General Periodic Functions Research Articles

Exact solutions, wave dynamics and conservation laws of a generalized geophysical Korteweg de Vries equation in ocean physics using Lie symmetry analysis

An evolution equation is a partial diferential equation that describes the time evolution of a physical system starting from given initial data. Evolution equations arise from many areas of applied and engineering sciences. To this end, this article investigates the analytical studies of a generalized geophysical Korteweg-de Vries equation in ocean physics. The examination of this model is conducted via the Lie group theory of diferential equations. In the first place, point symmetries, which are constituent elements of a four-dimensional Lie algebra, are systematically computed. Thereafter, one-parameter transformation groups for the algebra are calculated. Besides, going forward, a one-dimensional optimal system of subalgebras is derived in a procedural manner. Sequel to this, the subalgebras and combination of the achieved symmetries are invoked in the reduction proces which enables the derivation of nonlinear ordinary diferential equations associated with the generalized geophysical Korteweg-de Vries equation under study. Most of the achieved nonlinear ordinary diferential equations are further solved either via direct integration or using a power series approach. Furthermore, travelling wave solutions are initially obtained. This is attained via direct integration and the use of Jacobi elliptic function approach. These techniques enable the attainment of various exact soliton solutions, including non-topological soliton solutions as well as general periodic function solutions of note, such as cosine amplitude, sine amplitude, and delta amplitude solutions of the model. Furthermore, numerical simulations of the solutions are invoked to gain a gross knowledge of the physical phenomena represented by the under-study generalized geophysical Korteweg-de Vries equation in ocean physics. In the end, the investigation further gives attention to the calculation of conserved vectors for the model using Ibragimov's theorem for conservation laws, as well as Noether's theorem.

General Periodic Functions and Generalization of Fourier analysis

General Periodic Functions and Generalization of Fourier analysis

Wave-shape function model order estimation by trigonometric regression

The adaptive non-harmonic (ANH) model is a powerful tool to compactly represent oscillating signals with time-varying amplitude and phase, and non-sinusoidal oscillating morphology. Given good estimators of instantaneous amplitude and phase we can construct an adaptive model, where the morphology of the oscillation is described by the wave-shape function (WSF), a 2π-periodic more general periodic function. In this paper, we address the problem of estimating the number of harmonic components of the WSF, a problem that remains underresearched, by adapting trigonometric regression model selection criteria into this context. We study the application of these criteria, originally developed in the context of stationary signals, to the case of signals with time-varying amplitudes and phases. We then incorporate the order estimation to the ANH model reconstruction procedure and analyze its performance for noisy AM-FM signals. Experimental results on synthetic signals indicate that these criteria enable the adaptive estimation of the waveform of non-stationary signals with non-sinusoidal oscillatory patterns, even in the presence of considerable amount of noise. We also apply our reconstruction procedure to the task of denoising simulated pulse wave signals and determine that the proposed technique performs competitively to other denoising schemes. We conclude this work by showing that our adaptive order estimation algorithm takes into account the interpatient waveform variability of the electrocardiogram (ECG) and respiratory signals by analyzing recordings from the Fantasia Database.

Intermittent sliding locomotion of a two-link body.

We study the possibility of efficient intermittent locomotion for two-link bodies that slide by changing their interlink angle periodically in time. We find that the anisotropy ratio of the sliding friction coefficients is a key parameter, while solutions have a simple scaling dependence on the friction coefficients' magnitudes. With very anisotropic friction, efficient motions involve coasting in low-drag states, with rapid and asymmetric power and recovery strokes. As the anisotropy decreases, burst-and-coast motions change to motions with long power strokes and short recovery strokes, and roughly constant interlink angle velocity on each. These motions are seen in the spaces of sinusoidal and power-law motions described by two and five parameters, respectively. Allowing the duty cycle to vary greatly increases the motions' efficiency compared to the case of symmetric power and recovery strokes. Allowing further variations in the concavity of the power and recovery strokes improves the efficiency further only when friction is very anisotropic. Near isotropic friction, a variety of optimally efficient motions are found with more complex waveforms. Many of the optimal sinusoidal and power-law motions are similar to those that we find with an optimization search in the space of more general periodic functions (truncated Fourier series). When we increase the resistive force's power-law dependence on velocity, the optimal motions become smoother, slower, and less efficient, particularly near isotropic friction.

