Sturm-Liouville Eigenvalue Problem Research Articles

Analytic study of properties of holographic superconductors in Born-Infeld electrodynamics

In this paper, based on the Sturm-Liouville eigenvalue problem, we analytically investigate several properties of holographic s-wave superconductors in the background of a Schwarzschild-AdS spacetime in the framework of Born-Infeld electrodynamics. Based on a perturbative approach, we explicitly find the relation between the critical temperature and the charge density and also the fact that the Born-Infeld coupling parameter indeed affects the formation of scalar hair at low temperatures. Higher value of the Born-Infeld parameter results in a harder condensation to form. We further compute the critical exponent associated with the condensation near the critical temperature. The analytical results obtained are found to be in good agreement with the existing numerical results.

Analytic study of Gauss-Bonnet holographic superconductors in Born-Infeld electrodynamics

Using Sturm-Liouville (SL) eigenvalue problem, we investigate several properties of holographic s-wave superconductors in Gauss-Bonnet gravity with Born-Infeld electrodynamics in the probe limit. Our analytic scheme has been found to be in good agreement with the numerical results. From our analysis it is quite evident that the scalar hair formation at low temperatures is indeed affected by both the Gauss-Bonnet as well as the Born-Infeld coupling parameters. We also compute the critical exponent associated with the condensation near the critical temperature. The value of the critical exponent thus obtained indeed suggests a universal mean field behavior.

NOTE ON ANALYTICAL STUDIES OF ONE-DIMENSIONAL HOLOGRAPHIC SUPERCONDUCTORS

We employ the variational method for the Sturm–Liouville eigenvalue problem to analytically study phase transition of one-dimensional holographic superconductors. It is shown that this method is not a very powerful method to analytically calculate the properties of holographic superconductors. From the analytical treatment of scalar operator condensate at critical temperature, we also show that the mean-field critical exponent 1/2 results from the coupling term between scalar field and vector field, which may be an universal property of holographic superconductors with a similar coupling term in their equations of motion.

Galoisian approach for a Sturm-Liouville problem on the infinite interval

We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval. Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with two examples for a stationary Schroedinger equation having a generalized Hulthen potential and an eigenvalue problem for a traveling front in the Allen-Cahn equation.

A New Approach for Linear Eigenvalue Problems and Nonlinear Euler Buckling Problem

We propose a numerical Taylor’s Decomposition method to compute approximate eigenvalues and eigenfunctions for regular Sturm‐Liouville eigenvalue problem and nonlinear Euler buckling problem very accurately for relatively large step sizes. For regular Sturm‐Liouville problem, the technique is illustrated with three examples and the numerical results show that the approximate eigenvalues are obtained with high‐order accuracy without using any correction, and they are compared with the results of other methods. The numerical results of Euler Buckling problem are compared with theoretical aspects, and it is seen that they agree with each other.

Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

Abstract In this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.

Analytical investigation of the phase transition between holographic insulator and superconductor in Gauss-Bonnet gravity

We employ the variational method for the Sturm-Liouville eigenvalue problem to analytically study the phase transition between the holographic insulator and superconductor in the Gauss-Bonnet gravity. By investigating the s-wave and p-wave holographic insulator/superconductor models, we find that this analytic method is more effective to obtain the analytic results on the condensation and the critical phenomena in the AdS soliton background in Gauss-Bonnet gravity. Our analytic result can be used to back up the numerical computations in the AdS soliton with Gauss-Bonnet correction.

Analytical studies on holographic superconductors in Gauss-Bonnet gravity

We use the variational method for the Sturm-Liouville eigenvalue problem to analytically calculate some properties of holographic superconductors with Gauss-Bonnet gravity in probe limit. By studying the holographic p-wave and s-wave superconductors in (3+1)-dimensional boundary field theories, it is found that near the critical temperature, the critical exponent of the condensation is 1/2 which is the universal value in mean-field theory. We also find that when Gauss-Bonnet coefficients grow bigger the operators on the boundary field theory will be harder to condense. These are in good agreement with the numerical results.

An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate

This article is concerned with uniqueness for reconstructing a periodic inhomogeneous medium sitting on a perfectly conducting plate. We deal with the problem in the framework of time-harmonic Maxwell systems without TE or TM polarization. An orthogonal relation is obtained for two refractive indices and then used to prove that the refractive index can be uniquely identified from a knowledge of the incident fields and the total tangential electric field on a plane above the inhomogeneous medium, utilizing the eigenvalues and eigenfunctions of a quasi-periodic Sturm–Liouville eigenvalue problem.

