Series Of Eigenfunctions Research Articles

The influence of difference in the surface properties on the axisymmetric vibrations of an oblate drop in an AC field

The forced oscillations of an incompressible fluid drop in the non-uniform electric field are considered. The external electric field acts as an external force that causes motion of the contact line. In order to describe this contact line motion the modified Hocking boundary condition is applied: the velocity of the contact line is proportional to the deviation of the contact angle and the rate of the fast relaxation processes, whose frequency is proportional to twice the frequency of the electric field. The equilibrium drop has the form of a cylinder bounded by axially parallel solid planes. These plates have different surface (wetting etc.) properties. The solution of the problem is represented as a Fourier series in eigenfunctions of the Laplace operator. The resulting system of heterogeneous equations for unknown amplitudes was solved numerically. The amplitude-frequency characteristics and the evolution of the drop shape are plotted for different values of the problem parameters.

О корректности краевых задач с условиями периодичности для уравнения смешанного типа второго рода

В работе установлены промежутки изменения показателя степени вырождения уравнения смешанного типа с характеристическим вырождением, в которых первая краевая задача и видоизмененная краевая задача (аналог задачи Келдыша) с условиями периодичности поставлены корректно. В случае первой задачи установлен критерий единственности ее решения. Показано, что решение аналога задачи Келдыша единственно с точностью до слагаемого линейной функции. Решения построены в виде суммы ряда по системе собственных функций соответствующей одномерной спектральной задачи. При обосновании сходимости ряда, представляющего решение первой краевой задачи, в классе регулярных решений данного уравнения возникает проблема малых знаменателей более сложной структуры, чем в ранее известных работах, в связи с чем установлена оценка об отделенности от нуля с соответствующей асимптотикой. На основании этой оценки найдены достаточные условия относительно граничных функций для обоснования равномерной сходимости ряда и их производных до второго порядка включительно.

Multidimensional Paley–Zygmund theorems and sharp $L^p$ estimates for some elliptic operators

The goal of the paper is twofold. Firstly we study sufficient conditions of convergence for random series of eigenfunctions in $L^\infty$. The eigenfunctions are considered with respect to a reference elliptic operator like the Laplace-Beltrami operator or a Schrodinger operator with growing potential on the Euclidean space. That is a generalization of an old result due to Paley and Zygmund. Secondly, we obtain a few optimal $L^p$ bounds of eigenfunctions including a generalization of the Bernstein inequality. We show that the previous two themes are intimately linked.

An integral equation technique for the analysis of phased array antenna with matching step discontinuities

Applying the integral equation method of overlapping partial domains and the Schwartz alternating method to solving an electromagnetic wave diffraction problem is considered in this paper. The infinite rectangular waveguide phased array antenna scanning in H plane which waveguides have step matching discontinuities is represented. The whole field definition domain is sliced into three overlapping partial domains. The system of integral representations for unknown Ey components of the electrical field vector in each domain is set up using Greenʼs functions. Unknown functions in each domain are presented as orthogonal eigenfunction series. Using Galerkinʼs procedure, the system of integral representations is reduced to the system of linear equations for unknown expansion coefficients. For Schwartz method the system of integral representation is solved using iterative methods. The dependences of the reflection coefficient magnitude and phase on the value of scan angle are obtained. The comparison of obtained results for particular cases with known ones is performed.

Diffraction of hydroelastic waves by multiple vertical circular cylinders

The diffraction problem of hydroelastic waves beneath an ice sheet by multiple bottom-mounted circular cylinders is considered. The elastic thin-plate theory is adopted to model the ice sheet, while the linearized velocity potential theory adopted for the fluid flow. The velocity potential corresponding to each cylinder is expanded into a series of eigenfunctions, and the total potential is expressed as a summation of these expansions over the entire NC number of cylinders. For each cylinder, the Green’s second identity is used outside its domain to obtain a set of linear equations. For each different cylinder, the domain used is different. NC cylinders give NC sets of coupled linear equations. Investigations are made for different arrangements of cylinders, piercing through ice sheets. Results for the wave forces on the cylinders with clamped and free conditions of the ice edge are obtained. Physical phenomena corresponding to cylinders arranged in square, in an array, in a double-array and in a staggered double array are discussed.

