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In the mathematical description of various phenomena and processes arising in mathematical physics, electrical engineering, economics, have to deal with matrix differential equations. Therefore, such equations are relevant as for mathematicians and specialists in other fields natural sciences. This article considers quasilinear matrix differential equations with coefficients depicted in the form of absolutely and uniformly convergent Fourier series with slow variable in a sense coefficients and frequency (class F). The differences of the diagonal elements of the matrices of the linear part are pure imaginary, that is, we are dealing with a critical case. But between these diagonal elements assume certain relations that indicate the absence of resonance between the natural frequencies of the system and frequency of external excitation force. The problem is considered establishing signs of existence in such an equation of class solutions F . By means of a number of transformations the equation is reduced to the equation noncritical case, and the solution of the class F of this equation is sought by the method of successive approximations using the principle compression reflections. Then based on the properties of the solutions of the transformed equation, conclusions are drawn about the properties initial equation.

The paper explores the possibility for summing Fourier series nonlinearly via the Pythagorean harmonic mean. It reports on new results for this summability with introduction of new concepts like the smoothing operator and semi-harmonic summation. The smoothing operator is demonstrated to be Kalman filtering for linear summability, logistic processing for Pythagorean harmonic summability and linearized logistic processing for semi-harmonic summability. An emerging direct inapplicability of harmonic summability to seismic-like signals is shown to be resolvable by means of a regularizational asymptotic approach., Comment: 20 pages, 0 figures

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.

We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier coefficients of a suitable periodic function associated with its Abel transform. This allows us to compute the Legendre coefficients of the inverse Abel transform in an easy, fast and accurate way by means of a single Fast Fourier Transform. The algorithm is thus appropriate also for the inversion of Abel integrals given in terms of samples representing noisy measurements. Rigorous stability estimates are proved and the accuracy of the algorithm is illustrated also by some numerical experiments., 19 pages, 4 figures

Functionally graded materials (FGMs) structures are increasingly used in engineering due to their superior mechanical and material properties, and the FGMs plate with cutouts is a common structural form, but research on the vibration characteristics of FGMs plate with cutouts is relatively limited. In this paper, the three-dimensional exact solution for the vibration analysis of FGMs rectangular plate with circular cutouts subjected to general boundary conditions is presented based on the three-dimensional elasticity theory. The displacement field functions are expressed as standard cosine Fourier series plus auxiliary cosine series terms satisfying the boundary conditions in the global coordinate system. The plate with circular cutout is discretized into four curve quadrilateral sub-domains using the p-version method, and then the blending function method is applied to map the closed quadrilateral region to the computational space. The characteristic equation is obtained based on the Lagrangian energy principle and Rayleigh–Ritz method. The efficiency and reliability of proposed method are verified by comparing the present results with those available in the literature and FEM methods. Finally, a parametric study is investigated including the cutout sizes, the cutout positions, and the cutout numbers from the free vibration characteristic analysis and the harmonic analysis. The results can serve as benchmark data for other research on the vibration of FGMs plates with cutouts.

In this paper, a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier–Stokes equations on the three-torus is proposed. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton–Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. The required analytic estimates to verify the contractibility of the operator are presented in full generality and symmetries from the model are used to reduce the size of the problem to be solved. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier–Stokes equations with Taylor–Green forcing.

This article presents the convergence analysis of the fixed-point method (FPM) to model the nonlinear magnetic characteristics of a 2-D magnetostatic problem. In this study, FPM is used as the iterative nonlinear solver of the hybrid analytical modeling (HAM) technique for the accurate computation of the magnetic field distribution. The benchmark consists of a stator with excitation windings, an air gap, and a slotless mover. The relative errors between two successive iterations are calculated using different error estimators: the attraction force on the mover, the Fourier coefficients defined in the air gap, the magnetic flux density, and the magnetic scalar potential distributions. The effect of the number of mesh elements and harmonics on the accuracy and computational cost of the model is investigated for different levels of magnetic saturation. It is observed that the maximum rate of change in the relative difference of attraction force during the iterations is found to be 0.52 under the magnetic saturation. In addition, the absolute error of the attraction force between the developed hybrid model with FPM and the finite element method (FEM) is achieved to be 0.18%, while HAM has approximately three times less number of degrees-of-freedom when compared to FEM.

This paper deals with a new mathematical model of three-dimensional generalized thermoelasticity which has been improved using Lord–Shulman theory. The governing equations on non-dimensional forms have been applied to a three-dimensional half-space subjected to a rectangular moving heat source and traction-free surface by using the Laplace and double Fourier transform techniques. The inverses of the double Fourier and Laplace transforms have been calculated numerically by applying the complex formula of inversion of the transform by of the Fourier expansion method. The numerical results of the temperature increment, strain, stress, and displacement distributions have been represented in graphs for various values of the heat source speed parameter to show its effect on the thermo-mechanical waves. The heat source speed parameter leads to significant effects on both the thermal and mechanical waves.

