Dual Series Equations Research Articles

Hyperslip velocity of melting ice sliding down inclined parallel ridges

A geometric and physical model for melting ice sliding over inclined superhydrophobic (SH) surfaces with parallel ridges is presented. By analyzing the micro-shear flows of molten liquid films between the ice layer and SH surfaces, the hyperslip velocities of melting ice sliding are investigated. The stick-slip boundary condition of the SH surface is used to establish the dual-series equations analytically, and the numerical solutions are implemented by truncating Fourier series and transforming the dual-series equations into linear algebraic equations to determine the hyperslip velocities of melting ice sliding. The numerical results indicate that the non-dimensional hyperslip velocities increase nonlinearly from near 0 to approximately 1.1 for longitudinal sliding and from near 0 to approximately 0.55 for transverse sliding with an increasing air groove ratio (a). The hyperslip velocities increase with increasing δ at the beginning initially (δ < 1), after which they tend toward asymptotic solutions as δ = 1. The hyperslip velocity ratio (Wh/Uh) shows that longitudinal ridges are at least twice as effective as transverse ridges in enhancing the ice hyperslip velocity, with the velocities accounting for more than 60% of the ice sliding velocities for arbitrary θ at a = 0.95 and δ = 0.1. The relative deviations between the numerical and asymptotic solutions are less than 5% at δ = 1, with the maximum relative deviation occurring at a = 0.65 for arbitrary θ.

NATURAL ELECTROMAGNETIC MODES OF A COMPOSITE OPEN STRUCTURE INVOLVING A PERFECTLY CONDUCTING STRIP GRATING, AN INHOMOGENEOUS FERRITE LAYER, AND A MONOLAYER OF GRAPHENE

Subject and Purpose. Сonsidered are the natural modes and their correspondent eigenfrequencies of a composite structure which is nonuniform along one of the coordinates and consists of a lossy ferromagnetic layer placed in a static magnetic field. The layer involves a perfectly conducting strip grating at one of its boundaries and a graphene monolayer at the other. Methods and Methodology. The above stated problem can be approached within the analytical regularization procedure developed for dual series equations. The latter concern a broad class of diffraction problems which include, in particular, the diffraction of monochromatic plane waves on strip gratings placed at the boundary of a gyromagnetic medium. The amplitudes of the electromagnetic eigenmodes can be obtained from the infinite set of homogeneous linear algebraic equations solvable within a truncation technique. The roots of the system’s determinant represent complex-valued eigenfrequencies of the system under investigation. The material parameters adopted in our computations for the ferromagnetic layer correspond to such of yttrium iron garnet. Results. A number of numerical programs have been developed which permit analyzing the dependences of wave field eigenfunctions and complex eigenfrequencies upon geometrical parameters of the structure (such as grating slot width and period, and thickness of the lossy layer), as well as on electrodynamic parameters of the ferromagnet and graphene characteristics, specifically the chemical potential and relaxation energy of electrons. A number of behavioral regularities have been established, as well as the effect of non-uniformity of ferrite layer parameters upon the structure’s eigenfrequencies and wave field eigenfunctions. Conclusions. The structure under study has been shown to be is an open oscillatory system with a set of complex-valued natural frequencies demonstrating finite points of accumulation. The real parts of these eigenfrequencies lie in a certain interval determined by characteristic frequencies of the ferrite layer, while the imaginary parts are negative, such that the correspondent natural modes show an exponential decay with time. The grating edges represent the mirrors which the natural surface oscillations are reflected from, being supported at that by the ferromagnetic medium. The results obtained in this paper can be useful for creating the elemental base for microwave devices and the devices themselves.

