Fourth-order Eigenvalue Problem Research Articles

A Legendre spectral method based on a hybrid format and its error estimation for fourth-order eigenvalue problems

<abstract><p>In this paper, we developed and studied an efficient Legendre spectral method for fourth order eigenvalue problems with the boundary conditions of a simply supported plate. Initially, a new variational formulation based on a hybrid format and its discrete variational form were established. We then employed the spectral theory of complete continuous operators to establish the prior error estimates of the approximate solutions. By integrating approximation results of some orthogonal projection operators in weighted Sobolev spaces, we further gave the error estimation for the approximating eigenvalues and eigenfunctions. In addition, we developed an effective set of basis functions by utilizing the orthogonal properties of Legendre polynomials, and subsequently derived the matrix eigenvalue system of the discrete variational form for both two-dimensional and three-dimensional cases, based on a tensor product. Finally, numerical examples were provided to demonstrate the exponential convergence and efficiency of the algorithm.</p></abstract>

Nonconforming virtual element discretization for the transmission eigenvalue problem

A C0 nonconforming virtual element method (VEM) is employed for the discretization of nonselfadjoint fourth-order eigenvalue problems derived from the transmission eigenvalue problem on general polygonal meshes. Further, we linearize the model problem by introducing an auxiliary variable and thus the variational problem becomes double in size. We discretize the modified scheme by using suitable projection operators and acquire the convergence analysis under standard assumptions on the polygonal meshes. We establish that the resulting discretization provides a correct approximation of the spectrum and conclude the convergence of the eigenfunctions and eigenvalues. A variety of representative examples are examined to justify the theoretical estimates.

Forchheimer–Bénard Instability of the Non-Newtonian Fluid

The present paper examines the effect of vertical throughflow on the onset of convective instability in a horizontal porous layer filled with a non-Newtonian power-law fluid (PL). The permeable boundary layers are exposed to two different uniform constant temperature conditions, The Oberbeck-Boussinesq hypothesis is considered with the Darcy-Forchheimer model. A fourth-order eigenvalue problem is stemmed from the performance of the linear stability analysis, and the critical values are obtained using the shooting method combined with the Runge-Kutta method. The non-Newtonian Darcy-Rayleigh number (R), the Péclet number (Pe), the Forchheimer number (G), and the power-law index (n) are the parameters whose value play a crucial role in the onset of instability. The finding shows more stabilizing effects arise in pseudoplastic fluid than dilatant one at Peclet number Pe << 1 where the inverse behaviour takes place at large Peclet number even with the existence of the drag number or without it.

A [formula omitted]-nonconforming virtual element methods for the vibration and buckling problems of thin plates

In this work, we study the C0-nonconforming VEM for the fourth-order eigenvalue problems modeling the vibration and buckling problems of thin plates with clamped boundary conditions on general shaped polygonal domain, possibly even nonconvex domain. By employing the enriching operator, we have derived the convergence analysis in discrete H2 seminorm, and H1, L2 norms for both problems. We use the Babuška–Osborn spectral theory (Babuška and Osborn, 1991), to show that the introduced schemes provide well approximation of the spectrum and prove optimal order of rate of convergence for eigenfunctions and double order of rate of convergence for eigenvalues. Finally, numerical results are presented to show the good performance of the method on different polygonal meshes.

Hydrodynamic instability of flow through a rotating channel filled with isotropic porous media

Various geophysical and engineering applications have underlying physics, comprising system rotation's effects on the dynamics and transport phenomena in porous media flows. Comprehensive knowledge of the instability in a rotating fluid-saturated porous layer is beneficial for controlling the transport phenomena and the mixing process. The present study focuses on the temporal evolution of small disturbances in a pressure-induced fluid flow through a spanwise rotating channel filled with an isotropic porous material. A Darcy–Brinkman model, including the Coriolis force term in the momentum equation, is employed to describe the developed flow. A normal mode analysis is performed, and the coupled Orr–Sommerfeld–Squire eigenvalue problem is formulated to capture the linear instability of the perturbed flow. The Chebyshev collocation technique is used to solve the fourth-order eigenvalue problem to capture the transient behavior of the finite-amplitude disturbances. The temporal growth rate and marginal stability curves related to the Coriolis force-based instabilities are investigated. The rotating porous media flow is unstable at a much lower Reynolds number than the non-rotating configuration. The analysis confirms co-existing unstable modes and mode coalescence for a specific range of parameters, which can enhance the mixing and transport inside the porous layer. The neutral stability curves show the appearance of two different unstable zones corresponding to the long and moderate waves. Moreover, the higher permeability and porosity of the porous medium have a destabilizing influence.

