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Based on the linearized Euler equations and the normal-mode approach, the Pridmore-Brown equation for acoustics and the stability of parallel shear flows is solved for a boundary layer flow with an exponential velocity profile. The general solution is derived in terms of the confluent Heun function, and in turn, using the boundary conditions of vanishing disturbances at infinity and zero wall-normal velocity, the boundary value problem is converted to an algebraic eigenvalue problem. Solutions to the eigenvalue problem allow a comprehensive picture of the stability of compressible boundary layers. In particular, the complex eigenvalues ω are calculated as a function of the streamwise wavenumber α and the Mach number M. We observe that with growing wavenumbers, the number of eigenvalues increases discretely. Only for M > 1, unstable modes exist, and the boundary between neutrally stable and unstable modes is defined by the transonic line ω = α to be calculated from a degenerated eigenvalue equation. For large wall distances, the eigenfunctions converge exponentially toward zero. The spatial decay rate defines an "acoustic boundary layer thickness" δa, which indicates how far outside modes are still audible. For M > 1 and large α, δa diverges exponentially, i.e., in this parameter range, modes are perceptible even far from the boundary layer. A particularly steep rise of δa is observed when M > 2. Thus, there is a wavenumber range in which ωi and δa are both large and thus generates a particularly strong noise impact. From the eigenfunction for the unstable modes, a strong increase in the amplitude of acoustic waves can be identified in the vicinity of the wall, which indicates an accumulation and saturation of energy and thereafter leads to temporal instability and sound radiation in the free stream. [ABSTRACT FROM AUTHOR]

For rectangular thick plates with non-Levy boundary conditions, it is important to explore analytical free vibration solutions because the classical inverse and semi-inverse exact solution methods are not applicable to this category of problems. This work is to develop an extended separation-of-variable (SOV) method to find closed-form analytical solutions for the free vibration of rectangular Mindlin plates with arbitrary homogeneous boundary conditions. In the extended SOV method, characteristic differential equations and boundary conditions in two directions are obtained by employing the Rayleigh principle and the assumption that the mode functions are in the SOV form, and two transcendental eigenvalue equations are achieved through boundary conditions. But these two eigenvalue equations cannot be solved simultaneously since there are two equations and only the natural frequency is the unknown variable. Considering this, the second assumption in this method is that the natural frequencies corresponding to two-direction mode functions are independent of each other in the mathematical sense, thus there are two unknowns in two transcendental eigenvalue equations, and the closed-form solutions for plates with arbitrary boundary conditions can be obtained non-iteratively. From the physical sense, the natural frequencies pertaining to different direction mode functions should be the same, and this conclusion is validated analytically and numerically. The present natural frequencies and mode shapes agree well with those obtained by other analytical and numerical methods. Especially, for the plates with at least two opposite sides simply supported, the present solutions are exact. [ABSTRACT FROM AUTHOR]

We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions' systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem. [ABSTRACT FROM AUTHOR]

The suggested approach to maximizing the difference between the first and second eigenvalues of the Laplace operator is based on the introduction of nonlocal boundary conditions of a special form. It is shown that the difference can be arbitrarily large. [ABSTRACT FROM AUTHOR]

Incompatible discretization methods provide added flexibility in computation by allowing meshes to be unaligned with geometric features and easily accommodating non-interpolatory approximations. Such formulations that are based on Nitsche's approach to enforce surface constraints weakly, which shares features with stabilized methods, combine conceptual simplicity and computational efficiency with robust performance. The basic workings of the method are well understood, in terms of a bound on the parameter. However, its spectral behavior has not been explored in depth. Such investigations can shed light on properties of the operator that effect the solution of boundary-value problems. Furthermore, incompatible discretizations are rarely used for eigenvalue problems. The spectral investigations lead to practical procedures for solving eigenvalue problems that are formulated by Nitsche's approach, with bearing on explicit dynamics. [ABSTRACT FROM AUTHOR]

We consider the spectral problem y'''(x) = λy(x) with general two-point boundary conditions that do not contain the spectral parameter λ. We prove that the boundary conditions in this problem are degenerate if and only if their 3 × 6 coefficient matrix can be reduced by a linear row transformation to a matrix consisting of two diagonal 3 × 3 matrices one of which is the identity matrix and the diagonal entries of the other are all cubic roots of some number. Further, the characteristic determinant of the problem is identically zero if and only if that number is −1. We also show that the problem in question cannot have finite spectrum. [ABSTRACT FROM AUTHOR]