Method of Multiobjective Optimization of Controlled Systems with Distributed Parameters

The constructive method for multicriteria optimization of control processes of deterministic and not fully defined controlled systems with distributed parameters, described by linear multidimensional parabolic partial differential equations with internal and boundary control actions, is proposed. The optimization problem is considered when an uniform approximation accuracy of object’s final state to the required spatial distribution of controlled function is given. The suggested approach is based on the one-criterion option in the form of minimax convolution of the normalized quality criteria and the subsequent transition to the equivalent form of a typical variational problem with constraints. It is applied to the deterministic model of an object described by an infinite system of differential equations with respect to time-dependent modes of the controlled quantity expansion in a series of eigenfunctions of the initial-boundary value problem. Further procedures for the preliminary parametrization of control actions, based on analytical optimum conditions and reduction to semi-infinite programming problems, allow one to find the desired extremals using their Chebyshev properties and fundamental laws of the domain in typical application conditions of estimating the accuracy of approaching the object’s final state to the required one in a uniform metric. The obtained results are extended to the tasks of program control on the principle of guaranteed result by ensembles of object trajectories under conditions of interval uncertainty of the parametric characteristics of the distributed system and multiple external disturbances. A demonstrated example of a multicriteria optimization of an innovation technology of metal induction heating prior a hot forming is of special interest. The typical optimization criteria such as energy consumption, metal loss due to scale formation, and heating accuracy are considered as components of vector optimization criterion.

Plane wave diffraction from a finite soft cone at oblique incidence

The canonical problem of diffraction of the plane acoustic wave on a finite hollow soft cone in the general case of oblique incidence is solved. The corresponding diffraction problem is formulated as a first-kind boundary value problem for the Helmholtz equation with respect to the scattered velocity potential for each perturbed azimuthal harmonic. The diffracted field is presented as a series of eigenfunctions for the subregions formed by the cone. The pertinent diffraction problem is reduced to an infinite system of linear algebraic equations (ISLAE) for each azimuthal harmonic by using the continuity conditions and the orthogonality properties of the associated Legendre functions. The use of the analytical regularization approach transforms the ISLAE to the second kind and allows to justify the truncation method for obtaining the numerical solution in the required class of sequences. The limiting transition to the disc is considered and the simplified regularization operator for this particular case is derived. The low-frequency approximation of the problem is examined and the analytical solution in the explicit form is derived. By means of illustration, we establish the relationship between the wave size of the cone and the order of reduction of the polar and azimuthal harmonics. We examine the scattering features of a finite soft cone as a function of the incident angle. We validate the obtained results by comparing them to the ones known in literature.

Analytical solution to heat transfer in compressible laminar flow in a flat minichannel

Heat transfer in compressible laminar flow in mini-/micro-channels, a classical and general topic in fields of fuel cells, electronics, micro heat exchanger, etc., is revisited. Based on a two-dimensional continuum flow model, analytical solutions of the dimensionless model are achieved in closed-form symbolic algebras of Whittaker eigenfunctions, corresponding to two kinds of boundary conditions with arbitrarily prescribed wall temperature or wall heat flux. As the eigenvalues and eigenfunctions are independent on the dimensionless quantities, which influence the along-the-channel behaviors, the algorithm reveals the common features of compressible laminar thermal flows. The algorithms do not require the assumption of a linear pressure distribution, which is proved to be untenable in some cases (e.g. constant wall heat flux). The algorithms are validated well by the exact (numerical) computations in exemplary cases of both small and moderate Reynolds number, Mach number and Eckert number of air. Although expressed in a series of eigenfunctions, only several terms (sometimes one or two terms) of solutions are required for a practical computation.