The onset of instability in a hydromagnetic channel flow of Casson fluid: the accurate solutions

We study the temporal stability of linear two-dimensional disturbances of plane Poiseuille flow of a Casson fluid in a rigid parallel channel under the influence of a uniform magnetic field. When the disturbance is taken in normal mode it transforms the perturbed equations to a fourth-order eigenvalue problem which is then solved by the spectral method. We first obtain sufficient conditions for the hydrodynamic stability in the channel by analyzing the nature of the growth rate and the Reynolds number. For the chosen set of parameters, the eigenspectra of the flow exhibit the familiar Y-shaped structure with three distinct modes depending on the phase speed approach. The eigenfunctions corresponding to the least stable and most unstable eigenmodes have the symmetric and antisymmetric modes. For a certain range of Casson parameter β and the magnetic number M, instability always appears on the wall mode for which the eigenfunction is antisymmetric about the channel center-line. The neutral stability curves that correspond to various values of β and M are found to be an extension of the continuation of the Tollmien-Schlichting instability that is noticed for a Newtonian channel flow. The regions of stability are enlarged for small β and large M.

Characteristics of Eigenfunctions and Eigenvalues of a Fourth Order Differential Equations with Spectral Parameter in the Boundary Conditions

The characteristics of eigenvalues, and their positions on the real axis are discussed in this article for a fourth order eigenvalue problem with a spectral parameter in the boundary conditions. We have demonstrated that our described problem has a unique solution, as well as, it is shown the characteristics of eigenfunctions and their derivatives in the space

Approximate solutions of the Fourth-Order Eigenvalue Problem

In this paper, the differential transformation (DTM) and the Adomian decomposition (ADM) methods are proposed for solving fourth order eigenvalue problem. This fourth order eigenvalue problem has nonstrongly regular boundary conditions. This the fourth order problem has been examined for p(t) = t, B = 0, a = 0,01 where p(t) ≠ 0 is a complex valued and a ≠ 0 The differential transformation and the Adomian decomposition methods are briefly described. An approximate solution is obtained by performing seven iterations with the Adomian decomposition method. The same number of iterations have been made in the differential transformation method. The approximation results obtained by both methods have been compared with each other. These data have been presented in table. The ADM and the DTM approximation solutions have been shown by plotting in Figure 1. Here, the approaches obtained by using the two methods are found to be in high agreement. Consequently, highly accurate approximate solutions of fourth order eigenvalue problem are obtained. Such good results also revealed that the Adomian decomposition and the differential transformation methods are fast, economical and motivating. The exact solution of the fourth order eigenvalue problem for nonstrongly regular can not be found in the literature. Therefore, this study will give an important idea to determine approximate solution behavior of this fourth order problem.

Virtual Elements for the Transmission Eigenvalue Problem on Polytopal Meshes

The transmission eigenvalue problem is a challenging model in the inverse scattering theory and has important applications in this topic. The aim of this paper is to analyze a $C^1$ virtual element method on polytopal meshes in $\mathbb{R}^d$ $(d=2,3)$ for solving a quadratic and non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained by using the approximation theory for compact non-self-adjoint operators. Finally, a set of numerical tests illustrating the good performance of the virtual scheme are presented.

A High Accuracy Nonconforming Finite Element Scheme for Helmholtz Transmission Eigenvalue Problem

In this paper, we consider a cubic $H^2$ nonconforming finite element scheme $B_{h0}^3$ which does not correspond to a locally defined finite element with Ciarlet$'$s triple but admit a set of local basis functions. For the first time, we deduce and write out the expression of basis functions explicitly. Distinguished from the most nonconforming finite element methods, $(\delta\Delta_h\cdot,\Delta_h\cdot)$ with non-constant coefficient $\delta>0$ is coercive on the nonconforming $B_{h0}^3$ space which makes it robust for numerical discretization. For fourth order eigenvalue problem, the $B_{h0}^3$ scheme can provide $\mathcal{O}(h^2)$ approximation for the eigenspace in energy norm and $\mathcal{O}(h^4)$ approximation for the eigenvalues. We test the $B_{h0}^3$ scheme on the vary-coefficient bi-Laplace source and eigenvalue problem, further, transmission eigenvalue problem. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed scheme.