This paper studies controllability of the Poisson equation on the unit disk in C subject to the third boundary condition when the control is imposed on the boundary. We use complex analytic methods to prove existence and uniqueness of the control when the parameter λ is a nonzero complex number but not a negative integer (not an eigenvalue). Otherwise, due to multiplicity of solutions to the underlying problem, when λ is a negative integer, controllability could only be obtained if proper additional conditions on the boundary are imposed. [ABSTRACT FROM AUTHOR]

A new algorithm for eigenvalue problems for the fractional Jacobi-type ODE is proposed. The algorithm is based on piecewise approximation of the coefficients of the differential equation with subsequent recursive procedure adapted fromsome homotopy considerations. As a result, the eigenvalue problem (which is in fact nonlinear) is replaced by a sequence of linear boundary value problems (besides the first one) with a singular linear operator called the exact functional discrete scheme (EFDS). A finite subsequence of m terms, called truncated functional discrete scheme (TFDS), is the basis for our algorithm. The approach provides super-exponential convergence rate as m → ∞. The eigenpairs can be computed in parallel for all given indexes. The algorithm is based on some recurrence procedures including the basic arithmetical operations with the coefficients of some expansions only. This is an exact symbolic algorithm (ESA) for m =∞and a truncated symbolic algorithm (TSA) for a finite m. Numerical examples are presented to support the theory. [ABSTRACT FROM AUTHOR]

In this article we discuss a method for constructing trace formulae for the heat-volume potential of the time-fractional diffusion equation to lateral surfaces of cylindrical domains and use these conditions to construct as well as to study a nonlocal initial boundary value problem for the time-fractional diffusion equation. [ABSTRACT FROM AUTHOR]

In this article we consider eigenvalue problems for fourth-order ordinary differential equation with spectral parameter in boundary conditions. We study the location of eigenvalues on the real axis, find the multiplicities of eigenvalues, investigate the oscillation properties of eigenfunctions, and the basis properties in the space Lp, 1 < p < ∞, of the subsystems of eigenfunctions of this problem. [ABSTRACT FROM AUTHOR]

In this paper, we obtain accurate analytic solutions for free vibration of rectangular thick plates with an edge free. This is the first time that an up-to-date rational superposition method in the symplectic space is used for thick plates’ free vibration. The Hamiltonian system-based governing equation is first constructed. The eigenvalue problems of two fundamental vibration problems are formed for a thick plate with an edge free and the others clamped. By symplectic expansion, the fundamental solutions are obtained. Superposition of these solutions is equal to that of the original plate, which yields the analytic frequency equation. The mode shapes are then obtained. With simple treatments, the solution reduces to those of the plates with an edge free and the others subjected to combinations of the clamped and simply supported boundary conditions. The developed symplectic superposition method yields the analytic solutions with satisfactory accuracy as well as fast convergence by rigorous derivation, without assuming any trial solutions; thus, its extension to more boundary value problems of high-order partial differential equations represented by thick plates’ problems may be explored. [ABSTRACT FROM AUTHOR]

Schrödinger equation for an electron confined to a two-dimensional strip is considered in the presence of homogeneous orthogonal magnetic field. Since the system has edges, the eigenvalue problem is supplied by the boundary conditions (BC) aimed in preventing the leakage of matter away across the edges. In the case of spinless electrons the Dirichlet and Neumann BC are considered. The Dirichlet BC result in the existence of charge carrying edge states. For the Neumann BC each separate edge comprises two counterflow sub-currents which precisely cancel out each other provided the system is populated by electrons up to certain Fermi level. Cancelation of electric current is a good starting point for developing the spin-effects. In this scope we reconsider the problem for a spinning electron with Rashba coupling. The Neumann BC are replaced by Robin BC. Again, the two counterflow electric sub-currents cancel out each other for a separate edge, while the spin current survives thus modeling what is known as pure spin current – spin flow without charge flow. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of a reversed S-shaped component in the positive solutions set of the fourth-order boundary value problem where $\lambda>0$ is a parameter, $h\in C[0,1]$ and $f\in C[0,\infty )$ , $f(0)=0$ , $f(s)>0$ for all $s>0$ . By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to parameter λ, and especially, we obtain the existence of three distinct positive solutions for λ being in a certain interval. [ABSTRACT FROM AUTHOR]

