Eigenvalue Problems For Differential Equations Research Articles

Deep learning method for finding eigenpairs in Sturm-Liouville eigenvalue problems

Solving the eigenvalue problem for differential equations in inhomogeneous media poses a significant challenge across diverse scientific fields. While classical finite difference methods and finite element methods have produced numerous outcomes, they heavily rely on discretizing the computational domain, which can introduce complexities and limitations. In this study, we present an unsupervised neural network approach tailored for finding eigenpairs in Sturm-Liouville eigenvalue problems within inhomogeneous media. Our method introduces eigenvalues as trainable parameters, crafts a novel cost function, incorporates an adaptive hyper-parameter tuning strategy, and sequentially trains the eigenpairs. The simplicity, accuracy, and interpretability of our approach significantly expand its applicability across various domains. The method we present in this paper can easily tackle boundary value conditions with derivatives, resulting in orthogonal eigenfunctions. This is a very important advantage of deep learning methods that has not yet been noticed. Quantitative estimation of eigenpairs is given for the Sturm-Liouville eigenvalue problems. Furthermore, we extend the proposed methodology to tackle two-dimensional cases, periodic scenarios, demonstrating its versatility and broad potential. For more information see https://ejde.math.txstate.edu/Volumes/2024/53/abstr.html

Convection in a Horizontal Porous Layer with Vertical Pressure Gradient Saturated by a Power-Law Fluid

The onset of convection in a porous layer saturated by a power-law fluid is here investigated. The walls are considered to be isothermal, isobaric and permeable in such a way that a vertical throughflow is described. The threshold for a buoyancy-driven cellular flow is investigated by means of a linear stability analysis. This study consists in introducing disturbances with small amplitude. The disturbances are plane waves, i.e. a normal modes stability analysis of the basic stationary solution is performed. The resulting problem is an ordinary differential equation eigenvalue problem which is solved numerically by coupling the Runge–Kutta method with the shooting method. Results are presented in the form of marginal stability curves and their critical points representing the values of the control parameters such that the growth rate of the disturbances is zero. It is found that, among roll disturbances, the most unstable modes are stationary and uniform with infinite wavelength. For this reason, an asymptotic analysis for vanishing wave numbers is carried out. The results of this asymptotic analysis are obtained analytically displaying a very good agreement with the numerical solution. It is found that vertical throughflow plays a destabilising role for pseudoplastic fluids and a stabilising role for dilatant fluids.

Simplified variational iteration method for solving ordinary differential equations and eigenvalue problems

A simplified variational iteration method is proposed to solve high-order homogeneous or nonhomogeneous linear ordinary differential equation and ordinary differential equation eigenvalue problems more efficiently and conveniently. The simplification includes two aspects: (1) explicitly deducing the general form of the differential equation for the identification of the general Lagrange multiplier while avoiding the complexity of variational calculations during identification and (2) simplifying the iterative expressions to reduce the computational work of each iteration. Three ordinary differential equations in mechanics are solved by this simplified variational iteration method, which proves that it is valid and more concise than traditional methods. To make the method more practical, it is suggested that some complicated analytical derivations be executed numerically, thereby achieving a simplified semi-analytical variational iteration method that can be easily implemented by computer programs. The method is then used to numerically solve two complex ordinary differential equation problems derived from the continuum analysis of tall building structures: a sixth-order nonhomogeneous ordinary differential equation with complex boundary conditions and a sixth-order ordinary differential equation eigenvalue problem. Numerical computer programs are developed for these two problems, and corresponding examples are provided to verify the accuracy and efficiency of the simplified variational iteration method in solving complex ordinary differential equation problems.

A combinational efficient algorithm for elliptic eigenvalue problems

In this paper, we present a combinational efficient algorithm for solving second-order elliptic ordinary differential equation eigenvalue problems on four kinds of meshes. Firstly, a combinational scheme is proposed by leveraging the linear finite element scheme and the second-order finite difference scheme based on mass lumping method. An efficient algorithm is constructed by using the combinational scheme with a constant combined parameter. Then, we improve the efficient algorithm by computing the quasi-optimal combined parameters for different eigenvalues. Finally, numerical experiments tested on both uniform and nonuniform meshes are shown to illustrate the efficiency of our algorithms.

NON-BAROTROPIC LINEAR ROSSBY WAVE INSTABILITY IN THREE-DIMENSIONAL DISKS

Astrophysical disks with localized radial structure, such as protoplanetary disks containing dead zones or gaps due to disk-planet interaction, may be subject to the non-axisymmetric Rossby wave instability (RWI) that lead to vortex-formation. The linear instability has recently been demonstrated in three-dimensional (3D) barotropic disks. It is the purpose of this study to generalize the 3D linear problem to include an energy equation, thereby accounting for baroclinity in three-dimensions. Linear stability calculations are presented for radially structured, vertically stratified, geometrically-thin disks with non-uniform entropy distribution in both directions. Polytropic equilibria are considered but adiabatic perturbations assumed. The unperturbed disk has a localized radial density bump making it susceptible to the RWI. The linearized fluid equations are solved numerically as a partial differential equation eigenvalue problem. Emphasis on the ease of method implementation is given. It is found that when the polytropic index is fixed and adiabatic index increased, non-uniform entropy has negligible effect on the RWI growth rate, but pressure and density perturbation magnitudes near a pressure enhancement increases away from the midplane. The associated meridional flow is also qualitatively changed from homentropic calculations. Meridional vortical motion is identified in the nonhomentropic linear solution, as well as in a nonlinear global hydrodynamic simulation of the RWI in an initially isothermal disk evolved adiabatically. Numerical results suggest buoyancy forces play an important role in the internal flow of Rossby vortices.