Exact solution of the two-dimensional scattering problem for a class of δ-function potentials supported on subsets of a line

We use the transfer matrix formulation of scattering theory in two-dimensions (2D) to treat the scattering problem for a potential of the form where , and b are constants, is the Dirac δ function, and g is a real- or complex-valued function. We map this problem to that of and give its exact (nonapproximate) and analytic (closed-form) solution for the following choices of : (i) a linear combination of δ functions, in which case is a finite linear array of 2D δ functions; (ii) a linear combination of with real; (iii) a general periodic function that has the form of a complex Fourier series. In particular we solve the scattering problem for a potential consisting of an infinite linear periodic array of 2D δ functions. We also prove a general theorem that gives a sufficient condition for different choices of to produce the same scattering amplitude within specific ranges of values of the wavelength λ. For example, we show that for arbitrary real and complex parameters, a and , the potentials and have the same scattering amplitude for .

Analytical Method for Stroboscopically Sampling General Periodic Functions With Arbitrary Frequency Sweep Rates

Parameter sweeps are commonly used to explore the behavior of dynamical systems. This paper derives exact solutions for the instances in time to stroboscopically sample the response of a dynamical system subject to varying input excitations. This work will enable more accurate bifurcation diagrams and Poincaré sections in parameter regimes where numerical approaches may lead to incorrect behavior characterization. The simplest case of a linear frequency sweep is first considered before generalizing the results to include more complex functions with nonlinear sweep rates and arbitrary phase shifts.

Repetitive learning control for a class of partially linearizable uncertain nonlinear systems

The repetitive learning control for a class of partially feedback linearizable systems is considered. The signal to be tracked is a general periodic function with known period. The considered system, especially, the unlinearizable subsystem of it, contains unknown functions. Only the desired state variables of the linearizable subsystem can be used in control law design. First, some system constants related to the desired system and error system are presented. Then based on these system constants, a state feedback repetitive control algorithm is proposed, which guarantees that all the signals in the closed-loop systems are bounded and the tracking error converges to zero asymptotically. A simulation example is presented to show the utility of the proposed control algorithm.

Elliptic inflation: interpolating from natural inflation to R 2-inflation

We propose an extension of natural inflation, where the inflaton potential is a general periodic function. Specifically, we study elliptic inflation where the inflaton potential is given by Jacobi elliptic functions, Jacobi theta functions or the Dedekind eta function, which appear in gauge and Yukawa couplings in the string theories compactified on toroidal backgrounds. We show that in the first two cases the predicted values of the spectral index and the tensor-to-scalar ratio interpolate from natural inflation to exponential inflation such as R 2- and Higgs inflation and brane inflation, where the spectral index asymptotes to n s = 1 − 2/N ≃ 0.967 for the e-folding number N = 60. We also show that a model with the Dedekind eta function gives a sizable running of the spectral index due to modulations in the inflaton potential. Such elliptic inflation can be thought of as a specific realization of multi-natural inflation, where the inflaton potential consists of multiple sinusoidal functions. We also discuss examples in string theory where Jacobi theta functions and the Dedekind eta function appear in the inflaton potential.

Ground state solutions for a diffusion system

In this paper, we study the following diffusion system {∂tu−Δxu+V(x)u=g(t,x,v),−∂tv−Δxv+V(x)v=f(t,x,u) where z:=(u,v):R×RN→R2, V(x)∈C(RN,R) is a general periodic function. Under weaker conditions on nonlinearity, we establish the existence of ground state solutions via variational methods.

Solution of a one-dimensional torsion Schrödinger equation with a general periodic potential

Relations are found for the calculation of eigenvalues and eigenfunctions of the Hamiltonian of internal rotational motion in molecules in the basis of plane waves. The dependences of kinematic coefficient F(φ) and potential V(φ) on dihedral angle φ are represented by Fourier series for both symmetric and asymmetric functions, as well as for general periodic functions. If a molecule has symmetry elements, the found solution transforms to that previously known.

Homoclinic orbits for an infinite dimensional Hamiltonian system with periodic potential

In this paper, we study the following diffusion system Open image in new window where \(z:(u,v):\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{2}\), \(V(x)\in C(\mathbb{R}^{N},\mathbb{R})\) is a general periodic function, g(t,x,v), f(t,x,u) are periodic in t,x and superquadratic in v,u at infinity. By using much more direct methods to prove all Cerami sequences for the energy functional are bounded and establish the existence of homoclinic orbits, which are ground state solutions for above system.