We consider a fractional boundary value problem involving a fractional derivative with respect to a certain function g. A Hartman-Wintner-type inequality is obtained for such problem. Next, several Lyapunov-type inequalities are deduced for different choices of the function g. Moreover, some applications to eigenvalue problems are presented. [ABSTRACT FROM AUTHOR]

We derive a priori bounds for positive supersolutions of -Δpu = ρ(x)f(u), where p > 1 and Δp is the p-Laplace operator, in a smooth bounded domain of RN with zero Dirichlet boundary conditions. We apply our results to the nonlinear elliptic eigenvalue problem -Δpu = λf(u), Dirichlet boundary condition, where f is a nondecreasing continuous differentiable function on such that f(0) > 0, f(t)¹=(p-1) is superlinear at infinity, and give sharp upper and lower bounds for the extremal parameter λ*p. In particular, we consider the nonlinearities f(u) = eu and f(u) = (1 + u)m (m > p - 1) and give explicit estimates on λ*p. As a by-product of our results, we obtain a lower bound for the principal eigenvalue of the p-Laplacian that improves obtained results in the recent literature for some range of p and N. [ABSTRACT FROM AUTHOR]

Numerical methods for the transmission eigenvalue problems are hot topics in recent years. Based on the work of Lin and Xie (2015), we build a multigrid method to solve the problems. With our method, we only need to solve a series of primal and dual eigenvalue problems on a coarse mesh and the associated boundary value problems on the finer and finer meshes. Theoretical analysis and numerical results show that our method is simple and easy to implement and is efficient for computing real and complex transmission eigenvalues. [ABSTRACT FROM AUTHOR]

We study the lasing modes of microdisk lasers with ring-like electrodes or active regions, in two-dimensional (2-D) formulation. The considered eigenvalue problem is adapted to the extraction of both modal spectra and thresholds from the Maxwell equations with exact boundary conditions. We reduce it to a transcendental equation and solve it numerically. The obtained lasing frequencies and the associated values of threshold material gain of the ring-pumped laser are compared with similar quantities of the fully active microdisk. This comparison shows that the optimal position of the active ring is shifted inward from the disk rim. Its location and width can be used as an engineering instrument to manipulate the thresholds. This effect is explained using the optical theorem and overlap coefficients. [ABSTRACT FROM AUTHOR]

In this paper, a full (nested) multigrid scheme is proposed to solve eigenvalue problems. The idea here is to use a correction method to transform the eigenvalue problem solving to a series of corresponding boundary value problem solving and eigenvalue problems defined on a very low-dimensional finite element space. The boundary value problems which are defined on a sequence of multilevel finite element spaces can be solved by some multigrid iteration steps. The computational work of this new scheme can reach the same optimal order as solving the corresponding boundary value problem by the full multigrid method. Therefore, this type of full multigrid method improves the overfull efficiency of the eigenvalue problem solving. [ABSTRACT FROM AUTHOR]

Here are described four solvers for time-harmonic electromagnetic fields in checkerboard patterns. A pattern is built by four squares with constant permittivity [ABSTRACT FROM AUTHOR]

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In this paper, a new multilevel correction scheme is proposed to solve Stokes eigenvalue problems by the finite element method. This new scheme contains a series of correction steps, and the accuracy of eigenpair approximation can be improved after each step. In each correction step, we only need to solve a Stokes problem on the corresponding fine finite element space and a Stokes eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving Stokes eigenvalue problems by the finite element method. As applications of this multilevel correction method, a multigrid method and an adaptive finite element technique are introduced for Stokes eigenvalue problems. Some numerical results are given to validate our schemes. [ABSTRACT FROM AUTHOR]

The limit behavior of the discrete spectrum of the Sturm-Liouville problem whose potential is a polynomial with complex coefficients on an interval, on a half-axis, and on the entire axis is studied. It is shown that, at large parameter values, the eigenvalues are concentrated along the so-called limit spectral graph; the curves forming this graph are classified. Asymptotics of eigenvalues along curves of various types in the graph are calculated. [ABSTRACT FROM AUTHOR]