A streamline approach to two-dimensional steady non-ideal detonation: the straight streamline approximation

Detonations in non-ideal explosives tend to propagate significantly below the nominal one-dimensional detonation speed. In these cases, multi-dimensional effects within the reaction zone are important. A streamline-based approach to steady-state non-ideal detonation theory is developed. It is shown in this study that, given the streamline shapes, the two-dimensional problem reduces to an ordinary differential equation eigenvalue problem along each streamline, the solution of which determines the local shock shape that, in turn, leads to the solution of the detonation speed as a function of charge diameter. A simple approximation of straight but diverging streamlines is considered. The results of the approximate theory are compared with those of high-resolution direct numerical simulations of the problem. It is shown that the straight streamline approximation is remarkably predictive of highly non-ideal explosive diameter effects. It is even predictive of failure diameters. Given this predictive capability, one potential use of the method is in the determination of rate law parameters by fitting to data from unconfined rate stick experiments. This is illustrated by using data for ammonium nitrate fuel oil explosives.

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω ⊂ ℝn be a bounded domain, H = L2(Ω), L: D(L) ⊂ H → H be an unbounded linear operator, f ∈ C(\(\bar \Omega\) × ℝ, ℝ) and λ ∈ ℝ. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem $$Lu = \lambda f\left( {x,u} \right), u \in D\left( L \right),$$ which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a secondorder ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.

Trapped modes in elastic plates, ocean and quantum waveguides

Trapped modes and bound states occur in waveguides in a wide variety of physical situations, here we consider one class: the guide has one wall with Dirichlet (clamped) boundary conditions and the other Neumann (stress-free) boundary conditions. We consider the possibility of trapped modes in otherwise uniform elastic/ocean/quantum waveguides, that have a perturbation in thickness, or in curvature. These are all mathematically connected, so their study can be unified, and the same asymptotic procedure can be applied to them all. For bent waveguides, with such boundary conditions, the sign of the curvature function is shown to play an important role in the possibility, or otherwise, of trapping. An asymptotic scheme is developed to analyse whether trapped modes should be expected and to obtain the frequencies at which trapped modes are excited. A mathematical explanation and physical argument for the existence, or otherwise, of trapped modes, depending on different situations, is given. The asymptotic approach leads to an ordinary differential equation eigenvalue problem and its solutions are compared both with numerics, for the full governing equations, and with a further simplification that gives highly accurate estimates for the lowest eigenvalue.

Preconditioned iterative methods for a class of nonlinear eigenvalue problems

This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to nonlinear eigenvalue problems with very large sparse ill-conditioned matrices monotonically depending on the spectral parameter. To compute the smallest eigenvalue of large-scale matrix nonlinear eigenvalue problems, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix–vector multiplications, preconditioner-vector multiplications, linear operations with vectors, and inner products of vectors. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

Elasticity, shell theory and finite element results for the buckling of long sandwich cylindrical shells under external pressure

The buckling of a sandwich cylindrical shell under uniform external hydrostatic pressure is studied in three ways. The simplifying assumption of a long shell is made (or, equivalently, ‘ring’ assumption), in which the buckling modes are assumed to be two-dimensional, i.e. no axial component of the displacement field, and no axial dependence of the radial and hoop displacement components. All constituent phases of the sandwich structure, i.e. the facings and the core, are assumed to be orthotropic. First, the structure is considered a three-dimensional (3D) elastic body, the corresponding problem is formulated and the solution is derived by solving a set of two linear homogeneous ordinary differential equations of the second-order in r (the radial coordinate), i.e. an eigenvalue problem for differential equations, with the external pressure, p the parameter/eigenvalue. A complication in the sandwich construction is due to the fact that the displacement field is continuous but has a slope discontinuity at the face-sheet/core interfaces, which necessitates imposing ‘internal’ boundary conditions at the face-sheet/core interfaces, as opposed to the traditional two-end-point boundary value problems. Second, the structure is considered a shell and shell theory results are generated with and without accounting for the transverse shear effect. Two transverse shear correction approaches are employed, one based only on the core, and the other based on an effective shear modulus that includes the face-sheets. Third, finite element results are generated by use of the ABAQUS finite element code. In this part, two types of elements are used: a shear deformable shell element and a solid 3D (brick) element. The results from all these three different approaches are compared.