Mixed boundary value problems in the theory of elasticity in an infinite strip

We propose a method of solving mixed boundary value problems in the theory of elasticity in an infinite horizontal strip that has two points of change in the type of boundary conditions located on the upper and lower sides of the strip and lying at the ends of a vertical segment—the line of the joint between the left and right half-strips. The solutions to the right and the left of the joint line are represented as series in eigenfunctions that correspond to particular boundary conditions on the horizontal sides of the right and left half-strips. Solution continuity conditions or, on the contrary, discontinuities of the displacements or stresses can be specified at the joint between the half-strips. We illustrate the main idea of the method initially using a mixed boundary value problem for the harmonic equation as an example. Here, biorthogonal functions are constructed, their properties and the properties of the coefficients of expansions are investigated, and examples are given to illustrate the correctness of the obtained solutions. Subsequently we solve a mixed boundary value problem in which two trigonometric systems of functions with different eigenvalues are involved at the joint between the half-strips. We consider examples where (1) displacement and stress continuity conditions, (2) a discontinuity of the longitudinal displacements, and (3) a discontinuity of the transverse displacements are specified at the joint between the half-strips.

Mixed problem for the singular partial differential equation of parabolic type

The scheme for solving of a mixed problem is proposed for a differential equation \[a(x)\frac{\partial T}{\partial \tau}= \frac{\partial}{\partial x} \left(c(x)\frac{\partial T}{\partial x}\right) -g(x)\, T\] with coefficients $a(x)$, $g(x)$ that are the generalized derivatives of functions of bounded variation, $c(x)>0$, $c^{-1}(x)$ is a bounded and measurable function. The boundary and initial conditions have the form $$p_{1}T(0,\tau)+p_{2}T^{[1]}_x (0,\tau)= \psi_1(\tau), q_{1}T(l,\tau)+q_{2}T^{[1]}_x (l,\tau)= \psi_2(\tau), $$ $$T(x,0)=\varphi(x), $$ where $p_1 p_2\leq 0$, $q_1 q_2\geq 0$ and by $T^{[1]}_x (x,\tau)$ we denote the quasiderivative $c(x)\frac{\partial T}{\partial x}$. A solution of this problem seek by the reduction method in the form of sum of two functions $T(x,\tau)=u(x,\tau)+v(x,\tau)$. This method allows to reduce solving of proposed problem to solving of two problems: a quasistationary boundary problem with initial and boundary conditions for the search of the function $u(x,\tau)$ and a mixed problem with zero boundary conditions for some inhomogeneous equation with an unknown function $v(x,\tau)$. The first of these problems is solved through the introduction of the quasiderivative. Fourier method and expansions in eigenfunctions of some boundary value problem for the second-order quasidifferential equation $\big(c(x)X'(x)\big)' -g(x)X(x)+ \omega a(x)X(x)=0$ are used for solving of the second problem. The function $v(x,\tau)$ is represented as a series in eigenfunctions of this boundary value problem. The results can be used in the investigation process of heat transfer in a multilayer plate.

Frequency-dependent time decay of Schrödinger flows

We show that the presence of negative eigenvalues in the spectrum of the angular component of an electromagnetic Schrödinger Hamiltonian $H$ generically produces a lack of the classical time-decay for the associated Schrödinger flow $e^{-itH}$. This is in contrast with the fact that dispersive estimates (Strichartz) still hold, in general, also in this case. We also observe an improvement of the decay for higher positive modes, showing that the time decay of the solution is due to the first nonzero term in the expansion of the initial datum as a series of eigenfunctions of a quantum harmonic oscillator with a singular potential.\linebreak A completely analogous phenomenon is shown for the heat semigroup, as expected.

On Fourier Series in Generalized Eigenfunctions of a Discrete Sturm–Liouville Operator

For semicontinuous summation methods generated by Λ = {λn(h)} (n = 0, 1, 2,...; h > 0) of Fourier series in eigenfunctions of a discrete Sturm–Liouville operator of class B, some results on the uniform a.e. behavior of Λ-means are obtained. The results are based on strong- and weak-type estimates of maximal functions. As a consequence, some statements on the behavior of the summation methods generated by the exponential means λn(h) = exp(−uα(n)h) are obtained. An application to a generalized heat equation is given.