The eigenvalue problem of holomorphic functions on the unit disc for the third boundary condition with general coefficient is studied using Fourier analysis. With a general anti-polynomial coefficient a variable number of additional boundary conditions need to be imposed to determine the eigenvalue uniquely. An additional boundary condition is required to obtain a unique eigenvalue when the coefficient includes an essential singularity rather than a pole. In either case explicit solutions are derived. [ABSTRACT FROM AUTHOR]

The mixed-order eigenvalue problem -δΔ2u+Δu + λu = 0 with δ > 0, modeling small amplitude vibrations of a thin plate, is analyzed in a bounded two-dimensional domain Ω that contains a single small hole of radius ε centered at some x0 ϵ Ω. Clamped conditions are imposed on the boundary of Ω and on the boundary of the small hole. In the limit ε → 0, and for δ = O(1), the limiting problem for u must satisfy the additional point constraint u(x0) = 0. To determine how the eigenvalues of the Laplacian in a domain with a small hole are perturbed by adding the small fourth-order term -δΔ2u, together with an additional boundary condition on ϑΩ and on the hole boundary, the asymptotic behavior of the eigenvalues of the mixed-order eigenvalue problem are studied in the dual limit ε → 0 and δ → 0. Leading-order behaviors of eigenvalues are determined for three ranges of δ ⪡ 1: δ ⪡O(ε2), δ = O(ε2), and O(ε2) ⪡ δ ⪡ 1. In the first two of these regimes, the limiting behavior depends of the radius of the hole ε, while in the regime O(ε2) ⪡ δ ⪡ 1 the eigenvalue is asymptotically independent of ε. Therefore, it is this regime that provides a transition to the point constraint behavior characteristic of the range δ = O(1). The asymptotic results for the eigenvalues are validated by full numerical simulations of the PDE. [ABSTRACT FROM AUTHOR]

We present the variational method for finding the eigenfunctions and eigenvalues in the Sturm-Liouville problem with Dirichlet boundary conditions; the method is based on the proposed functional. As a test example, we consider the potential cos(4 x). Also computations for two functions sin(( x - π)/π) and a high nonisosceles triangle are given. [ABSTRACT FROM AUTHOR]

In this paper, an efficient technique based on the Chebyshev polynomial expansions for computing the eigenvalues of second- and fourth-order Sturm-Liouville boundary value problems is proposed. This technique reduces the given Sturm-Liouville problem to an integral equation. The resulting integral equation is then transformed into the eigenvalue equation by calculating the integrals at the grid points using the Chebyshev expansions. Thus, the required eigenvalues of the given problem are obtained by solving this eigenvalue equation. The excellent performance of this scheme is illustrated through some numerical examples, and comparison with other methods is presented. [ABSTRACT FROM AUTHOR]

Several technological issues, such as reactor start-up analysis or kinetics studies of accelerator-driven systems, demand the asymptotic time behaviour of neutron transport to be assessed. Typically, this amounts to solving an eigenvalue equation associated to the Boltzmann operator, whose precise nature depends on whether delayed neutrons are taken into account. The inverse of the dominant eigenvalue can be physically interpreted as the asymptotic reactor period. In this work, we propose a Monte Carlo method for determining the dominant alpha eigenvalue of the Boltzmann operator and the associated fundamental mode for arbitrary geometries, materials, and boundary conditions. Extensive verification tests of the algorithm are performed, and Monte Carlo calculations are finally validated against reactor period measurements carried out at the ORPHEE facility of CEA/Saclay. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to describe the dynamic transition of the incompressible MHD equations in a three dimensional (3D) rectangular domain from a perspective of pattern formation. We aim to classify the formations of roll, rectangle and hexagonal patterns at the first critical Rayleigh number. When the first eigenvalue of the linearized operator is real and simple, the critical eigenvector has either a roll structure or a rectangle structure. In both cases we find that the transition is continuous or jump depending on a non- dimensional number computed explicitly in terms of system parameters. When the critical eigenspace has dimension two corresponding to two real eigenvalues, we study the transitions of hexagonal pattern. In this case, we show that all three types of transitions--continuous, jump and mixed--can occur in eight different transition scenarios. Finally, we study the case where the first eigenvalue is complex, simple and corresponding eigenvector has a roll structure. In this case, we find that both continuous and jump transitions are possible. We give several bounds on the parameters which separate the parameter space into regions of different transition scenarios. [ABSTRACT FROM AUTHOR]

