider the case where disclination lines occur in the bulk of NLC strictly above local extrema of the modulated surface. These disclination lines run along the crests and troughs of the relief waves. In the approximation of planar director deformations we obtain analytical expressions describing a director distribution in the bulk of the cell. Equilibrium distances from disclination lines to the modulated surface are calculated and their dependences on the cell thickness and the period and depth of the surface relief are studied. It is shown that the distances from disclination lines to the modulated surface decrease as the depth of the relief increases.

Modelling of director equilibrium states in a nematic cell with relief surface

ABSTRACTWe study two-dimensional equilibrium configurations of nematic liquid crystal (NLC) director in a cell bounded by two parallel surfaces. One surface is planar and the other one is spatially modulated. The relief of the modulated surface is described by a smooth periodic sine-like function. The orientation of NLC director easy axis is assumed to be homeotropic at one cell surface and planar at the other one. The NLC director anchoring with cell surfaces is assumed to be strong. We consider the case where disclination lines occur in the bulk of NLC above the extrema of the modulated surface. These disclination lines run along the crests and troughs of the surface relief. If the orientation of director at both bounding surfaces is of the same type, then NLC director field is continuous. For both configurations mentioned above (with defects and without defects), we obtain analytical expressions for director distribution in the bulk of NLC in the approximation of planar director deformations. Equilibrium distances from disclination lines to the spatially modulated surface are calculated when the defects occur. The dependences of these equilibrium distances on the period and depth of surface relief and the cell thickness are investigated in detail.

Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions

We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connection between the worst-case errors of our algorithms in the cosine space and the worst-case errors of some related algorithms in the well-known weighted Korobov space of smooth periodic functions. By exploiting this connection, we are able to obtain constructive worst-case error bounds with good convergence rates for the cosine space.

On the Invalidity of Fourier Series Expansions of Fractional Order

Abstract The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions einωx are replaced by the Mittag-Leffler functions Eα (i(nωx)α) , over the interval [0,Mα/ω] where 0 < ω < ∞ and Mα > 0 is the period of the function Eα (ixα) , i.e., Eα (ixα) = Eα (i(x +Mα)α) . He showed that any smooth periodic function f with period Mα/ω can be expanded in a Fourier-type series. We will show that the only possible period of the function Eα (ixα) is Mα = 0; hence the invalidity of any Fourier-type series expansion of f.

Low-discrepancy sequences for piecewise smooth functions on the two-dimensional torus

We produce explicit low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a convex domain with positive curvature in R2. The proof depends on simultaneous diophantine approximation and a general version of the Erdős–Turán inequality.

On Lacunary Series with Random Gaps

Abstract We prove Strassen’s law of the iterated logarithm for sums \({\sum^{N}_{k=1} f(n_kx),}\) where f is a smooth periodic function on the real line and \({(n_k)_{k \geqq 1}}\) is an increasing random sequence. Our results show that classical results of the theory of lacunary series remain valid for sequences with random gaps, even in the nonharmonic case and if the Hadamard gap condition fails.

Optical under-sampling by using a broadband optical comb with a high average power.

We demonstrate a new method to improve the performance of photonic assisted analog to digital converters (ADCs) that are based on frequency down-conversion obtained by optical under-sampling. The under-sampling is performed by multiplying the radio frequency signal by ultra-low jitter broadband phase-locked optical comb. The comb wave intensity has a smooth periodic function in the time domain rather than a train of short pulses that is currently used in most photonic assisted ADCs. Hence, the signal energy at the photo-detector output can be increased and the signal to noise ratio of the system might be improved without decreasing its bandwidth. We have experimentally demonstrated a system for electro-optical under-sampling with a 6-dB bandwidth of 38.5 GHz and a spur free dynamic range of 99 dB/Hz(2/3) for a signal with a carrier frequency of 35.8 GHz, compared with 94 dB/Hz(2/3) for a signal at 6.2 GHz that was obtained in the same system when a pulsed optical source was used. The optical comb was generated by mixing signals from two dielectric resonator oscillators in a Mach-Zehnder modulator. The comb spacing is equal to 4 GHz and its bandwidth was greater than 48 GHz. The temporal jitter of the comb measured by integrating the phase noise in a frequency region of 10 kHz to 10 MHz around comb frequencies of 16 and 20 GHz was only about 15 and 11 fs, respectively.

Waiting time effect for motion by positive second derivatives and applications

We discuss the waiting time effect for the evolution of a planar graph governed by its positive part of second derivative. For any smooth periodic function which contains finitely many convex pieces in one period, we show that the waiting time is continuous by using comparison arguments. Moreover, we show that the convex parts keep expanding in size in a strict manner, which answers an open question posed by Kohn and Serfaty (Commun Pure Appl Math 59:344–407, 2006) in this special case. The results on waiting time effect are also applied to the stationary problem of mean curvature type on an unbounded nonconvex domain for our study of its game-theoretic interpretation.