Solving Sturm–Liouville problems by piecewise perturbation methods, revisited

We present the extension of the successful Constant Perturbation Method (CPM) for Schrödinger problems to the more general class of Sturm–Liouville eigenvalue problems. Whereas the original CPM can only be applied to Sturm–Liouville problems after a Liouville transformation, the more general CPM presented here solves the Sturm–Liouville problem directly. This enlarges the range of applicability of the CPM to a wider variety of problems and allows a more efficient solution of many problems. The CPMs are closely related to the second-order coefficient approximation method underlying the SLEDGE software package, but provide for higher order approximations. These higher order approximations can also be obtained by applying a modified Neumann method. The CPM approach, however, leads to simpler formulae in a more convenient form.

An improved Tau method for a class of Sturm–Liouville problems

We consider the numerical solution of Sturm–Liouville eigenvalue problems by Legendre–Gauss Tau method. The latter approximates the solution of differential equations as a finite sum of Legendre polynomials { L k ( x ) ; k ∈ N } . We propose an improved approach which seeks approximants in terms of a finite sum of exponentially weighted Legendre polynomials e ω k x L k ( x ) ; k ∈ N for some real or complex frequencies { ω k } . With the introduction of such exponentials, Legendre–Gauss Tau method can detect the sharp variations exhibited by the highly indexed Sturm–Liouville eigenfunctions. The efficiency of our results is illustrated through numerical examples.

Application of Gegenbauer polynomial expansions to mitigate Gibbs phenomenon in Fourier–Bessel series solutions of a dynamic sphere problem

AbstractWe utilize the inverse polynomial reconstruction (IPR) method to mitigate the Gibbs phenomenon observed in Fourier–Bessel (FB) series. Gibbs phenomenon is the oscillatory behavior that occurs near discontinuities when evaluating series solutions for Sturm–Liouville eigenvalue problems. We employ an approach that uses expansions of the solution in terms of Gegenbauer polynomials on each side of solution discontinuities, the location of which must be known in advance. The IPR solutions provide pointwise values that are more accurate than the truncated series solution, which are polluted by Gibbs phenomenon. We apply this method to discontinuous solutions of a time dependent, linear elastic spherical shell problem, for which a series solution is derived in terms of a FB expansion. For the loading conditions and material properties, we consider the Gibbs phenomenon in the FB solution for a perfectly elastic shell renders the numerically evaluated results unusable as an ‘exact’ solution for code verification analysis. We quantify the degree to which the IPR method eliminates the Gibbs phenomenon in the computed solution. Copyright © 2009 John Wiley & Sons, Ltd.

Comparison between the Variational Iteration Method and the Homotopy Perturbation Method for the Sturm-Liouville Differential Equation

We applied the variational iteration method and the homotopy perturbation method to solve Sturm-Liouville eigenvalue and boundary value problems. The main advantage of these methods is the flexibility to give approximate and exact solutions to both linear and nonlinear problems without linearization or discretization. The results show that both methods are simple and effective.

Solution of Sturm–Liouville Problems Using Modified Neumann Schemes

The main purpose of this paper is to describe the extension of the successful modified integral series methods for Schrödinger problems to more general Sturm–Liouville eigenvalue problems. We present a robust and reliable modified Neumann method which can handle a wide variety of problems. This modified Neumann method is closely related to the second-order Pruess method but provides for higher-order approximations. We show that the method can be successfully implemented in a competitive automatic general-purpose software package.

Efficient numerical solution of the one-dimensional Schrodinger eigenvalue problem using Magnus integrators

We discuss a new numerical method, based on a modified Magnus integrator, to solve the Sturm-Liouville eigenvalue problem in its Schrodinger form. A modified Magnus integrator has already been used by Degani & Schiff (2006, J. Comput. Appl. Math., 193, 413-436) to approximate the oscillating solution of a Schrodinger problem over the classically allowed region. Here we show that the technique can be successfully extended to the nonoscillatory classically forbidden region. This means that the modified Magnus integrator can be used to propagate the solution over the whole integration interval and is well suited for application in a shooting process to locate the eigenvalues. Such a shooting procedure is formulated and shown to allow the efficient computation of a range of eigenvalues and eigenfunctions.

