Form Of Infinite Series Research Articles

Incoherent Detection Performance Analysis of the Distributed Multiple-Input Multiple-Output Radar for Rice Fluctuating Targets

Utilizing spatial diversity, the distributed multiple-input multiple-output (MIMO) radar has the potential advantage of improving system detection performance. In this paper, the incoherent detection performance of distributed multiple-input multiple-output (MIMO) radars is investigated for Rice fluctuating targets. To calculate the incoherent detection probability, the moment generating function (MGF) of the Rice variable is expanded as the infinite series form. By inverting the product of MGFs of multiple independent Rice variables, new closed-form expressions for the probability density function (PDF) of the sum of independent and weighted squares of Rice variables are proposed. The proposed PDF expression for the sum of independent, non-identically distributed (i.n.i.d.) Rice variables involves an infinite series in terms of the confluent Lauricella function. Specially, the PDF for the sum of independent identically distributed (i.i.d.) Rice is expressed as the confluent hypergeometric function-based infinite series. In addition, the uniform convergence of the proposed PDF expression is also validated. Using this proposed expression, the closed-form and approximate expressions of the incoherent detection probability of MIMO radar are derived, respectively. Numerically evaluated results are illustrated and compared with Monte Carlo (MC) simulations to validate the accuracy of the derivations.

An Approach to Solve Gas Dynamic Equation by Fuzzy Hilfer Fractional Differential Equation

In this work, Adomian decomposition method (ADM) based on Hilfer derivative is proposed to solve nonlinear fuzzy fractional gas dynamic equation. We reform the provided fractional gas dynamic equation into fuzzy Hilfer fractional differential equation. Subsequently with description of Hilfer fractional derivative and suitable initial conditions under fuzzy sense, we obtain approximate solution to the given problem. This method provides an analytical solution in the form of infinite power series. The behavior of the solution is illustrated using graphical representation.

Incoherent phenomena in anisotropic periodic structures: from modeling to experimental demonstration.

The periodic structures are widely studied in numerous optical applications and there is a number of good tools for numerical modeling of such a structures (for example rigorous coupled-wave analysis, finite-difference time-domain, finite element method etc.). However, when it comes to the modeling of incoherent effects in many cases of practical interest, the current methods are not rigorous enough or depend on computationally demanding averaging of coherent response. In this paper, we present a novel approach to modeling of incoherent effects in structures with lateral periodicity based on scattering matrix formalism, as a way to describe optical response of a structure, and on application of incoherent wave summation in the form of infinite geometric series and generalized Mueller matrix calculus. This method can be combined with any of the existing coherent methods of modeling periodic structures and it offers significantly faster computational performance than partially coherent/incoherent methods based on averaging. It is compared with other methods for modeling of incoherent effects and also with experimental spectroscopic data. This method is then used to explain phenomena emerging from the complex interaction between diffraction grating and thick substrate.

Mathematical analysis of a real time dynamical system: Interaction between Agricultural Variables

In this study, we are interested in the investigation and analysis of a mathematical model for the growth of normal agricultural biomass. The model is divided into four compartments as normal Agricultural assets, auxiliary Agricultural assets, Industrial assets, and Ecospherical assets. We applied the Laplace Decomposition Method to obtain approximate solutions in the form of infinite series and numerical justification was performed on the model parameter values with the aid of Maple 18 software. We studied the effect of varying the growth rate of normal agriculture on the yield of agricultural assets and mathematical analyses of the model equation to its uniqueness and stability of equilibrium are carried out. From the results, it was observed that the yield of normal Agricultural assets decreases due to the decrease in the growth rate coefficient of normal agriculture from 10 % to 90 % and the yield of normal Agricultural assets increases due to the increase in the growth rate coefficient of normal agriculture from 110 % to 190 %. We observed a loss in biodiversity when the growth rate coefficients of normal agriculture decreased from 90 %, 20 %, and 10 %, and a biodiversity gain was observed due to the increase of the growth rate coefficients of normal Agriculture from 110 % to 190 %.

Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases.

