Ordinary Fourier Series Research Articles

A modified variant of coyote optimization algorithm for solving ordinary differential equations and oscillatory mechanical problems

A vital subject of engineering structures is mechanical oscillations. If the mechanical oscillation is uncontrolled, it can lead to structural failure due to large dynamic stresses developed, as the collapse occurred on the Broughton Suspension bridge due to soldiers walking in step. This paper addresses mechanical oscillation problems employing a proposed modified variant of coyote optimizer. Coyote optimization algorithm (COA) is a new meta-heuristic which mimics the social behavior of coyotes. COA suffers with stagnation problems and immature convergence while solving optimization problems. In this paper, the COA is hybridized with Laplace Crossover operator and new culture tendency strategies are adapted, modified variants of coyote optimization algorithm (MvCOA). The proposed MvCOA is presented to approximately solve mechanical oscillation problems independently of their order, form, and stated conditions. With the fundamental concepts of ordinary differential equations and Fourier series expansion, mechanical oscillation problems can be modeled as a problem of optimization whereby the optimization task is achieved using the proposed MvCOA. Five different ordinary differential equations and four mechanical oscillation problems are solved approximately and then compared with their corresponding exact solutions. The statistical analysis validates that the presented MvCOA is an effective algorithm for different optimization problems. However, from the empirical results, it is visible that the suggested MvCOA approximate approach was able to reach a successful performance solving different mechanical oscillation problems.

Pressure Control of High Pressure Tubing Based on Fluid Mechanics

This paper studies the problem of the constant pressure control for high pressure tubing. The main factors affecting the pressure change in high pressure oil pipe are the opening time of check valve and the cam angular velocity. Based on the theory of fluid mechanics, ordinary differential equation and Fourier series, the flow-pressure model and pressure-cam angular velocity model are established, respectively. Furthermore, the above model is extended to the case of multiple nozzles. These three models accurately express the functional relationship between the fuel flow rate, the pressure in the tubing and the cam angular velocity. According to the established models, the pressure in the high-pressure oil pipe can be accurately controlled by adjusting some parameters. Finally, a simulation example is given to verify the effectiveness of the proposed method.

Highly nonlinear wind waves in Currituck Sound: dense breather turbulence in random ocean waves

We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of nonlinear Fourier analysis (NLFA) methods (Osborne 2010) to analyze the data set: The approach spectrally decomposes a nonlinear wave field into sine waves, Stokes waves, and phase-locked Stokes waves otherwise known as breather trains. Breathers are nonlinear beats, or packets which “breathe” up and down smoothly over cycle times of minutes to hours. The maximum amplitudes of the packets during the cycle have a largest central wave whose properties are often associated with the study of “rogue waves.” The mathematical physics of the nonlinear Schrodinger (NLS) equation is assumed and the methods of algebraic geometry are applied to give the nonlinear spectral representation. The distinguishing characteristic of the NLFA method is its ability to spectrally decompose a time series into its nonlinear coherent structures (Stokes waves and breathers) rather than just sine waves. This is done by the implementation of multidimensional, quasi-periodic Fourier series, rather than ordinary Fourier series. To determine preliminary estimates of nonlinearity, we use the significant wave height Hs, the peak period Tp, and the length of the time series T. The time series analyzed here have 8192 points and T =1677.72 s = 27.96 min. Near the peak of the storm, we find Hs ≈ 0.55 m, Tp ≈ 2.4 s so that for the wave steepness of a near Gaussian process, \({S} = \left (\pi ^{5/2}/g\right )H_{s}/{T}_{p}^{2}\), we find S ≈ 0.17, quite high for ocean waves. Likewise, we estimate the Benjamin-Feir (BF) parameter for a near Gaussian process, \({I_{BF}} = \left (\pi ^{5/2}/g \right ) H_{s} T/{T}_{p}^{3}\), and we find IBF ≈ 119. Since the BF parameter describes the nonlinear behavior of the modulational instability, leading to the formation of breather packets in a measured wave train, we find the IBF for these storm waves to be a surprisingly high number. This is because IBF, as derived here, roughly estimates the number of breather trains in a near Gaussian time series. The BF parameter suggests that there are roughly 119 breather trains in a time series of length 28 min near the peak of the storm, meaning that we would have average breather packets of about 14 s each with about 5-6 waves in each packet. Can these surprising results, estimated from simple parameters, be true from the point of view of the complex nonlinear wave dynamics of the BF instability and the NLS equation? We analyze the data set with the NLFA to verify, from a nonlinear spectral point of view, the presence of large numbers of breather trains and we determine many of their properties, including the rise time for the breathers to grow to their maximum amplitudes from a quiescent initial state. Energetically, about 95% of the NLFA components are found to consist of breather trains; the remaining small amplitude components are sine and Stokes waves. The presence of a large number of densely packed breather trains suggests an interpretation of the data in terms of breather turbulence, highly nonlinear integrable turbulence theoretically predicted for the NLS equation, providing an interesting paradigm for the nonlinear wave motion, in contrast to the random phase Gaussian approximation often considered in the analysis of data.