Linear stability analysis of blood flow in small vessels

We investigate the linear stability of unidirectional Poiseuille flow of blood modeling the fluid as a spatially inhomogeneous fluid in which viscosity depends on the red blood cell concentration (RBCs). We consider small vessels like arteries terminal branches, arterioles or venules, where the inhomo-geneity is due to the non uniform distribution of RBCs on the vessel cross section. The stability analysis is performed applying the classical normal-mode linear analysis which results in a fourth-order eigenvalue problem that is solved numerically. The results obtained indicates that the flow is unconditionally unstable. However, those patterns in which the RBCs concentration decreases towards the vessel walls show growth rates so small that the observability of the instability requires a very large time. Conversely, the growth rates associated to the profiles in which the RBCs concentration increases toward the vessel walls are at least three order of magnitude larger than the previous case. We therefore believe that those distributions in which the RBCs are more concentrated around the vessel center are to be considered more “stable” than those in which RBCs accumulate towards the vessel walls.

Approximation of the Zero-Index Transmission Eigenvalues with a Conductive Boundary and Parameter Estimation

In this paper, we present a spectral-Galerkin method to approximate the zero-index transmission eigenvalues with a conductive boundary condition. This is a new eigenvalue problem derived from the scalar inverse scattering problem for an isotropic media with a conductive boundary condition. In our analysis, we will consider the equivalent fourth-order eigenvalue problem where we establish the convergence when the approximation space is the span of finitely many Dirichlet eigenfunctions for the Laplacian. We establish the convergence rate of the spectral approximation by appealing to Weyl’s law. Numerical examples for computing the eigenvalues and eigenfunctions for the unit disk and unit square are presented. Lastly, we provide a method for estimating the refractive index assuming the conductivity parameter is either sufficiently large or small but otherwise unknown.

On global continua of solutions bifurcating from zero and infinity of some nonlinear fourth order eigenvalue problems

On global continua of solutions bifurcating from zero and infinity of some nonlinear fourth order eigenvalue problems

Spectral properties of a Fourth-Order eigenvalue problem with quadratic spectral parameters in a boundary condition

Consider the linear eigenvalue problem of fourth-order $ y^{(4)}(x)-(q(x)y'(x))' = \lambda y(x), 0 \lt x \lt l,\\ y(0) = y'(0) = 0,\\ (a_0+a_1\lambda+a_2\lambda^2)y'(l)+(b_0+b_1\lambda+b_2\lambda^2)y''(l) = 0,\\ y(l)\cos\delta-Ty(l)\sin\delta = 0, $ where \lt i \gt λ \lt /i \gt is a spectal parameter, $\delta\in[\frac{\pi}{2}, \pi]$, \lt i \gt Ty \lt /i \gt = \lt i \gt y \lt /i \gt ''' - \lt i \gt qy \lt /i \gt ', \lt i \gt q \lt /i \gt ( \lt i \gt x \lt /i \gt ) is a positive absolutely continuous function on the interval [0, \lt i \gt l \lt /i \gt ], \lt i \gt δ \lt /i \gt , \lt i \gt a \lt /i \gt \lt sub \gt \lt i \gt i \lt /i \gt \lt /sub \gt and \lt i \gt b \lt /i \gt \lt sub \gt \lt i \gt i \lt /i \gt \lt /sub \gt ( \lt i \gt i \lt /i \gt = 0, 1, 2) are real constants. We obtain not only the existence, simplicity and interlacing properties of the eigenvalues, the oscillation properties of the eigenfunctions, but also the asymptotic formula of the eigenvalues and the corresponding eigenfunctions for sufficiently large \lt i \gt n \lt /i \gt . Moreover, a new inner Hilbert space and a new sufficient conditions will be given to discuss the basis properties of the system of the eigenfunctions in \lt i \gt L \lt /i \gt \lt sub \gt \lt i \gt p \lt /i \gt \lt /sub \gt (0, \lt i \gt l \lt /i \gt ).