Dual Series Equations to Solve the Laplace Equation with Mixed Boundary Conditions

Dual Series Equations to Solve the Laplace Equation with Mixed Boundary Conditions

A Formal Solution of Quadruple Series Equations

It cannot be overstated how significant Series Equations are to the fields of pure and applied mathematics respectively. The majority of mathematical topics revolve around the use of series. Virtually, in every subject of mathematics, series play an important role. Series solutions play a major role in the solution of mixed boundary value problems. Dual, triple, and quadruple series equations are useful in finding the solution of four part boundary value problems of electrostatics, elasticity and other fields of Mathematical physics. Cooke devised a method for finding the solution of quadruple series equations involving Fourier-Bessel series and obtained the solution using operator theory. Several workers have devoted considerable attention to the solutions of various equations involving for instance, trigonometric series, The Fourier-Bessel series, The Fourier Legendre series, The Dini series, series of Jacobi and Laguerre polynomials and series equations involving Bateman K-functions. Indeed, many of these problems arise in the investigation of certain classes of mixed boundary value problems in potential theory. There has been less work on quadruple series equations involving various polynomials and functions. In light of the significance of quadruple series solutions, proposed work examines quadruple series equations that include the product of r generalised Bateman K functions. Solution is formal, and there has been no attempt made to rationalise many restricting processes that have been encountered.

Advancing contact of a 2D elastic curved beam indented by a rigid pin with clearance

A two-dimensional analytical solution is presented for stresses and displacements in an elastic curved beam forming an incomplete ring in frictionless and unbonded contact with a rigid pin loaded by a point force and in the presence of clearance. The circular beam is modeled as an incomplete elastic thick ring, constrained at both ends and in a plane stress state. The stress and displacement fields within the beam are derived from a biharmonic Airy stress function, according to the Michell solution in polar coordinates. The mixed boundary value problem is reduced to a set of dual series equations and then to a non-homogeneous linear system of infinite equations, which is then solved by truncation. The non-linear relations between the applied load and the contact angle or the pressure distribution are obtained by using an inverse method. The analytical results are compared with finite element predictions for a pin-lug connection and a reasonable agreement is observed for several typical geometries. The peaks of contact pressure and von Mises equivalent stress and their location within the curved beam are evidenced.

Simultaneous Dual Series Equations

Simultaneous Dual Series Equations

Coupling impedance of a PEC angular strip in a vacuum pipe

This study deals with the evaluation of the longitudinal and transverse coupling impedance of a charge travelling in a drift tube with a perfectly electric conductive angular strip. The problem is formulated in the particle frame as dual series equations, and efficiently solved through the representation of the unknown in terms of Neumann series. Then, the electric parameters are obtained in the pipe frame through the Lorentz transforms. The presented solution can be adopted as the general methodology in case of angular discontinuities in particle accelerators.

Numerical solution of a Fredholm integral equation of the second kind involving some dual series equations

In this article, the Laplace equation with nonhomogenous mixed boundary conditions, defined on a surface of semi-infinite cylinder is solved by using dual series equations (DSE). The DSE method is based on transforming the mixed problem to a Fredholm integral equation of second kind, where the resulted integral equation is equipped with kernel defined of infinite integral. We solved the equation by using numerical quadrature method and obtain the numerical solution for the Laplace equation.

H-polarized plane-wave scattering by a PEC strip grating on top of a dielectric substrate: analytical regularization based on the Riemann-Hilbert Problem solution

ABSTRACTWe consider the H-polarized plane wave scattering from an infinite flat grating of perfectly electrically conducting strips, placed on the interface of a dielectric slab. We reduce this problem to a dual series equation for the complex amplitudes of the Floquet spatial harmonics. Then, we perform analytical regularization of this equation, based on the inversion of the static part of the problem with the aid of the Riemann-Hilbert Problem. This yields a Fredholm second-kind infinite matrix equation, numerical solution of which has a guaranteed convergence. Numerical results obtained demonstrate how the rate of convergence depends on the geometrical parameters and then concentrate on the resonance effects in the reflection and transmission. We reveal and discuss ultra-high-Q resonances on the lattice modes of such a composite grating, overlooked in earlier studies.