Basis properties of the eigenfunctions of two-interval Sturm–Liouville problems

In this paper a Sturm–Liouville equation together with eigenparameter-dependent boundary-transmission conditions are considered on two disjoint intervals. We construct the resolvent operator and Green’s function and obtain asymptotic approximate formulas for eigenvalues and corresponding eigenfunctions. The obtained results are implemented to the investigation of the basis properties of the system of eigenfunctions in the Lebesgue space $$L_2$$ with new measures. In particular, we show that the eigenfunction expansion series regarding the convergence behaves in the same way as an ordinary Fourier series.

An equiconvergence theorem for an integral operator with a variable upper limit of integration

We suggest simple sufficient conditions on the kernel of the integral operator {fx188-01} providing expansion with respect to the root functions to be equiconvergent with ordinary Fourier series.

On equiconvergence of spectral expansions of self-adjoint integral operators

The paper gives simple sufficient conditions on the kernel of the self-adjoint integral operator considered ensuring the equiconvergence of expansions in eigenfunctions and associated functions with the ordinary trigonometric Fourier series expansions. The presented arguments correct inaccuracies in [1].

How well does the finite Fourier transform approximate the Fourier transform?

AbstractWe show that the answer to the question in the title is “very well indeed.” In particular, we prove that, throughout the maximum possible range, the finite Fourier coefficients provide a good approximation to the Fourier coefficients of a piecewise continuous function. For a continuous periodic function, the size of the error is estimated in terms of the modulus of continuity of the function. The estimates improve commensurately as the functions become smoother. We also show that the partial sums of the finite Fourier transform provide essentially as good an approximation to the function and its derivatives as the partial sums of the ordinary Fourier series. Along the way we establish analogues of the Riemann‐Lebesgue lemma and the localization principle. © 2004 Wiley Periodicals, Inc.

The asymmetrical mixed temperature problem for a transversely isotropic elastic layer

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic elastic layer (soft plate) investigated is in ideal contact with an absolutely rigid layer, deformable only by thermal expansion. The generalized plane temperature problem reduces to determining the stress-strain state of the soft anisotropic layer investigated using the equations of the mixed problem of elasticity theory. At the ends of the boundary layer of the soft plate (a thin contact layer), no conditions are imposed. On the remaining part of the ends of the soft plate, the boundary conditions correspond to a free boundary. The problem has a bounded smooth solution. Unlike the approach described earlier [1], it is proposed to seek an accurate solution in the form of ordinary Fourier series with respect to a single longitudinal coordinate. Solutions in polynomials are also used. It is shown that the existence of these solutions in polynomials enables the convergence of the Fourier series to be improved considerably.

The mixed symmetrical temperature problem for a transversely isotropic elastic layer

The problem of the uniform heating of a symmetrical three-layer plate with absolutely rigid outer layers, deformed solely due to thermal expansion, is solved. The generalized plane temperature problem is reduced to determining the stress-strain state, which is symmetrical with respect to two coordinates, of the inner layer (a soft filler) of transversely isotropic material using the equations of the mixed problem of elasticity theory. The layers are in ideal mutual contact. The conditions at the ends of the filler boundary layer (a thin contact layer) are not specified. On the remaining part of the ends of the filler the boundary conditions correspond to a free boundary. The problem has a finite smooth solution. The construct the exact solution a modification of Mathieu's method [1] is proposed, which consists of the fact that, in addition to ordinary Fourier series, solutions in polynomials are used. It is shown that the presence of these solutions in polynomials enables the convergence of the Fourier series to be accelerated considerably.

The generalized-plane problem of the theory of elasticity on the rotation of a right prism with a square cross-section

The problem of the generalized-plane deformation of a rotating long square elastic prism is solved. The ends of the prism are load free. A modification of Mathieu's method for constructing the solution of this problem with mass forces is proposed. The proposed modification involves the use of the polynomial solutions of a biharmomic equation in addition to ordinary Fourier series. The existence of these polynomial solutions enables one to increase the convergence of the Fourier series substantially. The stressed state of the prism and the distortion of its faces are investigated.