An analytical model for crosswind integrated concentrations released from a continuous source in a finite atmospheric boundary layer

An analytical model for the crosswind integrated concentrations released from a continuous source in a finite atmospheric boundary layer is formulated by considering the wind speed as a power law profile of vertical height above the ground and eddy diffusivity as an explicit function of downwind distance from the source and vertical height. A closed form analytical solution of the resulting advection–diffusion equation for these profiles of wind speed and eddy diffusivity with the physically relevant boundary conditions is derived using the separation of variables technique that leads to a Sturm–Liouville eigen value problem. Various particular cases of the model are deduced. The model is evaluated with the observations obtained from Prairie Grass experiment in various stability classes varying from very unstable to neutral and stable conditions and Hanford diffusion experiment in stable conditions. The agreement is found to be good between the computed and observed concentrations in both the diffusion experiments. For Prairie Grass experiment, the model is predicting 78% cases with in a factor of two and gives a value of NMSE as 0.075. On the other hand, for Hanford observations in stable conditions, it predicts 70% cases with in factor of two. An extensive analysis of statistical measures with the downwind distances from the source reveals that the model is performing well close to the source.

Applications of Prüfer transformations in the theory of ordinary differential equations

This article is a review article on the use of Prufer Transformations techniques in proving classical theorems from the theory of Ordinary Dierential Equations. We consider self-adjoint second order linear dierential equations of the form Lx = (p(t)x 0 (t)) 0 + g(t)x(t) = 0, t 2 (a,b). (?) We use Prufer transformation techniques (which are a gener- alization of Poincare phase-plane analysis) to obtain some of the main theorems of the classical theory of linear dieren- tial equations. First we prove theorems from the Oscillation Theory (Sturm Comparison theorem and Disconjugacy theo- rems). Furthermore we study the asymptotic behavior of the equation (?) when t ! 1 and we obtain necessary and suf- ficient conditions in order to have bounded solutions for (?). Finally, we consider a certain type of regular Sturm-Liouville eigenvalue problems with boundary conditions and we study their spectrum via Prufer transformations.

The Riesz Basis Property of an Indefinite Sturm-Liouville Problem with a Non-Odd Weight Function

For the Sturm-Liouville eigenvalue problem − f′′ = λrf on [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r changing its sign at 0 we discuss the question whether the eigenfunctions form a Riesz basis of the Hilbert space L 2 |r|[− 1, 1]. So far a number of sufficient conditions on r for the Riesz basis property are known. However, a sufficient and necessary condition is only known in the special case of an odd weight function r. We shall here give a generalization of this sufficient and necessary condition for certain generally non-odd weight functions satisfying an additional assumption.

On the estimate of mixing length in interdigital micromixers

Abstract The multilamination process that characterizes interdigital micromixers is an efficient and technologically feasible method for maximizing and controlling mass and/or heat transfer between two or more segregated fluid streams. We analyze the dynamics of mixing that takes place in the mixing channel downstream the interdigital apparatus. Specifically, we investigate, for different flow profiles, how the channel length necessary to achieve a prescribed level of mixedness depends on the degree of lamination (number and thickness of lamellae) of the feed stream. As a case study, we consider plug, shear and Poiseuille flow, and compare steady-state profiles resulting from the numerical simulation of the full advection–diffusion problem with the analytical solution stemming from the one-dimensional Sturm–Liouville eigenvalue problem along the spanwise coordinate, obtained neglecting streamwise diffusion. We find that (i) the mixing length can be significantly affected by the flow profile, especially at high degree of lamination of the feed stream, and (ii) in general, no obvious scaling between mixing length and lamellar thickness can be assumed. A rigorous way to approach the design of these micromixers is proposed.

Computation of eigenvalues and solutions of regular Sturm–Liouville problems using Haar wavelets

The paper presents a novel method for the computation of eigenvalues and solutions of Sturm–Liouville eigenvalue problems (SLEPs) using truncated Haar wavelet series. This is an extension of the technique proposed by Hsiao to solve discretized version of variational problems via Haar wavelets. The proposed method aims to cover a wider class of problems, by applying it to historically important and a very useful class of boundary value problems, thereby enhancing its applicability. To demonstrate the effectiveness and efficiency of the method various celebrated Sturm–Liouville problems are analyzed for their eigenvalues and solutions. Also, eigensystems are investigated for their asymptotic and oscillatory behavior. The proposed scheme, unlike the conventional numerical schemes, such as Rayleigh quotient and Rayleigh–Ritz approximation, gives eigenpairs simultaneously and provides upper and lower estimates of the smallest eigenvalue, and it is found to have quadratic convergence with increase in resolution.