In this article, a mathematical model representing solution of the nonlinear generalized equal width (GEW) equation has been considered. Here we aim to investigate solutions of GEW equation using a numerical scheme by using sextic B-spline Subdomain finite element method. At first Galerkin finite element method is proposed and a priori bound has been established. Then a semi-discrete and a Crank-Nicolson Galerkin finite element approximation have been studied respectively. In addition to that a powerful Fourier series analysis has been performed and indicated that our method is unconditionally stable. Finally, proficiency and practicality of the method have been demonstrated by illustrating it on two important problems of the GEW equation including propagation of single solitons and collision of double solitary waves. The performance of the numerical algorithm has been demonstrated for the motion of single soliton by computing L ∞ and L 2 norms and for the other problem computing three invariant quantities I 1 , I 2 and I 3 . The presented numerical algorithm has been compared with other established schemes and it is observed that the presented scheme is shown to be effectual and valid.

We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems., Comment: 24 pages, 4 figures, submitted to Computational Methods in Applied Mathematics

An analytical model of laminated composite double-plate system (LCDPS) is established, which efficiently analyzes the common 3D plate structure in engineering applications. The proposed model combines the first-order shear deformation theory (FSDT) and the classical delamination theory, and then the LCDPS’s vibration characteristics are investigated. In the process of analysis, the improved Fourier series method (IFSM) is used to describe the displacement admissible function of the LCDPS, which can remove the potential discontinuities at the boundaries. Five sets of artificial springs are introduced to simulate the elastic boundary constraints, and the restraints of the Winkler elastic layer can be adjustable. The improved Fourier series is substituted into the governing equations and boundary conditions; then, applying the Rayleigh–Ritz method, we take all the series expansion coefficients as the generalized coordinates. After that, a set of standard linear algebraic equations was obtained. On this basis, the natural frequency and mode shapes of the LCDPS can be obtained by solving the standard eigenvalue problem. By the discussion of numerical examples and the comparison with those of the reports in the literature, the convergence and the reliability of the present approach are validated. Finally, the parametric investigations of the free vibration with complex boundary conditions are carried out, including the influence of boundary conditions, lamination scheme, plate geometric parameters, and elastic coefficient between two plates.

We give asymptotic estimates of the variance of the number of integer points in translated thin annuli in any dimension.

It is well known that for higher order elliptic equations, the positivity preserving property (PPP) may fail. In striking contrast to what happens under Dirichlet boundary conditions, we prove that the PPP holds for the biharmonic operator on rectangular domains under partially hinged boundary conditions, i.e., nonnegative loads yield positive solutions. The result follows by fine estimates of the Fourier expansion of the corresponding Green function.

We find two-sides estimates for the best uniform approximations of classes of convolutions of $2\pi$-periodic functions from unit ball of the space $L_p, 1 \le p, Comment: 19 pages

The main purpose of this paper is to establish an analysis model of laminated composite rectangular plate with/without cutouts on the basis of three-dimensional elasticity theory and provide the exact three-dimensional solution. In the present work, the effect of the cutout is considered by subtracting the energies of cutouts from the total energy of the entire plate. The standard three-dimensional trigonometric cosine Fourier series and auxiliary Fourier series are chosen as admissible functions, and the Hamilton&rsquo, s principle and Rayleigh&ndash, Ritz procedure are used to obtain the exact solution. In order to verify the effectiveness and reliability of the proposed method, some numerical results are obtained, and the results are compared with the ones available in the literature or finite element analysis. Finally, the effects of some key parameters which will affect the vibration characteristics are analyzed, and the non-dimensional frequency parameters are obtained, which can serve as benchmark data for the future research.

In this paper, we consider a boundary value problem for a nonlinear partial differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. With respect to the first variable, this equation is a nonlinear fractional differential equation in the positive part of the considering segment and is a second-order nonlinear differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of nonlinear boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the classical solution of the problem are proved for regular values of the spectral parameter. For irregular values of the spectral parameter, an infinite number of solutions of the mixed equation in the form of a Fourier series are constructed.

The conventional frequency domain method (CFDM) and dual-force-based time domain method (DTDM) are often used to solve the steady-state response of system with complex damping under an arbitrary force. However, the calculation efficiency of the DTDM is low due to the straightforward summation operation of series even if the solution of the DTDM is the exact real part of the solution. In addition, since the CFDM only can obtain the real part of solution not the complete solution, it gives misleading information that the solution does not have an imaginary part. In this paper, a fast frequency domain method (FFDM) is proposed to calculate the complete response of complex damping system including the imaginary part with a higher accuracy in a much faster manner. The new FFDM uses half of the Fourier series of the discrete Fourier transform of the actual arbitrary force to construct the Fourier series of the dual force, followed by calculating the time history response using the inverse fast Fourier transform. The new developed method is validated through three numerical examples with harmonic and seismic excitations. The numerical results show that the accuracy of the new FFDM is compatible to the DTDM but with much higher computational efficiency.