CERTAIN QUADRUPLE SERIES EQUATIONS INVOLVING LAGUERRE POLYNOMIALS

Srivastava ([13], [15]) has solved dual series equations involving Bateman-k functions and Jacobi polynomials. Srivastava [16] has obtained more results for the Konhauser-biorhogonal set. Lowndes ([3], [4]), Srivastava [12], Lowndes and Srivastava [5], Srivastava[14], Srivastava and Panda [17] have obtained the solution of dual series equations involving Jacobi and Laguerre polynomials and also solved triple series equations involving Laguerre polynomials. Singh, Rokne and Dhaliwal [10] have find out the solution of triple series equations involving Laguerre polynomials in a closed form. Kuldeep Narain ([7], [8]), Rajnesh Krishnan Mudaliar and Kuldeep Narain [6] have solved Certain dual and quadruple series equations involving generalized Laguerre polynomials and Jacobi polynomials as kernels. In the present paper, an exact solution has been obtained for the quadruple series equations involving Laguerre polynomials by Noble [9] modified multiplying factor technique.

Modelling approach for multi‐carrier‐based pulse‐width modulation techniques utilised in asymmetrical cascaded H‐bridge inverters

The increasing use of medium-voltage drives and high-power equipment requires a detailed study on the switching method and topology of multi-level inverters. Asymmetric cascaded H-bridge topologies that have unequal input dc voltages with different devices in various parts of the cascaded H-bridge inverter (CHB) are representative of significant improvements in medium-voltage industrial drives. Various modulation strategies are used in multi-level power conversion applications. In the multi-carrier switching method for asymmetric CHB, the number of switches is less than carriers and is usually used in the off-line switching method. These methods of switching are not applicable to online systems. In this study, equations for combining different pulse-width modulation (PWM) to use the controllability advantage of the multi-carrier method are introduced in asymmetric topologies. Then, the dual Fourier series equations are applied to model each inverter switching relation of the asymmetric cascaded blocks. To model each inverter switching relation of the asymmetric blocks using sinusoidal PWM, the dual Fourier series equations were applied. The presented analytical modelling switching method was validated using simulation and experimental case studies.

Singular integral equations analysis of THz wave scattering by an infinite graphene strip grating embedded into a grounded dielectric slab.

We consider the scattering and absorption of a plane H-polarized THz wave by an infinite periodic graphene strip grating embedded into a grounded dielectric slab. The problem is reduced to the dual series equations, which, in turn, are reduced to the singular integral equation with additional conditions. The numerical method of the solution is based on the Nystrom-type algorithm and has guaranteed convergence. We use the isotropic model of graphene with conductivity described by a scalar function obtained from Kubo formalism. The dependences of the absorption coefficient on the frequency as well as the near-field distribution are presented. They show a variety of plasmon and grating-mode resonances, which are identified and studied. Almost perfect absorption is obtained near these resonances.

Regularization method for solving dual series equations involving heat equation with mixed boundary conditions

Regularization method for solving dual series equations involving heat equation with mixed boundary conditions

Dual Series Method for Solving a Heat Equation with Mixed Boundary Conditions

The paper is devoted to solving a nonhomogeneous nonstationary heat equation in cylindrical coordinates with a nonaxial symmetry. This equation is subjected to nonhomogeneous, mixed, and discontinuous boundary conditions of the second and third kinds that are specified on the disk of a finite cylinder surface. In fact, the solution of the given problem is obtained by using a new type of dual series equations (DSEs) with a Bessel function of the first kind. In particular, DSEs reduce the problem to a Fredholm integral equation of the second kind which can be solved numerically by several efficient methods.

