Eigenvalues Of Boundary Value Problem Research Articles

Investigation of conditions necessary for inception of positive corona in air based on differential formulation of photoionization

Sharp point electrodes generate significant electric field enhancements where electron impact ionization leads to the formation of electron avalanches that are seeded by photoionization. Photoionization of molecular oxygen due to extreme ultraviolet emissions from molecular nitrogen is a fundamental process in the inception of a positive corona in air. In a positive corona system, the avalanche of electrons in the bulk of the discharge volume is initiated by a specific distribution of photoionization far away from the region of maximum electron density near the electrode where these photons are emitted. Here, we present a new approach to finding the inception conditions for a positive corona, which is based on a differential formulation of the photoionization problem. The proposed iterative solution considers the same inception problem that has been solved in the existing literature by using either an integral approach to photoionization or a differential formulation of photoionization and considering the inception problem as a boundary-value eigenvalue problem. The results are validated by comparisons with previous integral formulations and time dynamic plasma fluid solutions in planar and spherical geometries. The results illustrate ideas advanced in Kaptzov (1950 Elektricheskiye Yavleniya v Gazakh i Vacuume p 610) providing a physically transparent connection between an effective secondary electron emission coefficient due to volume photoionization in a positive corona system and the secondary electron emission in conventional Townsend discharge theory. The results also demonstrate the significance of boundary conditions for accurate corona solutions that are based on a differential formulation of photoionization.

ASYMPTOTIC ANALYSIS OF STURM-LIOUVILLE PROBLEM WITH DIRICHLET AND NONLOCAL TWO-POINT BOUNDARY CONDITIONS

In this study, we obtain asymptotic expansions for eigenvalues and eigenfunctions of the one–dimensional Sturm–Liouville equation with one classical Dirichlet type boundary condition and two-point nonlocal boundary condition. We analyze the characteristic equation of the boundary value problem for eigenvalues and derive asymptotic expansions of arbitrary order. We apply the obtained results to the problem with two-point nonlocal boundary condition.

АЛГОРИТМЫ ВЫЧИСЛЕНИЯ СОБСТВЕННЫХ ЗНАЧЕНИЙ ДИСКРЕТНЫХ ПОЛУОГРАНИЧЕННЫХ ОПЕРАТОРОВ, ЗАДАННЫХ НА КВАНТОВЫХ ГРАФАХ

Spectral problems for differential operators defined on quantum graphs are of great scientific interest related to problems in quantum mechanics, computer network modeling, image processing, ranking algorithms, modeling of electrical, and mechanical and acoustic processes, in networks of a diverse nature, in designing nano systems with prescribed properties and in other areas. Theoretical solutions of direct and inverse spectral problems on quantum graphs have been developed, but computational algorithms based on these methods are computationally inefficient. We have not seen any published works that consider examples of numerical solutions of spectral problems on finite connected graphs with a large number of vertices and edges. Therefore, the development of new computationally effective algorithms for numerical solution of spectral problems given on finite connected graphs is urgent. This paper develops a technique for finding the eigenvalues of boundary value problems on finite connected graphs with a required number of vertices and edges. To use this technique, it is necessary to know the eigenvalues and vectors of the eigenfunctions of corresponding unperturbed vector operators which are usually self-adjoint. Finding them manually, if the graph has a large number of vertices and edges, is difficult. This led to writing a package of programs in the mathematical environment Maple to find transcendental equations in the symbolic mode to calculate eigenvalues and find the eigenfunctions of unperturbed boundary value problems. Examples of calculating eigenvalues for a quantum graph which models an anthracene aromatic compound molecule are presented.

Slow Magnetoacoustic Oscillations in Stellar Coronal Loops

Slow magnetoacoustic oscillations in stellar coronal loops with gravitational stratification are analyzed with a numerical solution of the boundary value problem for eigenvalues and eigenfunctions. In this study, we only focus on the resonant periods. The effects of the gravitational stratification, star mass, loop temperature, and loop length on the properties of slow magnetoacoustic oscillations are investigated. It is shown that the discrepancy between stratified and nonstratified loops is higher in density perturbations than in velocity perturbations. When the star has a larger mass, higher coronal temperature, and longer loop, the density perturbations in the stratified loop are significantly different from the harmonic functions. The periods in the stratified loop are slightly longer than in the nonstratified loop. The periods calculated in our model (14–644 minutes) are consistent with the periods of stellar quasi-periodic pulsations observed in both soft X-rays (2–70 minutes) and white lights (8–390 minutes).