In this paper we study an elliptic eigenvalue problem with non-local boundary condition. We prove the existence of the principal eigenvalue and its main properties. As consequence, we show the existence and uniqueness of positive solution of a nonlinear problem arising from population dynamics. [ABSTRACT FROM AUTHOR]

The existence of positive solutions for a predator-prey model with cross-diffusion under homogeneous Dirichlet boundary conditions is studied. By the maximum principle, some apriori estimates of positive solution are obtained. Then, by considering the related eigenvalue problems, two unbounded neutral curves are given. Finally, using Crandall-Rabinowitz bifurcation theory, with the growth rate of prey as a bifurcation parameter, the positive solutions is emanated from the semi-trivial solutions are derived. [ABSTRACT FROM AUTHOR]

In this paper, we extend some spectral properties of regular Sturm-Liouville problems to those which consist of a Sturm-Liouville equation with discontinuous weight at two interior points together with spectral parameter-dependent boundary conditions. By modifying some techniques of [C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 293-308; O. Sh. Mukhtarov and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal, 46 (2005) 681-694], we give an operator-theoretic formulation for the considered problem and obtain asymptotic formulas for the eigenvalues and eigenfunctions. [ABSTRACT FROM AUTHOR]

The paper deals with the study of flux density through a newly proposed twisted clad guide containing DB medium. The inner core and the outer clad sections are usual dielectrics, and the introduced twisted windings at the core-clad interface are treated under DB boundary conditions.The pitch angle of twist is supposed to greatly contribute towards the control over the dispersion characteristics of the guide.The eigenvalue equation for the guiding structure is deduced, and the analytical investigations aremade to explore the propagation patterns of flux densities corresponding to the sustained low-order hybrid modes under the situation of varying pitch angles. The emphasis has been put on the effects due to the DB twisted pitch on the propagation of energy flux density through the guide. [ABSTRACT FROM AUTHOR]

The free vibration analysis of rotating ring-stiffened cylindrical shells with arbitrary boundary conditions is investigated by employing the Rayleigh-Ritz method. Six sets of characteristic orthogonal polynomials satisfying six classical boundary conditions are constructed directly by employing Gram-Schmidt procedure and then are employed to represent the general formulations for the displacements in any axial mode of free vibrations for shells. Employing those formulations during the Rayleigh-Ritz procedure and based on Sanders' shell theory, the eigenvalue equations related to rotating ringstiffened cylindrical shells with various classical boundary conditions have been derived. To simulate more general boundaries, the concept of artificial springs is employed and the eigenvalue equations related to free vibration of shells under elastic boundary conditions are derived. By adjusting the stiffness of artificial springs, those equations can be used to investigate the vibrational characteristics of shells with arbitrary boundaries. By comparing with the available analytical results for the ring-stiffened cylindrical shells and the rotating shell without stiffeners, the method proposed in this paper is verified. Strong convergence is also observed from convergence study. Further, the effects of parameters, such as the stiffness of artificial springs, the rotating speed of the ring-stiffened shell, the number of ring stiffeners and the depth to width ratio of ring stiffeners, on the natural frequencies are studie. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a second-order resonance anti-periodic boundary value problem [ABSTRACT FROM AUTHOR]

For a closed connected manifold N, we establish the existence of geometric structures on various subgroups of the contactomorphism group of the standard contact jet space J1N, as well as on the group of contactomorphisms of the standard contact T*N×S1 generated by compactly supported contact vector fields. The geometric structures are biinvariant partial orders (for J1N and T*N×S1) and biinvariant integer-valued metrics (T*N×S1 only). We prove some forms of contact rigidity in T*N×S1, namely that certain (possibly singular) subsets of the form X×S1 cannot be disjoined from the zero section by a contact isotopy, and in addition that there are restrictions on commuting products of contactomorphisms. Finally, we prove multiplicity results for orbits of certain contact flows in T*N×S1 with Legendrian boundary conditions, which in particular apply to Reeb chords. The method is that of generating functions for Legendrians in jet spaces. [ABSTRACT FROM PUBLISHER]