Properties of the Fourth-Order Boundary Value Problem with Transmission Conditions

We investigate the fourth-order eigenvalue problem with transmission conditions. We apply the Fourier method for solving of this problem arising in finding of transverse vibration of a string. We derive the eigenvalues and eigenfunctions of Sturm–Liouville problems and their properties. We obtain a solution of the considered equation subject to initial, boundary and transmission conditions at the point of discontinuity.

A Non-normal-Mode Marginal State of Convection in a Porous Rectangle

The fourth-order Darcy–Benard eigenvalue problem for onset of thermal convection in a 2D rectangular porous box is investigated. The conventional type of solution has normal-mode dependency in at least one of the two spatial directions. The present eigenfunctions are of non-normal-mode type in both the horizontal and the vertical direction. A numerical solution is found by the finite element method, since no analytical method is known for this non-degenerate fourth-order eigenvalue problem. All four boundaries of the rectangle are impermeable. The thermal conditions are handpicked to be incompatible with normal modes: The lower boundary and the right-hand wall are heat conductors. The upper boundary has given heat flux. The left-hand wall is thermally insulating. The computed eigenfunctions have novel types of complicated cell structures, with intricate internal cell walls.

A virtual element method for the transmission eigenvalue problem

In this paper, we analyze a Virtual Element Method (VEM) for solving a non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a [Formula: see text]-conforming discretization by means of the VEM. We use the classical approximation theory for compact non-self-adjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.

Analytic growth rate of gravitational instability in self-gravitating planar polytropes

Gravitational instability is a key process that may lead to fragmentation of gaseous structures (sheets, filaments, haloes) in astrophysics and cosmology. We introduce here a method to derive analytic expressions for the growth rate of gravitational instability in a plane stratified medium. First, the main strength of our approach is to reduce this intrinsically fourth-order eigenvalue problem to a sequence of second-order problems. Second, an interesting by-product is that the unstable part of the spectrum is computed by making use of its stable part. Third, as an example, we consider a pressure-confined, static, self-gravitating slab of a fluid with an arbitrary polytropic exponent, with either free or rigid boundary conditions. The method can naturally be generalised to analyse the stability of richer, more complex systems. Finally, our analytical results are in excellent agreement with numerical solutions. Their second-order expansions provide a valuable insight into how the rate and wavenumber of maximal instability behave as functions of the polytropic exponent and the external pressure (or, equivalently, the column density of the slab).

Eigenvalues of a Linear Fourth-Order Differential Operator with Squared Spectral Parameter in a Boundary Condition

We consider the spectrum of the following linear fourth-order eigenvalue problem: $$\begin{aligned} (py'')''(x)-(q(x)y'(x))'= & {} \lambda y(x),\quad x\in (0,l),\\ y'(0)\cos \alpha -y''(0)\sin \alpha= & {} 0,\\ y(0)\cos \beta +Ty(0)\sin \beta= & {} 0,\\ y'(l)\cos \gamma +y''(l)\sin \gamma= & {} 0,\\ (c_0+c_1\lambda +c_2\lambda ^2) y(l)= & {} (d_0+d_1\lambda +d_2\lambda ^2)Ty(l), \end{aligned}$$ where $$\lambda $$ is a spectral parameter, $$Ty=(py'')'-qy'$$ , p(x) has absolutely continuous derivative, q(x) is absolutely continuous on [0, l]; $$\alpha ,$$ $$\beta $$ , $$\gamma $$ , $$c_i$$ and $$d_i$$ ( $$i=0,1,2$$ ) are real constants; $$0\le \alpha ,\beta ,\gamma \le \pi /2$$ . By giving a new condition to guarantee the self-definiteness of the corresponding operator L, we obtain the simplicity and interlacing properties of the eigenvalues and the oscillation properties of the corresponding eigenfunctions. Meanwhile, some exceptional cases are also discussed when the self-definiteness condition does not hold. These results extend some existing results of the linear fourth-order eigenvalue problems with linear parameter in the boundary conditions and some existing results of the classical eigenvalue problems.

Existence and nonexistence results for weighted fourth order eigenvalue problems with variable exponent

This paper presents sufficient conditions for the existence and nonexistence of eigenvalues for a $p(x)$-biharmonic equation with Navier boundary conditions and weight function on a bounded domain in $\mathbb{R}^N$. Our approach is mainly based on a adequate variational techniques.