ВЛАСНІ ЧАСТОТИ КОЛИВАНЬ ПЛАСТИНИ, ЯКА ПОДІЛЯЄ ДВОШАРОВУ ІДЕАЛЬНУ РІДИНУ З ВІЛЬНОЮ ПОВЕРХНЕЮ У ПРЯМОКУТНОМУ КАНАЛІ

The frequency spectrum of plane vibrations of an elastic plate separating a two-layer ideal fluid with a free surface in a rectangular channel is investigated analytically and numerically. For an arbitrary fixing of the contours of a rectangular plate, it is shown that the frequency spectrum of the problem under consideration consists of two sets of frequencies describing the vibrations of the free surface of the liquid and the elastic plate. The equations of coupled vibrations of the plate and the fluid are presented using a system of integro-differential equations with the boundary conditions for fixing the contours of the plate and the condition for the conservation of the volume of the fluid. When solving a boundary value problem for eigenvalues, the shape of the plate deflection is represented by the sum of the fundamental solutions of a homogeneous equation for a loose plate and a partial solution of an inhomogeneous equation by expanding in terms of eigenfunctions of oscillations of an ideal fluid in a rectangular channel. The frequency equation of free compatible vibrations of a plate and a liquid is obtained in the form of a fourth-order determinant. In the case of a clamped plate, its simplification is made and detailed numerical studies of the first and second sets of frequencies from the main mechanical parameters of the system are carried out. A weak interaction of plate vibrations on vibrations of the free surface and vice versa is noted. It is shown that with a decrease in the mass of the plate, the frequencies of the second set increase and take the greatest value for inertialess plates or membranes. A decrease in the frequencies of the second set occurs with an increase in the filling depth of the upper liquid or a decrease in the filling depth of the lower liquid. Taking into account two terms of the series in the frequency equation, approximate formulas for the second set of frequencies are obtained and their efficiency is shown. With an increase in the number of terms in the series of the frequency equation, the previous roots of the first and second sets are refined and new ones appear.

Asymptotic analysis of Sturm–Liouville problem with nonlocal integral-type boundary condition

In this study, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the one-dimensional Sturm–Liouville equation with one classical-type Dirichlet boundary condition and integral-type nonlocal boundary condition. We investigate solutions of special initial value problem and find asymptotic formulas of arbitrary order. We analyze the characteristic equation of the boundary value problem for eigenvalues and derive asymptotic formulas of arbitrary order. We apply the obtained results to the problem with integral-type nonlocal boundary condition.

Fiedler vector analysis for particular cases of connected graphs

In this paper, we consider a p-Laplacian eigenvalue boundary value problem involving both right Caputo and left Riemann-Liouville types fractional derivatives. To prove the existence of solutions, we apply the Schaefer’s fixed point theorem. Furthermore, we present the Lyapunov inequality for the corresponding problem.

Lyapunov-type inequality for higher order left and right fractional p-Laplacian problems

In this paper, we consider a p-Laplacian eigenvalue boundary value problem involving both right Caputo and left Riemann-Liouville types fractional derivatives. To prove the existence of solutions, we apply the Schaefer’s fixed point theorem. Furthermore, we present the Lyapunov inequality for the corresponding problem.

Smallest eigenvalues for boundary value problems of two term fractional differential operators depending on fractional boundary conditions

Let \(n\ge 2\) be an integer, and let \(n-1<\alpha \le n\). We consider eigenvalue problems for two point \(n-1 ,1\) boundary value problems $$\displaylines{ D_{0+}^{\alpha} u+a(t)u+\lambda p(t)u=0, \quad 0<t<1, \cr u^{(i)}(0)=0, \quad i=0,1,\dots,n-2, \quad D_{0^+}^{\beta} u(1)=0, }$$ where \(0\le \beta \le n-1\) and \(D_{0+}^{\alpha}\) and \(D_{0+}^{\beta}\) denote standard Riemann-Liouville differential operators. We prove the existence of smallest positive eigenvalues and then obtain comparisons of these smallest eigenvalues as functions of both \(p\) and \(\beta\).
 For more information see https://ejde.math.txstate.edu/Volumes/2021/62/abstr.html

An existence result for three solutions to a Dirichlet problem with real parameter

In this article, we establish the existence of solutions for partial differential equations (PDE) with boundary value conditions. For studying the existence result, suppose that some condition to find at least three nontrivial solutions for Dirichlet doubly eigenvalue boundary value problem by using Theorem of three-critical points.