Some special cases of dispersion characteristics of a uniaxial crystalline optical fiber with a helical winding on the core-cladding boundary are investigated theoretically. In the present case, the optical fiber is doubly unconventional: (i) in the choice of uniaxial crystalline optical fiber and (ii) in the choice of sheath helix between the core and cladding. Field components, boundary conditions, and eigenvalue equations for HE and EH modes are obtained. We consider two special cases, i.e.ψ= 0° andψ= 90° to obtain simpler eigenvalue equations which contain Bessel and modified Bessel functions and their derivatives. The nature of the dispersion curve remains unaffected with the anisotropy and only cutoff frequency is lowered for the positive uniaxial crystals. The helix pitch angle does not affect the modal cutoffs but it effectively controls the number of guided modes. This property has promising importance in long-distance communication where only a few modes are desired to be guided in order to minimize the inter-modal dispersion. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, the existence of positive solutions for a singular third-order three-point boundary value problem is investigated. By using Krasnoselskii’s fixed point theorem, several sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exists a positive solution are obtained. [Copyright &y& Elsevier]

In this paper, we consider a Sturm-Liouville problem which contains an eigenparameter appearing linearly in two boundary conditions, in addition to an internal point of discontinuity. Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. We apply the sinc method, which is based on the sampling theory to compute approximations of the eigenvalues. An error analysis is exhibited involving rigorous error bounds. Using computable error bounds we obtain eigenvalue enclosures in a simple way. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. [ABSTRACT FROM AUTHOR]

A theoretical study of cylindrical dielectric waveguide having plasma in the cladding region is investigated. The eigen-value equations for the proposed waveguide and its power calculation have been evaluated using the usual boundary conditions. By using these equations modal dispersion, normalized group velocity, power confinement factor, and fractional modal power curves are drawn for some lower-order TE and TM modes. The nature and shape of the dispersion curve is usual but the group velocity of TM01mode is largely affected in the presence of plasma. It may also be noted that for a fixed value of plasma density, the fractional modal power sustained by a particular fiber mode can be tuned at the optimum level by adjusting the operating frequency properly. Also, the maximum power is concentrated at the interface instead, at the waveguide axis. [ABSTRACT FROM AUTHOR]

The paper describes the propagation of electromagnetic waves through a cylindrical waveguide composed of uniaxial anisotropic chiral medium that is treated under the DB-boundary. The emphasis is given on the energy flux density in the guide considering varieties of chiral mediums and the waveguide dimensions. For this purpose, the eigenvalue equation is obtained using the appropriate boundary conditions, and the modal propagation constants of the guide are deduced. It has been investigated that the flux density mostly remains confined in the central regions of guide which, however, strongly depends on the type of waveguide medium as well as the dimensional measures. Both forward and backward wave propagations are observed in such guides – the feature essentially governed by the characteristics of the guiding medium. [ABSTRACT FROM AUTHOR]

In this paper, we study eigenvalue problem for a class of nonlinear fractional differential equations Dα0+u(t) = λf(u(t)); 0 < t < 1; u(0) = u(1) = u'(0) = u'(1) = 0; where 3 < α ⩽ 4 is a real number, Dα0+ is the Riemann-Liouville fractional derivative, λ is a positive parameter and f : (0,+∞) → (0,+∞) is continuous. By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. As an application, some examples are presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following new nonlocal Dirichlet boundary value problem: {−(a−b∫Ω|∇u|2dx)Δu=λu+g(x,u),x∈Ω,u=0,x∈∂Ω, where a and b are positive, λ is a positive parameter, 0≤λ, λ1 is the first eigenvalue of operator −Δ. Under appropriate assumptions on the function g which is of subcritical growth, we obtain a nontrivial solution. [ABSTRACT FROM AUTHOR]

This work deals with a model of a topographically trapped vortex appearing over isolated topography in a geophysical flow. The main feature of the study is that we pay special attention to the vertical structure of a topographically trapped vortex. The model considered allows one to study the vertical motion which is known not to be negligible in many cases. Given topography in the form of an isolated cylinder, and radial symmetry and stationarity of a uniform flow, in the linear approximation, we formulate a boundary value problem that determines all the components of the velocity field through a six-order differential operator, and nonincreasing boundary conditions at the center of the topography, and at infinity. The eigenvalues of the boundary value problem correspond to bifurcation points, in which the flow becomes unstable, hence non-negligible vertical velocities occur. We formulate a condition for the boundary value problem to have a discrete spectrum of these bifurcation points, and hence to be solvable. Conducting a series of test calculations, we show that the resulting vortex lies in the vicinity of topography, and can attain the distance up to half of the topography characteristic radius. [ABSTRACT FROM AUTHOR]