Three-dimensional analysis of thermoelastic damping in couple stress-based rectangular plates with nonlocal dual-phase-lag heat conduction

Since thermoelastic damping (TED) plays a decisive role in the functioning of micro/nanoresonators, accurate assessment of its magnitude in these tiny systems is imperative. Successfully simulating the thermomechanical behavior of various mechanical elements requires consideration of heat transfer in all three principal directions. Furthermore, there is no denying the impact of size on the mechanical and thermal domains. In light of these reasons, the current study strives to create a three-dimensional (3D) size-dependent TED model for rectangular micro/nanoplates via the use of the modified couple stress theory (MCST) and nonlocal dual-phase-lag (NDPL) heat transfer model concurrently for the first time. To accomplish this, by applying the Galerkin method to the 3D NDPL-based heat equation, the relation of temperature field is derived. Moreover, the coupled thermoelastic constitutive relations are extracted by including the couple stress effect. Lastly, by exploiting the obtained relations in the energy dissipation approach, an analytical TED formula in the form of infinite trigonometric series is attained. To appraise the rightness of derived formulation, the method of model reduction is applied. Moreover, an elaborate convergence analysis is implemented to realize the number of sufficient terms of the provided solution that result in well-grounded outcomes. A rigorous simulation study is also accomplished on the role of 3D heat conduction, scale parameters of MCST and NDPL model, boundary constraints, geometry and material in the variations of TED. Findings imply the definite impact of heat conduction dimension on TED value in rectangular plates with a large ratio of thickness to length and width.

Analysis of power grid operation modes considering the influence of tooth harmonics

A harmonic analysis of the magnetic field of an asynchronous motor was carried out, taking into account bilateral gearing and magnetic asymmetry of the air gap. The spectrum is obtained in the form of infinite series of spatio-temporal harmonic components. Magnetic asymmetry of an asynchronous motor takes into account static and dynamic eccentricity, magnetic asymmetry of steel. 27 different spatial harmonics of the field in the air gap of the induction motor were obtained. The considered spectrum of harmonics of the magnetic field will be induced in the stator windings by the electromotive force with the corresponding frequencies. The fields with γр ± κ pairs of poles rotate asynchronously with the rotor and are damped by the fields created by the currents of the rotor cage. Thus, the magnetic asymmetry affects the spatial harmonics of the magnetic field, changing the angular velocities of their rotation in the air gap. The appearance of magnetic asymmetry causes a redistribution of the spatio-temporal distribution of the harmonics of the magnetic field of the tooth frequency, which leads to a change in the amplitude and phase relationships between the quantities. This leads to the appearance of an asymmetric system of electromotive force of tooth frequency in the stator windings, and, therefore, to a different effect of the asymmetry of the air gap at the level of symmetrical components. In the general case, at the level of symmetric components, the electromotive force of the gear frequency is influenced by two groups of factors that determine asymmetric or different from the nominal operating modes of an asynchronous motor of the same size. An increase in the magnetic asymmetry of the air gap causes a nonlinear, close to parabolic character of the growth of the existing symmetrical components. Symmetric components of the electromotive force of the toothed frequency cause currents of the corresponding components to flow through the closed circuits of the stator windings of the asynchronous motor and the entire power supply system. The calculation scheme for replacing a three-phase network for toothed harmonics of electromotive force is given. The phase resistances of the windings of rotating electric machines for asymmetric three-phase networks are not the same for different sequences. The main method of calculating the considered network is the method of symmetrical components. The calculation is made for one main phase according to the substitution schemes, which corresponds to the symmetrical components of the electromotive force of the tooth frequency. As a result of theoretical studies, functional dependences of the influence of magnetic asymmetry on the level and distribution of phase currents of the toothed frequency were obtained. The nature of the influence at the level of symmetric components is close to parabolic, and the most sensitive to changes in magnetic asymmetry is the minor symmetric component of the zero sequence. The method of calculating the currents and voltages of the toothed frequencies of asynchronous motors in the conditions of power consumption has been developed. Theoretical propositions are confirmed by experimental studies. It has been established that the levels of toothed harmonics in power supply systems, as a rule, do not exceed 10–15 % of the level of the fundamental frequency. This leads to a decrease in the reliability of the operation of electrical networks and a shortening of the service life of the main energy equipment at the enterprise. However, frequency-amplitude resonance at tooth frequencies causes the greatest threat to the power plants of industrial enterprises. This is due to the specific nature of gear harmonics of asynchronous motors (parity, dependence on load and various technological factors). The biggest threat arises when using the same type of asynchronous motors with similar operating modes. Oscillograms of experimental studies are given.