International audience; The paper presents methodologies for tracing modal responses of a translating cable. Governing system equations are obtained based on perturbation of the response around the stationary position of the translating catenary. Then, after separation of spatio-temporal system variables, a multiple scale method is endowed for treatment of equations in space. Frequencies and mode shapes of the system are detected by imposing solvability conditions on the system equations and/or via expansion of system variables in the form of Fourier series accompanied by imposing the existence of non-trivial solutions. Finally, obtained results are confronted with those which are obtained by finite difference method.

This paper is devoted to investigating the uniform convergence conditions of Fourier series expansions of continuous functions in terms of eigenfunctions of a Sturm-Liouville problem with eigenparameter in one of the boundary conditions on a closed interval. Such problems are quite common in mathematical physics problems.

A wide range of wave energy applications relies on the accurate estimation of extreme wave conditions, while some of them are frequently affected by directionality. For example, attenuators and terminators are the most common types of wave energy converters whose performance is highly influenced by their orientation with respect to the prevailing wave direction. In this work, four locations in the eastern Mediterranean Sea are selected with relatively high wave energy flux values and extreme wave heights are examined with wave direction as a covariate. The Generalized Pareto distribution is used to model the extreme values of wave height over a pre-defined threshold, with its parameters being expressed as a function of wave direction through Fourier series expansion. In order to be consistent with the analysis obtained from the independent fits for directional sectors, the estimation of parameters is based on a penalized maximum likelihood criterion that ensures a good agreement between the two approaches. The obtained results validate the integration of directionality in extreme value models for the examined locations and design values of significant wave height are provided with respect to direction for the 50- and 100-year return period with bootstrap confidence intervals.

This paper deals with non-self-adjoint second-order differential operators with two constant delays from [π∕2,π)and two potentials from L20,π. We study the inverse spectral problem of recovering operators from their spectra. Four boundary value problems under Robin boundary conditions are considered. It has been proved that delays and the Fourier coefficients of potentials are uniquely determined from the spectra. Finally, potentials are constructed. MSC: 34B24, 34A55, Keywords: Differential operators with delays, Inverse spectral problems, Fourier coefficients

Let f be a continuous function which is periodic with respect to the hexagon lattice, and let A be a lower triangular infinite matrix of nonnegative real numbers with nonincreasing rows. The degree of approximation of the function f by matrix means T A n f of its hexagonal Fourier series is estimated in terms of the modulus of continuity of f.

In this paper, a semianalytical solution for the in-plane vibration analysis of annular panel with arbitrary distribution of internal point constraints is established for the first time. In-plane dynamic behavior of such panel structure is described via energy principle. A modified version of Fourier series is constructed for the in-plane vibration displacement expansion supplemented with the boundary smoothed terms, and the arbitrarily concentrated constraint in each field point is described in conjunction with Dirac delta function. A standard matrix eigenvalue problem containing various in-plane modal information of such annular panel is derived and solved through Rayleigh–Ritz procedure. Several numerical examples are presented to demonstrate the correctness and effectiveness of the proposed model by comparing the results with those from other approaches. Three representative types of point constraints, including point, line, and area configurations, are considered by collection of point constraints, and it is shown that the current model can make an accurate and efficient modal parameter prediction for annular panel with such most general case of point constraints.

We study the observability of the wave equation when the observation set changes over time.

A novel dynamic stiffness formulation which includes the effects of shear deformation and rotatory inertia is proposed to carry out the free vibration analysis of thick rectangular orthotropic plates i.e. the formulation is based on an extension of the Mindlin theory to orthotropic plates in a dynamic stiffness context. The modified trigonometric basis is used to construct the general solution for the free vibration problem of the plate in series form, permitting the derivation of an infinite system of linear algebraic equations which connect the Fourier coefficients of the kinematic boundary conditions of its four edges. The boundary conditions are essentially the deflections and angles of rotations constituting the displacement vector and shear forces and bending moments constituting the force vector. The force-displacement relationship is then obtained by relating the two vectors via the dynamic stiffness matrix. Essentially the ensuing infinite system forms the fundamental basis of the paper, which defines the dynamic stiffness matrix for an orthotropic Mindlin plate in an exact sense. Numerical results are obtained by using the Wittrick-Williams algorithm as solution technique. The theory is first validated using published results and then further illustrated by a series of numerical examples. The paper concludes with significant conclusions.

We study the reciprocity gap method (Andrieux et al., 1999) for the identification of planar cracks in the framework of three dimensional linear elasticity. In order to achieve better accuracy on thick domains, we propose to use polynomial test functions instead of Fourier series. We also propose to improve the solution by various regularization techniques that are tested and compared. Numerical assessments are provided in 2D and 3D.

In this paper, a main theorem dealing with | N ̄ , p n | k summability method has been generalized for φ − | N ̄ , p n ; δ | k summability by using different and general summability factors of Fourier series. Keywords: Summability factors, Absolute matrix summability, Fourier series, Infinite series, Hölder inequality, Minkowski inequality

In this paper, we study the Fourier series related to higher-order Bernoulli functions and give new identities for higher-order Bernoulli functions which are derived from the Fourier series of them.