Nonlinear dynamics in problems of stability of complex media

The problem of stability of nonlinear viscoelastic bodies with respect to finite perturbations is considered in this article. The analysis of the basic process of deformation of a viscoelastic medium is reduced to solution of a nonlinear boundary value problem with variable coefficients. Solutions for perturbations of displacements are in the form of series in eigenfunctions. Using the principle of possible displacements the question on stability of the ground state of a variational nonlinear problem is reduced to investigation of stability of the zero solution of an infinite system of ordinary differential equations with constant coefficients. For the resulting system of equations we construct a function, which under certain restrictions on the initial perturbations is a Lyapunov function. The dimension of the strange attractor of the dynamical system which allows to limit the number of terms in a Bubnov-Galerkin set is found.

Oscillations of oblate drop between heterogeneous plates under uniform electric field

The forced oscillations of the incompressible fluid drop under the action of the uniform electric field are considered. In equilibrium, the drop has the form of a circular cylinder bounded axially by the parallel solid planes; the contact angle is right. An incompressible fluid of different density surrounds the drop. The external electric field acts as an external force that causes motion of the contact line. In order to describe this contact line motion, the modified Hocking boundary condition is applied: the velocity of the contact line is proportional to the deviation of the contact angle and the speed of the fast relaxation processes, whose frequency is proportional to twice the frequency of the electric field. The case of heterogeneous plates is investigated. We assume that the Hocking parameter depends on the polar angle in this case. The function describing the change in the coefficient of the interaction between the plate and the fluid (the contact line) is expanded in a series of the Laplace operator eigenfunctions.

The Dynamics of Oblate Drop Between Heterogeneous Plates Under Alternating Electric Field

The dynamics of the incompressible fluid drop under the non-uniform electric field are considered. The drop is bounded axially by two parallel solid planes and the case of heterogeneous plates is investigated. The external electric field acts as an external force that causes motion of the contact line. We assume that the electric current is alternative current and the AC filed amplitude is a spatially non-uniform function. In equilibrium, the drop has the form of a circular cylinder. The equilibrium contact angle is 0.5π. In order to describe this contact line motion the modified Hocking boundary condition is applied: the velocity of the contact line is proportional to the deviation of the contact angle and the speed of the fast relaxation processes, which frequency is proportional to twice the frequency of the electric field. The Hocking parameter depends on the polar angle, i.e. the coefficient of the interaction between the plate and the fluid (the contact line) is a function of the plane coordinates. This function is expanded in a series of the Laplace operator eigenfunctions.

On the role of axial wall conduction in mini/micro counterflow heat exchangers

Understanding the role of multi-dimensional conjugate heat transfer on the thermal efficiency of mini/micro counterflow heat exchangers is critical for the design of high efficient equipment. This topic is addressed here by considering a simple model with exact solution: the laminar counterflow parallel-plate heat exchanger. Using as starting point the eigenfunction series solution recently obtained by the authors, a thorough parametric study is carried out to investigate the role of the two dimensionless parameters involved in multi-dimensional wall conduction: the dimensionless wall thickness, Δw, and the dimensionless wall thermal resistance, κw-1. The analytical eigencondition is first presented and discussed, and the associated eigenvalue spectrum is analyzed using contour plots of the lowest-order eigenvalues in the (Δw,κw-1) plane. The complex task of determining the eigenvalues numerically is largely facilitated by approximate expressions obtained from the asymptotic analysis of the singularities that appear in the eigencondition. The fast evaluation of the eigenvalues makes it possible to obtain contour plots of the heat exchanger effectiveness in the (Δw,κw-1) plane, which exhibit distinguished regimes corresponding to limiting cases with and without axial and transverse wall conduction effects, with smooth transitions occurring for moderately small values of Δw and κw-1. The analysis provides conditions for neglecting axial and transverse wall conduction, and shows that an optimum wall conductivity always exists in heat exchangers with sufficiently thin walls.

On one boundary-value problem for an equation of higher even order

We study boundary-value problem for an equation of even order in a rectangular domain. By the spectralmethod we obtain necessary and sufficient conditions of uniqueness of a solution. The solution is constructed in the formof infinite series in eigenfunctions. We obtain sufficient conditions under which this series is a regular solution.