Small vibrations of mechanical systems in the case of degenerate eigenvalues

Free vibrations of mechanical and hydrodynamic systems are considered. Some well-known works in this field are summarized and some aspects of computational nature are discussed. The eigenvalue boundary value problem is solved in the case of multiple roots for a system with n degrees of freedom. The case of the zero root of a semidefinite system is considered. The question of orthogonality of forms of natural oscillations is considered. It is shown that in some cases, the proximity of eigenvalues occurs in problems of dynamics of an aircraft containing elastic containers with a liquid. On the example of a mechanical analog of transverse vibrations of a cylindrical tank with a liquid, the possibility of rotation of the free surface of the liquid is shown.

Numerical Analysis of Dynamical Processes in Inhomogeneous Piezoceramic Cylinders (Review)*

The review is devoted to the numerical solution of new problems of electroasticity, namely, determination of the dynamical characteristics of inhomogeneous piezoceramic waveguides of circular cross-section and inhomogeneous piezoceramic cylinders of finite length. To solve these problems, an effective numerical–analytical approach is used. The approach employs various transformations (special functions, Fourier series expansion, and spline-collocation method), which make it possible to reduce the original three-dimensional partial differential equations of electroelasticity to a boundary-value eigenvalue problem for a system of ordinary differential equations. The system is solved by the method of discrete orthogonalization. Using the results obtained, the features of spectral characteristics in an inhomogeneous structure are studied considering the coupled electric field of the piezoceramic layers. The effect of the inhomogeneity and coupled electric field on the dynamical characteristics of the bodies is studied as well. Much attention is paid to the reliability of the numerical results.

Возникновение вращения жидкости при охлаждении свободной границы

For the Navier-Stokes system and the heat equation with vanishing viscosity, the stationary thermocapillary flow of an incompressible fluid in a horizontal layer of infinite thickness is calculated. The layer is bounded above by a free non-deformable boundary on which an uneven temperature distribution is specified. In the approximation of the boundary layer, asymptotic expansions of the solution of the problem are constructed. The main terms of the asymptotics satisfy the Prandtl equations of the boundary layer. Two types of regimes in the boundary layer are calculated - non-swirling and rotational fluid flows. Rotational regimes of fluid flows arise as a result of bifurcation of non-swirling regimes in the boundary layer. Bifurcation points are found by solving a boundary value problem for eigenvalues. It is shown that rotational regimes arise only when the free boundary is locally cooled. When the boundary is heated, fluid rotation does not occur. Two types of rotational modes are numerically calculated. All modes exist only if the velocity of the external fluid flow does not exceed the bifurcation value. The modes of the first type have axial symmetry. There are only two such modes. Other modes do not have axial symmetry. For modes of the second type, an exact solution in cylindrical coordinates is obtained. These modes depend on two independent parameters that fill the circle of unit radius. So, at the bifurcation point, many rotational regimes arise, which are a two-parameter family.

Recovering of two-point boundary conditions by finite set of eigenvalues of boundary value problems for higher order differential equations

Recovering of two-point boundary conditions by finite set of eigenvalues of boundary value problems for higher order differential equations

Asymptotic methods for solving boundary value eigenvalue problems

The aim of the study is an approximate construction with a given accuracy of solutions of boundary value problems for eigenvalues under various types of boundary conditions. It is shown that the problem of finding approximate large eigenvalues of boundary value problems is reduced to the analysis and solution of singularly perturbed differential equations with variable coefficients. Methods used: asymptotic diagram method developed to construct the asymptotic behavior of solutions of singularly perturbed differential equations and systems; methods of numerical integration of boundary value problems. The main results obtained are: the asymptotics of the required accuracy are constructed in the analytical form for the eigenvalues and eigenfunctions of the boundary value problems under various boundary conditions; analysis of the computational capabilities of the practical use of the constructed asymptotics in comparison with the results of numerical integration.