Analogs of Fourier series for a relativistic string model with massive ends

A description of the motions of a relativistic string with massive ends (for the model with a finite mass on the first end and an infinite mass on the second end) is proposed. This approach uses the expansion of the string world surface into a series that is not reduced to the ordinary Fourier series due to the nonlinearity of the problem. The state equation of the string is derived from the mass shell condition for its end. The string motions are classified, allowing linearization of the boundary condition by a natural parametrization of the trajectory of the moving end. The set of such world surfaces is shown to be limited; for the special cases of 2+1 - and 3+1-dimensional Minkowski spaces, all of them reduce to a helicoid.

INTEGRATION OF FOURIER–LEGENDRE SERIES

It is well known than an ordinary (trigonometric) Fourier series “can” be integrated term by term; that is, whether the Fourier series of an integrable function \(f\) is convergent or not, the series obtained by term by term integration of it is convergent to the corresponding integral of \(f\). There is also a more general theorem in which \(f\) and its Fourier series are first multiplied by a function of bounded variation before integrating. This paper aims to obtain similar theorems for the Fourier-Legendre series discussed by Love and Hunter in Proc. London Math. Soc (3) 64 (1992) 579-601 and by Love in that journal (3) 69 (1994) 629-672. Most of the paper is occupied by lemmas which lead to establishing the dominated convergence of the partial sums of the Fourier-Legendre series under certain conditions. For several reasons one of these conditions is that the interval of integration must be a closed subinterval of \((-1,1)\), the interval on which the Legendre functions \(P_\nu^\mu (x)\) are defined; further, I have only had success with integration over subintervals of \([- \frac{1}{2} \sqrt{3}, \frac{1}{2}\sqrt{3}]\).

Bogolyubov averaging and normalization procedures in nonlinear mechanics. III

We describe the technique of normalization based on the method of asymptotic decomposition in the space of representation of a finite-dimensional Lie group. The main topics of the theory necessary for understanding the method are outlined. Models based on the Van der Pol equation are investigated by the method of asymptotic decomposition in the space of homogeneous polynomials (the space of representation of a general linear group in a plane) and in the space of representation of a rotation group on a plane (ordinary Fourier series). The comparison made shows a dramatic decrease in the necessary algebraic manipulations in the second case. We also discuss other details of the technique of normalization based on the method of asymptotic decomposition.

Recent contributions to the theory of non-absolutely convergent fourier transforms

It's an old idea which goes back essentially to Fourier's pioneering works on the suject, to establish a direct connection between Fourier's double integral and the ordinary fourier series of the pertinent function. But this point of view remained a heuristic expedient up to now, and the rigorous foundation of Fourier transforms was built on the whole independently of the convergence theory of Fourier series-though naturally on the analogy of this field. The situation is somewhat similar also about the elegant results of Plancherel concerning square integrable functions and convergence in mean. In th paper [4] written in German, the author has dealed with some exact results of this kind, which will appear in more detailed form in the chapter on Fourier integrals of the monograph [5]. The present paper aims to summing up the most general devlopments found recently about the problem, together with certain applications. (See Theorems 1.-3).

Numerical construction of nonlinear wave-train solutions of the periodic Korteweg-de Vries equation.

I discuss a general approach for the numerical construction of exact, nonlinear wave-train solutions to the periodic Korteweg--de Vries (KdV) equation. The method is based upon the periodic inverse scattering transform (IST), a nonlinear generalization of ordinary Fourier series. In this approach, the solution to the KdV equation is represented by a linear superposition of nonlinearly interacting ``hyperelliptic functions'' which are the nonlinear ``oscillation modes'' or ``degrees of freedom'' of the equation; the amplitudes of the nonlinear modes are the constants of the motion for KdV evolution. Using the periodic IST formulation, I numerically construct several low-degree-of-freedom wave trains and discuss some of their physical properties. The approach given here depends explicitly on the application of methods from the field of algebraic geometry. Most of the examples presented are solutions to the KdV equation which have not been previously considered; the solutions are ``complex'' in the sense that, instead of being a single cnoidal wave, they are ``multicnoidal'' or ``polycnoidal.'' The IST spectrum often provides a much simpler interpretation of the wave motion than that given by the linear Fourier transform. This occurs primarily because the nonlinear wave trains constructed herein have a small number of IST modes; on the other hand, these wave trains generally require a large number of linear Fourier modes for their description.