Asymptotic behavior of the mean square displacement of the Brownian parametric oscillator near the singular point

A parametric oscillator with damping driven by white noise is studied. The mean square displacement (MSD) in the long-time limit is derived analytically for the case that the static force vanishes, which was not treated in the past work (Tashiro and Morita 2007 Physica A 377 401). The formula is asymptotic but is applicable to a general periodic function. On the basis of this formula, some periodic functions reducing MSD remarkably are proposed.

Dynamics of Nutrient-Phytoplankton Interaction in the Presence of Viral Infection and Periodic Nutrient Input

Chattopadhyay et al. [Biosystems (2003), 68, pp. 5-17] proposed and analyzed an N – P model in the presence of viral infection on phytoplankton population. They studied the dynamics under the constant nutrient input. The present paper deals with the problem with seasonal variability on nutrient input. We use a general periodic function for nutrient input. We observe the dynamics of the system by considering (i) the infected phytoplankton consumes nutrient and (ii) the infected phytoplankton is not in a state to consume nutrient. Conditions for the persistence and extinction of populations are worked out. Our numerical experiments show that if the infected phytoplankton does not take nutrient then susceptible phytoplankton coexists with the infected ones. But if the infected phytoplankton consumes nutrient then there is a chance for extinction of susceptible phytoplankton for high rate of infection. We also observe that periodic nutrient input enforces the system to enter into chaotic region.

On generic frequency decomposition. Part 1: Vectorial decomposition

The famous Fourier theorem states that, under some restrictions, any periodic function (or real world signal) can be obtained as a sum of sinusoids, and hence, a technique exists for decomposing a signal into its sinusoidal components. From this theory an entire branch of research has flourished: from the short-time or windowed Fourier transform to the wavelets, the frames, and lately the generic frequency analysis. The aim of this paper is to take the generic frequency analysis a step further. When a broader paradigm of functional analysis is defined, a much larger class of functions than those initially introduced in [Y. Wei, N. Chen, Square wave analysis, J. Math. Phys. 39 (8) (1998); Y. Wei, Frequency analysis based on general periodic functions, J. Math. Phys. 40 (7) (1999); Y. Wei, Frequency analysis based on easily generated functions, Appl. Comput. Harmon. Anal. (2000)] can be used to assemble bases in L^2. New methods and algorithms can be employed in function decomposition on such generic bases. It will be shown that these algorithms are a generalization of the Fourier analysis, i.e., they are reduced to the familiar Fourier tools when using orthogonal bases. The differences between the generic frequency analysis and the wavelets and frames theories will be discussed. Examples of analysis and reconstruction of functions using the given algorithms and generic bases will be given. In this first part the focus will be on vectorial decomposition, while the second part will be on polar decomposition with its consequences and applications.

Frequency analysis based on general periodic functions

Following the sine–cosine function, the sawtooth wave, square wave, triangular wave, trapezoidal wave, and so on become new easily generated periodic functions in modern electronics. Similar to Fourier’s idea, a natural question is whether a signal can be considered as a superposition of easily generated functions with different frequencies. Therefore it is necessary to generalize Fourier analysis based on sine–cosine functions into frequency analysis based on general periodic functions. In this paper, we introduce the frequency series and frequency transformation based on general periodic functions. We discuss when a frequency system is a complete system or an unconditional basis in L2[−π,π], and when a frequency transformation can be carried out in L2(−∞,+∞). For practical convenience almost all easily generated functions in electronics are considered carefully as examples. As a new and practical generalization of classical Fourier analysis, these results will become a theoretical foundation for the technique of easily generated function analysis in signal processing.

Analytical and numerical modeling of steady periodic heat transfer in extended surfaces

This paper deals with the analytical and numerical approaches that have been used to study periodic or oscillatory heat transfer processes occurring in extended surfaces. The details pertain to harmonic oscillations but many of the methods can be applied to more general periodic functions. For linear problems, the techniques include complex combination, Laplace transforms, finite differences, and boundary elements. For the nonlinear situations, approaches such as finite differences, finite elments, and different combinations of complex temperature, perturbation, series expansions, straightline, and finite differences have proved effective. Following a brief introduction, the applications of each approach are discussed in detail. Both straight and annular fin configurations are covered and the profile shapes include rectangular, trapezoidal, triangular, and convex parabolic. The periodic conditions involve oscillating base temperature, oscillating base heat flux, oscillating environment temperature, convection at the fin's base through a fluid with oscillating temperature, and some combinations of these conditions. The nonlinear problems discussed cover radiating and convecting-radiating fins, fins with variable thermal conductivity and coordinate dependent heat transfer coefficients, and systems with fin-to-fin, fin-to-base, and fin-to-environment radiative interactions