Boundary problem for the singular heat equation

The scheme for solving of a mixed problem with general boundary conditions is proposed for a heat equation \[a(x)\frac{\partial T}{\partial \tau}= \frac{\partial}{\partial x} \left(\lambda(x)\frac{\partial T}{\partial x}\right)\] with coefficient $a(x)$ that is thegeneralized derivative of a function of bounded variation, $\lambda(x)>0$, $\lambda^{-1}(x)$ is a bounded and measurable function. The boundary conditions have the form $$\left\{ \begin{array}{l}p_{11}T(0,\tau)+p_{12}T^{[1]}_x (0,\tau)+ q_{11}T(l,\tau)+q_{12}T^{[1]}_x (l,\tau)= \psi_1(\tau),\\p_{21}T(0,\tau)+p_{22}T^{[1]}_x (0,\tau)+ q_{21}T(l,\tau)+q_{22}T^{[1]}_x (l,\tau)= \psi_2(\tau),\end{array}\right.$$ where by $T^{[1]}_x (x,\tau)$ we denote the quasiderivative $\lambda(x)\frac{\partial T}{\partial x}$. A solution of this problem seek by thereduction method in the form of sum of two functions $T(x,\tau)=u(x,\tau)+v(x,\tau)$. This method allows to reduce solving of proposed problem to solving oftwo problems: a quasistationary boundary problem with initialand boundary conditions for the search of the function $u(x,\tau)$ and a mixed problem with zero boundaryconditions for some inhomogeneous equation with an unknown function $v(x,\tau)$. The first of these problems is solved through the introduction of the quasiderivative. Fourier method andexpansions in eigenfunctions of some boundary value problem forthe second-order quasidifferential equation $(\lambda(x)X'(x))'+ \omega a(x)X(x)=0$ are used for solving of the second problem. The function $v(x,\tau)$ is represented as a series in eigenfunctions of this boundary value problem. The results can be used in the investigation process of heat transfer in a multilayer plate.

Landau quantized dynamics and spectra for group-VI dichalcogenides, including a model quantum wire

This work is concerned with the derivation of the Green’s function for Landau-quantized carriers in the Group-VI dichalcogenides. In the spatially homogeneous case, the Green’s function is separated into a Peierls phase factor and a translationally invariant part which is determined in a closed form integral representation involving only elementary functions. The latter is expanded in an eigenfunction series of Laguerre polynomials. These results for the retarded Green’s function are presented in both position and momentum representations, and yet another closed form representation is derived in circular coordinates in terms of the Bessel wave function of the second kind (not to be confused with the Bessel function). The case of a quantum wire is also addressed, representing the quantum wire in terms of a model one-dimensional δ(x)-potential profile. This retarded Green’s function for propagation directly along the wire is determined exactly in terms of the corresponding Green’s function for the system without the δ(x)-potential, and the Landau quantized eigenenergy dispersion relation is examined. The thermodynamic Green’s function for the dichalcogenide carriers in a normal magnetic field is formulated here in terms of its spectral weight, and its solution is presented in a momentum/integral representation involving only elementary functions, which is subsequently expanded in Laguerre eigenfunctions and presented in both momentum and position representations.

An analytical symplectic approach to the vibration analysis of orthotropic graphene sheets

A nonlocal continuum orthotropic plate model is proposed to study the vibration behavior of single-layer graphene sheets (SLGSs) using an analytical symplectic approach. A Hamiltonian system is established by introducing a total unknown vector consisting of the displacement amplitude, rotation angle, shear force, and bending moment. The high-order governing differential equation of the vibration of SLGSs is transformed into a set of ordinary differential equations in symplectic space. Exact solutions for free vibration are obtianed by the method of separation of variables without any trial shape functions and can be expanded in series of symplectic eigenfunctions. Analytical frequency equations are derived for all six possible boundary conditions. Vibration modes are expressed in terms of the symplectic eigenfunctions. In the numerical examples, comparison is presented to verify the accuracy of the proposed method. Comprehensive numerical examples for graphene sheets with Levy-type boundary conditions are given. A parametric study of the natural frequency is also included.