On sliding interface contact in layered smart structures

• A dynamic, rough contact model is developed for smart structures. • Partial contact conditions and limiting cases are concerned. • The electric field and magnetic field are given explicitly. • The critical loading to achieve the full contact is determined. • Influences of the velocity and the gap are demonstrated. Based on the linear piezoelectricity and magnetoelectroelasticity theories, this study presents an exact solution for a frictionless, dynamic periodic contact problem of smart structures. Anisotropic piezoelectric materials and anisotropic magneto-electro-elastic materials both with wavy surfaces moves smoothly. The Galilean transformation and the dual series equation methods are used to derive exact solutions for dynamic wavy surface contact problem. The stresses, displacements, electric field and magnetic field are given explicitly for partial contact conditions. The influences of the velocity and the gap at the crests of the wavy surfaces on various physical quantities are detailed for the selected material coefficients of piezoelectric materials PVDF and magneto-electro-elastic materials BaTiO 3 − CoFe 2 O 4 . The illustrative examples imply that a higher critical loading leads to a full contact under two conditions: (1) at a higher velocity or (2) with an increased gap at the crests of the wavy surfaces. The largest magnitude of the surface normal stresses can be relieved by either lowering the velocity or reducing the gap at the crests of the wavy surfaces. The largest magnitudes of the surface electrical displacements and magnetic inductions occur at the edges of the contact region with spikes occurring, which suggests that the edges of the contact region are the likely locations of potential crack initiation and propagation.

Over-Determined Boundary Value Problem Method in the Theory of Mixed Problems for Acoustic Equations in Spherical Regions

The over-determined boundary value problem method is extended to solve some mixed problems for acoustic equations in spherical coordinates. The solvability conditions of auxiliary overdetermined problems for the coordinate regions are obtained. These conditions are used to move from initial mixed boundary value problems to dual series equations of the standard form and then to infinite sets of linear algebraic equations.

Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges for Arbitrary Meniscus Curvature

We numerically compute Nusselt numbers for laminar, hydrodynamically, and thermally fully developed Poiseuille flow of liquid in the Cassie state through a parallel plate-geometry microchannel symmetrically textured by a periodic array of isoflux ridges oriented parallel to the flow. Our computations are performed using an efficient, multiple domain, Chebyshev collocation (spectral) method. The Nusselt numbers are a function of the solid fraction of the ridges, channel height to ridge pitch ratio, and protrusion angle of menisci. Significantly, our results span the entire range of these geometrical parameters. We quantify the accuracy of two asymptotic results for Nusselt numbers corresponding to small meniscus curvature, by direct comparison against the present results. The first comparison is with the exact solution of the dual series equations resulting from a small boundary perturbation (Kirk et al., 2017, “Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges Accounting for Meniscus Curvature,” J. Fluid Mech., 811, pp. 315–349). The second comparison is with the asymptotic limit of this solution for large channel height to ridge pitch ratio.

Кручение растущего вала

Рассматривается задача о кручении растущего вязкоупругого вала жесткими дисками. Вал имеет форму кругового цилиндра. С его торцами соединены жесткие диски. Изучается процесс непрерывного роста поверхности вала под действием крутящих моментов, приложенных к дискам. Получены и исследованы парные сумматорные уравнения, отражающие математическое содержание задачи на различных стадиях процесса наращивания. Обсуждаются результаты численного анализа и особенности качественного поведения основных механических характеристик.

Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature

We investigate forced convection in a parallel-plate-geometry microchannel with superhydrophobic walls consisting of a periodic array of ridges aligned parallel to the direction of a Poiseuille flow. In the dewetted (Cassie) state, the liquid contacts the channel walls only at the tips of the ridges, where we apply a constant-heat-flux boundary condition. The subsequent hydrodynamic and thermal problems within the liquid are then analysed accounting for curvature of the liquid–gas interface (meniscus) using boundary perturbation, assuming a small deflection from flat. The effects of this surface deformation on both the effective hydrodynamic slip length and the Nusselt number are computed analytically in the form of eigenfunction expansions, reducing the problem to a set of dual series equations for the expansion coefficients which must, in general, be solved numerically. The Nusselt number quantifies the convective heat transfer, the results for which are completely captured in a single figure, presented as a function of channel geometry at each order in the perturbation. Asymptotic solutions for channel heights large compared with the ridge period are compared with numerical solutions of the dual series equations. The asymptotic slip length expressions are shown to consist of only two terms, with all other terms exponentially small. As a result, these expressions are accurate even for heights as low as half the ridge period, and hence are useful for engineering applications.

Dual and triple equations and q-orthogonal polynomials

In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also introduce and solve a system of dual series equations when the kernel is the q-Laguerre polynomials. Examples are included.