A study of heat and mass transfer flow of a variable viscosity couple stress fluid between inclined plates

In this paper, the Poiseuille flow of non-isothermal couple stress fluid of Reynolds model between two heated parallel inclined plates has been studied. The strongly nonlinear coupled ordinary differential equations have been investigated using the Asymptotic Homotopy Perturbation method (AHPM) and the Optimal Homotopy Asymptotic Method with DJ Polynomials (OHAM-DJ). Employing both these techniques, the approximate solutions for velocity profile, temperature distributions, shear stress, average velocity and volumetric flow rate have been achieved. The results obtained using both these methods are in the form of infinite series, hence the results can be obtained easily. Comparison of various results, obtained through these methods, is carried out. Here, it is worth to mention that the results gained by both these approaches are in excellent resemblance. The effect of different parameters on the flow problem has been studied numerically as well as presented graphically.

Scattering of moving atmospheric pressure induced tsunamis by bathymetry and coastline

Tsunamis can be generated by a moving atmospheric pressure disturbance. The 2022 Tonga volcanic eruption and tsunami demonstrated the global relevance of such a tsunami generation mechanism. The generated locked and free waves are often scattered by bathymetric variations and coastlines, generating more free waves and resulting in a complex wave field. The physical processes involved can be described by the forced linear shallow water equations. Analytical solutions are sought after for simplified bathymetric variations and coastline configurations. In this paper, a moving atmospheric pressure front, which is uniform in the direction normal to its propagation direction, is considered. The water wave motion is assumed to start from the quiescent condition, so that the incident locked and free waves are related. However, since the wave scattering processes are linear, the solutions obtained in this paper can be used for analyzing scattering of the incident locked and free waves with uncorrelated amplitudes. For one-dimensional horizontal (1DH) problems, the water depth is either a constant or has a sudden change (i.e., a step) with or without a vertically walled coastline and the atmospheric pressure front always moves in the direction normal to the depth contours. In the case where a coastline is considered, the atmospheric pressure front can either move from the land to the sea or from the sea to the land. In the 2DH (two-dimensional horizontal) example, the wave field generated by a moving atmospheric pressure front, sweeping over the a circular island surrounded by a circular shelf, is investigated and discussed. For all the cases considered, analytical solutions in the integral form are obtained using the Fourier transform method. For the 1DH cases, analytical solutions are also presented in the form of infinite series, summing over infinite number of scattered and reflected waves from the bathymetric variations and coastlines. The new solutions reveal both the complexity of the wave scattering process and the significant differences between locked and free waves.

On a boundary-value problem in a bounded domain for a time-fractional diffusion equation with the Prabhakar fractional derivative

We aim to study a unique solvability of a boundary-value problem for a time-fractional diffusion equation involving the Prabhakar fractional derivative in a Caputo sense in a bounded domain. We use the method of separation of variables and in time-variable, we obtain the Cauchy problem for a fractional differential equation with the Prabhakar derivative. Solution of this Cauchy problem we represent via Mittag-Leffler type function of two variables. Using the new integral representation of this two-variable Mittag-Leffler type function, we obtained the required estimate, which allows us to prove uniform convergence of the infinite series form of the solution for the considered problem.

ელექტროგამტარ სითხეში წრიული ფოროვანი ფირფიტის ბრუნვის სტაციონარული ამოცანა გაჟონვის სიჩქარის დიდი მნიშვნელობებისათვის თბოგადაცემით, მაგნიტური ველისა და ჯოულის სითბოს გათვალისწინებით

The stationary problem of rotation of a circular porous plate in an electrically conductive fluid is studied by the sequential approximation method (Green's function and small parameter method) for large values of the suction velocity with heat transfer, taking into account a weak magnetic field. The problem considers the case when the influence of dissipative members on heat transfer is very small, the energy equation takes into account the influence of Joule's heat on heat transfer, and it is assumed that the temperature along with the radius of the plate changes according to the square law. For solving the problem, the Navier-Stokes and energy partial differential equations of fluid motion in a magnetic field are reduced to ordinary nonlinear differential equations using generalized Karman embeddings, the solutions of which are sought in the form of infinite series for small values of the suction parameter. The solution of the problem by the means of Green's function is reduced to the solution of integral-differential equations. Recurrence formulas have been obtained, by the means of which it is possible to calculate the solution with any approximation. The first two approximations are clearly calculated. All kinematic characteristics of fluid flow are calculated. Images for heat transfer and pressure distribution on a circular plate are obtained. The moment of resistance to rotation of the plate and the heat transfer coefficient are also calculated.