The numerical inverse scattering transform for the periodic Korteweg-de Vries equation

We introduce inverse scattering transform (IST) algorithms for obtaining the spectrum of complex wave trains governed by the periodic Korteweg-de Vries (KdV) equation and for constructing solutions to this equation. A scattering matrix formulation is implemented to determine the Floquet spectrum and an iterated similarity transformation is exploited to compute the hyperelliptic osscilation modes. A linear superposition law of the hyperelliptic modes, a generalization of ordinary Fourier series, is used to construct general wave train solutions to the KdV equation. The algorithms should be useful for nonlinear Fourier analysis of computer generated wave trains and experimentally measured space or time series.

A generalization of nonharmonic Fourier series

It is shown that the theory of nonharmonic Fourier series is a special case of a general method for perturbing bases in Banach spaces, and that even in the classical case, there are many ways of perturbing ordinary Fourier series while still preserving the norm and pointwise convergence properties.

Spectral methods using rational basis functions on an infinite interval

By using the map y = L cot( t) where L is a constant, differential equations on the interval y ϵ [− ∞, ∞] can be transformed into t ϵ [0, π] and solved by an ordinary Fourier series. In this article, earlier work by Grosch and Orszag ( J. Comput. Phys. 25, 273 (1977)), Cain, Ferziger, and Reynolds ( J. Comput. Phys. 56, 272 (1984)), and Boyd ( J. Comput. Phys. 25, 43 (1982); 57, 454 (1985); SIAM J. Numer. Anal. (1987)) is extended in several ways. First, the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y-plane. Second, Galerkin's method will convert differential equations with polynomial or rational coefficients into banded matrix problems. Third, with orthogonal rational functions it is possible to obtain exponential convergence even for u( y) that asymptote to a constant although this behavior would wreck alternatives such as Hermite or sinc expansions. Fourth, boundary conditions are usually “natural” rather than “essential” in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions. Fifth, mapping a finite interval to an infinite one and then applying the rational Chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities. These concepts are illustrated by five numerical examples.

Nonharmonic Fourier Series and Spectral Theory

We consider the problem of using functions ${g_n}(x): = exp(i{\lambda _n}x)$ to form biorthogonal expansions in the spaces ${L^p}( - \pi , \pi )$, for various values of $p$. The work of Paley and Wiener and of Levinson considered conditions of the form $\left | {{\lambda _n} - n} \right | \leq \Delta (p)$ which insure that $\{ {g_n}\}$ is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for $p = 2$. In this paper, rather than imposing an explicit growth condition, we assume that $\{ {\lambda _n} - n\}$ is a multiplier sequence on ${L^p}( - \pi , \pi )$. Conditions are given insuring that $\{ {g_n}\}$ inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, ${\lambda _n}$ and ${g_n}$ are shown to be the eigenvalues and eigenfunctions of an unbounded operator $\Lambda$ which is closely related to a differential operator, $i\Lambda$ generates a strongly continuous group and $- {\Lambda ^2}$ generates a strongly continuous semigroup. Half-range expansions, involving ${\text {cos}}{\lambda _n}x$ or ${\text {sin}}{\lambda _n}x$ on $(0, \pi )$ are also shown to arise from linear operators which generate semigroups. Many of these results are obtained using the functional calculus for well-bounded operators.

Nonharmonic Fourier series and spectral theory

We consider the problem of using functions g n ( x ) := e x p ( i λ n x ) {g_n}(x): = exp(i{\lambda _n}x) to form biorthogonal expansions in the spaces L p ( − π , π ) {L^p}( - \pi ,\,\pi ) , for various values of p p . The work of Paley and Wiener and of Levinson considered conditions of the form | λ n − n | ≤ Δ ( p ) \left | {{\lambda _n} - n} \right | \leq \Delta (p) which insure that { g n } \{ {g_n}\} is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for p = 2 p = 2 . In this paper, rather than imposing an explicit growth condition, we assume that { λ n − n } \{ {\lambda _n} - n\} is a multiplier sequence on L p ( − π , π ) {L^p}( - \pi ,\,\pi ) . Conditions are given insuring that { g n } \{ {g_n}\} inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, λ n {\lambda _n} and g n {g_n} are shown to be the eigenvalues and eigenfunctions of an unbounded operator Λ \Lambda which is closely related to a differential operator, i Λ i\Lambda generates a strongly continuous group and − Λ 2 - {\Lambda ^2} generates a strongly continuous semigroup. Half-range expansions, involving cos λ n x {\text {cos}}{\lambda _n}x or sin λ n x {\text {sin}}{\lambda _n}x on ( 0 , π ) (0,\,\pi ) are also shown to arise from linear operators which generate semigroups. Many of these results are obtained using the functional calculus for well-bounded operators.