Boundary Eigenvalue Problems for Differential Equations Nη=λPη with λ-Polynomial Boundary Conditions

The present paper deals with the spectral properties of boundary eigenvalue problems for differential equations of the form Nη=λPη on a compact interval with boundary conditions which depend on the spectral parameter polynomially. Here N as well as P are regular differential operators of order n and p, respectively, with n>p⩾0. The main results concern the completeness, minimality, and Riesz basis properties of the corresponding eigenfunctions and associated functions. They are obtained after a suitable linearization of the problem and by means of a detailed asymptotic analysis of the Green's function. The function spaces where the above properties hold are described by λ-independent boundary conditions. An application to a problem from elasticity theory is given.

A Numerically Rigorous Proof of Curve Veering in an Eigenvalue Problem for Differential Equations

We consider parameter-dependent self-adjoint eigenvalue problems for differential equations. Frequently the eigenvalue curves show the interesting phenomenon of curve veer-ing. We propose a numerically rigorous procedure for proving this phenomenon in concrete situations.

Solving eigenvalue problems by implicit decomposition

This paper presents three innovative methods for solving eigenvalue problems for differential equations based upon the techniques of implicit decomposition developed by Luo and Friedman. An eigenvalue problem can be written as an approximate algebraic system of the form [K]{X} + λ[M]{X} = 0 by employing finite elements. These methods provide robust techniques to compute the real eigenpair, λ and {X}, where [K] and [M] can be asymmetric, indefinite, and even singular.

The distribution function of the solution of the random eigenvalue problem for differential equation

By means of the prüfer substitution and the Liouville equation, this paper gives an efficient way to evaluate the distribution functions of the solutions of the random eigenvalue problem.

Operators on L2(I) ⊕ Cr

Several authors have considered eigenvalue problems for differential equations where the eigenvalue parameter also appears in the boundary conditions. Such problems do not appear to arise from any spectral problem associated with a linear operator on a Hilbert space. However, it is possible to reset such problems in this context. This has been done for certain second order cases by Walter [4] using a special measure on the interval in question, and by Fulton [1, 2] using the type of space indicated in the title of this article.It is our purpose here to consider a general class of operators on L2(I) ⊕ Cr, which are based on a differential expression τ of order n on I. We shall first investigate adjoints, boundary conditions, and self-adjointness for such operators. We shall then show that all eigenvalue problems of the form τy = λy, with boundary conditions which involve λ in a linear fashion, can be reset in the context of such operators.

A Note on the Relationship between Finite-Difference and Shooting Methods for ODE Eigenvalue Problems

Finite-difference methods and shooting methods are two standard classes of techniques for solving ordinary differential equation eigenvalue problems. We first show that a Sturm sequence method often used for solving the finite-difference eigenvalue problem may be interpreted in the shooting framework. Then we find that the Sturm sequence procedure is easily extended to more complicated problems. This enables the shooting method to guarantee convergence to specified subsets of the modes of the problem without the usual requirement of an initial guess for the eigenvalues.

Deferred correction for the ordinary differential equation eigenvalue problem

Deferred correction for the ordinary differential equation eigenvalue problem

Stochastic eigenvalue problems for differential equations

We consider an eigenvalue problem for stochastic ordinary differential operators with stochastic boundary conditions. By replacing the random variables and stochastic processes by their mean values one gets the so-called averaged problem corresponding to the given stochastic problem. Let λ i (ω) denote the stochastic eigenvalues and μ i the eigenvalues of the averaged problem. In this paper the differences 〈 λ〉− μ i are calculated. The motive of such investigations was the effect of “bowing” in mixed crystal theory.

Numerical Treatment of Eigenvalue Problems for Differential Equations with Discontinuous Coefficients

The eigenvalues of a second order differential equation are approximated by ''factoring'' the second order equations into a first order system and then applying the Ritz-Galerkin method to this system. Convergence results and error estimates are derived. These error estimates are derived. These error estimates are based on the application of Sobolev spaces with variable order.

Pressure anisotropy stabilization of axisymmetric mirror machines

The stability of a two species, anisotropic pressure, axisymmetric plasma is studied using the guiding center plasma model. Successively, asymptotic expansions are applied appropriate to a long, thin plasma, and to a plasma with flux surfaces close to cylinders. The resultant stability problem may be cast as an ordinary differential equation eigenvalue problem or as a problem in the calculus of variations. It is shown that low beta plasmas cannot be confined and be stable although plasmas may be stable in which the pressure gradients are nonzero where the pressure tends to zero. Stable profiles are given; these profiles include the possibility of field reversed regions. These stable profiles require the anisotropic species to be cold near the axis. Rather than absolute stability, a weaker condition is also considered which for fixed azimuthal mode number ‖m‖ puts the point of accumulation of the spectrum of modes on the stable side. It is hoped that such a condition may yield systems stable to ‖m‖ small modes although not all values of ‖m‖. This condition is more readily satisfied and allows more reasonable profiles near the axis.