Searching for Laurent Solutions of Systems of Linear Differential Equations with Truncated Power Series in the Role of Coefficients

Systems of linear ordinary differential equations with the coefficients in the form of infinite formal power series are considered. The series are represented in a truncated form, with the truncation degree being different for different coefficients. Induced recurrent systems and literal designations for unspecified coefficients of the series are used as a tool for studying such systems. An algorithm for constructing Laurent solutions of the system is proposed for the case where the determinant of the leading matrix of the induced system is not zero and does not contain literals. The series included in the solutions are still truncated. The algorithm finds the maximum possible number of terms of the series that are invariant with respect to any prolongations of the truncated coefficients of the original system. The implementation of the algorithm as a Maple procedure and examples of its usage are presented.

On Some New Properties for the Gamma Functions

In this paper will be presented some new properties in form of the inequalities related to gamma function, product of two gamma functions and some of them will be presented in the form of infinite series and limit where are included the Euler number and the Euler-Mascheroni constant.

The Frequency Spectrum of a Harmonically Radiating Monopole Moving along Arbitrary Trajectories with Application to Helical Orbits

The sound field of a harmonically radiating monopole moving along arbitrary trajectories in three-dimensional space is studied in the frequency range and represented as a convolution integral in Cartesian coordinates. The observation time of the motion of the source can be finite or infinite. This convolution integral is referred to as the “Cartesian Convolution Integral ” and is applied to point sources moving on special orbits described by circular and conical helices. For such helical orbits, an alternative approach in cylindrical or spherical coordinates leads to closed form expressions in the form of infinite series for the frequency spectra. The comparison of the results of the Cartesian Convolution Integral with those of the series expansion shows a very good agreement. Finally, the Cartesian Convolution Integral is used to calculate the spectral sound field of a source moving along an elliptical helix.

Finite-time stability of singular time-delay systems based on a new weighted integral inequality

In this work, we study the finite-time stability for singular time-delay systems. Particularly, we analyze the effects from the exponential weighted function eη(t−s). A new weighted integral inequality in the form of infinite series for solving finite-time stability problems has been derived. Based on that, some sufficient conditions have been obtained to guarantee that the underlying system is finite-time stable. A numerical example is provided to demonstrate the efficiency of our methods.

Analytical Model for Thermoelastic Dissipation in Oscillations of Toroidal Micro/Nanorings in the Context of Guyer–Krumhansl Heat Equation

Thermoelastic dissipation or thermoelastic damping (TED) can restrict the quality factor of micro/nanoring resonators seriously. This paper employs the non-Fourier model of Guyer–Krumhansl (GK model) to render a size-dependent formulation and analytical solution for approximating the amount of TED in micro/nanorings with circular cross-section by inclusion of nonlocal and single-phase-lagging effects. To fulfill this objective, the equation of heat conduction in the ring is first established according to GK model. Then, by placing the temperature distribution obtained from the heat conduction equation in the TED relation defined on the basis of thermal energy approach, an expression in the form of infinite series is given for TED, which includes non-classical parameters of GK model. Finally, after checking the validity of the model through a comparative study, several simulation results are prepared to emphasize on the influence of different factors such as non-classical parameters of GK model, geometry of ring, vibrational mode and ambient temperature on TED value. Numerical examples reveal that the mentioned factors along with the two- or three-dimensional heat transfer (2D or 3D) model have major influences on TED variations.