Numerical Methods in Calculating Eigenvalues: Case Studies in Stability of Euler Columns

A comparative analysis of well renowned “Shooting Method” with another numerical method “Complementary Functions Method” (CFM) is presented for calculating eigenvalue (λ). Contrary to the shooting method hit and trial approach, CFM exploits the properties of linear ordinary differential equation (LODE). In the case of linear eigenvalue Boundary value Problem (BVP), CFM generates an algebraic equation system with one unknown “λ” and, alone root finding method is sufficient to give required eigenvalue. However, the Shooting Method create a system of algebraic equations containing two unknowns “λ” and “missing initial conditions”, that demands an additional numerical technique along with root finding method. These radical differences between two approaches, sets the basis for this comparative investigation. As a case study in Linear Elastic Stability, different cases of Euler columns are investigated by finding eigenvalues for each case numerically, under both methods. Comparison is performed on the basis of results accuracy and cost effectiveness for both numerical techniques while solving linear stability problems.

Bounds for distance between eigenvalues of boundary value problems with retarded argument

Abstract In this study we are concerned with spectrum of boundary value problems with retarded argument with discontinuous weight function, two supplementary transmission conditions at the point of discontinuity, spectral and physical parameters in the boundary condition and we obtain bounds for the distance between eigenvalues. We extend and generalize some approaches and results of the classical regular and discontinuous Sturm-Liouville problems. In the special case that ω (x) ≡ 1, the transmission coefficients γ 1 = δ 1, γ 2 = δ 2 and retarded argument Δ ≡ 0 in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

INTEGRAL METHOD OF SOLVING HEAT-CONDUCTION PROBLEMS WITH THE SECOND-KIND BOUNDARY CONDITION. 1. BASIC STATEMENTS

On the basis of systems of identical equalities and integral boundary characteristics, a new algorithm of solving a boundary-value problem on the nonstationary heat conduction in a canonical body with boundary condition of the second kind has been developed. The scheme proposed for finding approximate analytical solutions of boundary-value problems on nonstationary heat conduction with boundary conditions of the second kind involves the introduction into consideration of a temperature-disturbance front and separation of the whole heating process into two stages. For the first stage of this process, on the basis of the differentiation of the heat-conduction equation over a space variable and the application of symmetric integral and differential operators to the expressions obtained, two sequences of integral and differential identical equalities have been constructed. Each of these sequences includes integral or differential limiting characteristics for a definite boundary condition of the second kind. For the second stage, by way of introduction of a boundary function, differentiation of the heat-conduction equation with respect to a spatial coordinate, and application of integral operators to the expression obtained, a sequence of integral identical equalities involving integral boundary characteristics for the second-kind boundary condition has been constructed. On the basis of the integral and differential identical equalities obtained, closed systems of equations, allowing one to find polynomial coefficients of the temperature profile for the first and second stages of the heating process, have been constructed. A general scheme of determining approximate eigenvalues of boundary-value problems with boundary conditions of the second kind on the basis of construction of an ordinary differential equation and transformation of it into the characteristic equation is proposed. For each of the two stages of the heating process, special integral operators, reducing the boundary-value heat-conduction problem to the ordinary differential equation, are proposed.

Computing eigenfunctions and eigenvalues of boundary-value problems with the orthogonal spectral renormalization method

The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schr\"odinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR) method to compute ground and excited states (and their respective eigenvalues) of linear and nonlinear eigenvalue problems. The implementation of the algorithm follows four simple steps: (i) reformulate the underlying eigenvalue problem as a fixed point equation, (ii) introduce a renormalization factor that controls the convergence properties of the iteration, (iii) perform a Gram-Schmidt orthogonalization process in order to prevent the iteration from converging to an unwanted mode; and (iv) compute the solution sought using a fixed-point iteration. The advantages of the OSR scheme over other known methods (such as Newton's and self-consistency) are: (i) it allows the flexibility to choose large varieties of initial guesses without diverging, (ii) easy to implement especially at higher dimensions and (iii) it can easily handle problems with complex and random potentials. The OSR method is implemented on benchmark Hermitian linear and nonlinear eigenvalue problems as well as linear and nonlinear non-Hermitian $\mathcal{PT}$-symmetric models.

Systems of Functions Orthogonal Over the Domain and Their Application in Boundary-Value Problems of Mathematical Physics

We formulate a boundary-value problem for eigenvalues and eigenfunctions of the Helmholtz equation in a complex domain with the use of mutually conjugated complex variables. The obtained systems of functions are orthogonal in this domain and constructed by using the Bessel functions and powers of the conformal mappings of the analyzed domains onto a circle. The solutions of boundary-value problems for the principal equations of mathematical physics (hyperbolic, parabolic and elliptic types) are obtained in the form of the sums of series in the systems of functions orthogonal over the domain.