Study of nonlinear generalized Fisher equation under fractional fuzzy concept

<abstract><p>Fractional calculus can provide an accurate model of many dynamical systems, which leads to a set of partial differential equations (PDE). Fisher's equation is one of these PDEs. This article focuses on a new method that is used for the analytical solution of Fuzzy nonlinear time fractional generalized Fisher's equation (FNLTFGFE) with a source term. While the uncertainty is considered in the initial condition, the proposed technique supports the process of the solution commencing from the parametric form (double parametric form) of a fuzzy number. Next, a joint mechanism of natural transform (NT) coupled with Adomian decomposition method (ADM) is utilized, and the nonlinear term is calculated through ADM. The obtained solution of the unknown function is written in infinite series form. It has been observed that the solution obtained is rapid and accurate. The result proved that this method is more efficient and less time-consuming in comparison with all other methods. Three examples are presented to show the efficiency of the proposed techniques. The result shows that uncertainty plays an important role in analytical sense. i.e., as the uncertainty decreases, the solution approaches a classical solution. Hence, this method makes a very useful contribution towards the solution of the fuzzy nonlinear time fractional generalized Fisher's equation. Moreover, the matlab (2015) software has been used to draw the graphs.</p></abstract>

Displacement-function analysis of local stresses in thick beams on simple supports

A new boundary modeling scheme is introduced for the solution of thick beams on simple supports using the displacement-function elasticity approach. The performance of two sets of mixed boundary conditions at the opposing lateral ends of a quasi-infinitely long beam is investigated for the analytical solution of the thick beam. In this approach, a cropped solution strategy is implemented to obtain the elastic field from the central region of the long beam, in which the solution of the beam is obtained in terms of a single scalar function in the form of infinite trigonometric series. The appropriateness and accuracy of the solutions is verified through the comparison with corresponding solutions obtained by FEM and classical beam theory. The convergence characteristics of the series solution are analyzed in the perspective of required computational effort with varying load distribution factor, beam aspect ratio and far-field span ratio. Finally, the method is applied to analyze the local stress and displacement fields, especially in the immediate neighborhood of the discretely applied loads and supports of thick beams. Significant concentrations of bending and shear stresses are observed, especially at the terminal regions of the discrete loading, which are captured in details and verified to be highly sensitive to the loading span and beam aspect-ratio.

Artificial Intelligence-Aided Computation for Green’s Function of the Spherical Layered Earth

Currently, there are considerable challenges associated with the theoretical derivation and numerical calculation of Green’s function for spherical layered earths. In order to study the coupling effect between wide-area interconnected systems and the earth, this earth model is widely used. Artificial intelligence methods have been applied in order to overcome these challenges. First, Green’s function in the form of infinite Legendre series is derived using intelligent symbolic operations, and the weight function of Legendre series is recursively computed by means of a recursive algorithm. It is on this basis that the intelligent complex image method has been developed for solving spherical layered Green’s functions, which transforms the summation of infinite series of Green’s functions into the superposition of complex image potentials. Additionally, the upper error limit of the algorithm has been determined, and examples have been provided to demonstrate the accuracy of the algorithm. A solution based on multiple precision algorithms has been proposed to resolve the numerical singularity problem and the extremely slow convergence of the spherical layered Green’s function at the earth’s scale. It has been demonstrated that the intelligent complex image method has advantages in terms of computing speed and accuracy. A spherical layered Green’s function at the earth scale is presented in this paper as an effective and intelligent solution.

ელექტროგამტარ სითხეში ფოროვანი წრიული ფირფიტის ბრუნვით წარმოქმნილი სასაზღვრო ფენის სტაციონარული ამოცანა სუსტი მაგნიტური ველისა და თბოგადაცემის გათვალისწინებით ცვლადი გაჟონვის სიჩქარის შემთხვევაში

The paper investigates the stationary task of the boundary layer generated by the rotation of a circular porous plate rotating in an electrically conductive fluid with respect a weak magnetic field and heat transfer when the suction velocity into the plate changes by a linear law. The problem is solved by using the method of successive approximation (Shvec method). For solving it, in the magnetic field electrically conductive fluid flow Navier-Stokes and energy nonlinear differential equations in partial derivatives is used, which is recorded as a system of ordinary nonlinear differential equations using Karman general insertions. The solutions of the obtained system of equations are sought in the form of infinite series. The first two approaches are clearly found. Appropriate equations are obtained to determine the thickness of dynamic and thermal boundary layers and their exact solutions are obtained. All the kinematic characteristics of the fluid flow are calculated and images are obtained for the heat transfer and pressure distribution on the circular plate. The torque of rotation of the plate and the coefficient of heat transfer are also calculated. In the particular case, the relationship between the thicknesses of the dynamic and thermal boundary layers is established.