Sturm-Liouville Eigenvalue Problem Research Articles

A Subspace Pursuit Method to Infer Refractivity in the Marine Atmospheric Boundary Layer

Inferring electromagnetic (EM) propagation characteristics within the marine atmospheric boundary layer (MABL) from data in real time is crucial for modern maritime navigation and communications. The propagation of EM waves is well modeled by a partial differential equation (PDE): a Helmholtz equation. A natural way to solve the MABL characterization inverse problem is to minimize what is observed and what is predicted by the PDE. However, this optimization is difficult because it has many local minima. We propose an alternative solution that relies on the properties of the PDE but does not involve solving the full forward model. Ducted environments result in an EM field which can be decomposed into a few propagating, trapped modes. These modes are a subset of the solutions to a Sturm–Liouville eigenvalue problem. We design a new objective function that measures the distance from the observations to a subspace spanned by these eigenvectors. The resulting optimization problem is much easier than the one that arises in the standard approach, and we show how to solve the associated nonlinear eigenvalue problem efficiently, leading to a real-time method.

Sampling eigenvalues by Hermite revisited

ABSTRACTWe show how the Hermite sampling method can be used to approximate the eigenvalues of Sturm-Liouville problems. Because it involves the derivatives with respect to the eigenvalue parameter, it requires more smoothness on the coefficients and integrating a one parameter family of matrix Sturm-Liouville operators to generate the required samples.

Note on a nonlocal Sturm–Liouville problem with both right and left fractional derivatives

In this paper, we research the geometric multiplicity of eigenvalues for a nonlocal Sturm–Liouville eigenvalue problem. To this end, we study the uniqueness of solutions for a nonlocal Sturm–Liouville problem under some initial value conditions.

Holographic superconductor with nonlinear Born–Infeld-type electrodynamics

Holographic s-wave superconductors in the framework of nonlinear Born–Infeld-type electrodynamics are investigated in the background of Schwarzschild anti-de Sitter black holes. As particular cases, at some model parameters, we obtain results for Born–Infeld and exponential electrodynamics. We explore the analytical Sturm–Liouville eigenvalue problem in the probe limit where the scalar and electromagnetic fields do not affect the background metric. The critical temperatures of phase transitions and the order parameter are calculated which depend on the model parameters. We show that the critical exponent near the critical temperature is 1/2. Making use of the matching method, we derive analytical expressions for the condensation values and the critical temperature. The conductivity by the analytical method is calculated.

Uniform local Lipschitz continuity of eigenvalues with respect to the potential in L^1[a,b

The present paper shows that the eigenvalue sequence $\{\lambda_n(q)\}_{n\geqslant 1}$ of regular Sturm-Liouville eigenvalue problem with certain monotonic weights is uniformly Lipschitz continuous with respect to the potential $q$ on any bounded subset of $L^1([a,b],\mathbb{R})$.

One-dimensional backreacting holographic p-wave superconductors

We analytically and numerically study the properties of one-dimensional holographic p-wave superconductors in the presence of backreaction. We employ the Sturm–Liouville eigenvalue problem for the analytical calculation and the shooting method for the numerical investigations. We apply the hbox {AdS}_{{3}}/hbox {CFT}_{{2}} correspondence and determine the relation between the critical temperature T_{c} and the chemical potential mu for different values of the mass m of a charged spin-1 field rho _{mu } and backreacting parameters. We observe that the data of both analytical and numerical studies are in good agreement. We find that increasing the backreaction and the mass parameter causes the greater values for {T_{c}}/{mu }. Thus, it makes the condensation harder to form. In addition, the analytical and numerical approaches show that the value of the critical exponent beta is 1 / 2, which is the same as in the mean field theory. Moreover, both methods confirm the existence of a second order phase transition.

On generalized and fractional derivatives and their applications to classical mechanics

A generalized differential operator on the real line is defined by means of a limiting process. When an additional fractional parameter is introduced, this process leads to a locally defined fractional derivative. The study of such generalized derivatives includes, as a special case, basic results involving the classical derivative and current research involving fractional differential operators. All our operators satisfy properties such as the sum, product/quotient rules, and the chain rule. We study a Sturm–Liouville eigenvalue problem with generalized derivatives and show that the general case is actually a consequence of standard Sturm–Liouville Theory. As an application of the developments herein we find the general solution of a generalized harmonic oscillator. We also consider the classical problem of a planar motion under a central force and show that the general solution of this problem is still generically an ellipse, and that this result is true independently of the choice of the generalized derivatives being used modulo a time shift. The generic nature of phase plane orbits modulo a time shift is extended to the classical gravitational n-body problem of Newton to show that the global nature of these orbits is independent of the choice of the generalized derivatives being used in defining the force law. Finally, restricting the generalized derivatives to a special class of fractional derivatives, we consider the question of motion under gravity with and without resistance and arrive at a new notion of time that depends on the fractional parameter. The results herein are meant to clarify and extend many known results in the literature and intended to show the limitations and use of generalized derivatives and corresponding fractional derivatives.

A unified treatment of polynomial solutions and constraint polynomials of the Rabi models

General concept of a gradation slicing is used to analyze polynomial solutions of ordinary differential equations (ODE) with polynomial coefficients, , where , pl(z) are polynomials, z is a coordinate, and . It is not required that ODE is either (i) Fuchsian or (ii) leads to a usual Sturm–Liouville eigenvalue problem. General necessary and sufficient conditions for the existence of a polynomial solution are formulated involving constraint relations. The necessary condition for a polynomial solution of nth degree to exist forces energy to a nth baseline. Once the constraint relations on the nth baseline can be solved, a polynomial solution is in principle possible even in the absence of any underlying algebraic structure. The usefulness of theory is demonstrated on the examples of various Rabi models. For those models, a baseline is known as a Juddian baseline (e.g. in the case of the Rabi model the curve described by the nth energy level of a displaced harmonic oscillator with varying coupling g). The corresponding constraint relations are shown to (i) reproduce known constraint polynomials for the usual and driven Rabi models and (ii) generate hitherto unknown constraint polynomials for the two-mode, two-photon, and generalized Rabi models, implying that the eigenvalues of corresponding polynomial eigenfunctions can be determined algebraically. Interestingly, the ODE of the above Rabi models are shown to be characterized, at least for some parameter range, by the same unique set of grading parameters.

An accurate method for solving a class of fractional Sturm-Liouville eigenvalue problems

Abstract This article is devoted to both theoretical and numerical study of the eigenvalues of nonsingular fractional second-order Sturm-Liouville problem. In this paper, we implement a fractional-order Legendre Tau method to approximate the eigenvalues. This method transforms the Sturm-Liouville problem to a sparse nonsingular linear system which is solved using the continuation method. Theoretical results for the considered problem are provided and proved. Numerical results are presented to show the efficiency of the proposed method.

Multiple eigenmodes of the Rayleigh-Taylor instability observed for a fluid interface with smoothly varying density.

In this article, multiple eigen-systems including linear growth rates and eigen-functions have been discovered for the Rayleigh-Taylor instability (RTI) by numerically solving the Sturm-Liouville eigen-value problem in the case of two-dimensional plane geometry. The system called the first mode has the maximal linear growth rate and is just extensively studied in literature. Higher modes have smaller eigen-values, but possess multi-peak eigen-functions which bring on multiple pairs of vortices in the vorticity field. A general fitting expression for the first four eigen-modes is presented. Direct numerical simulations show that high modes lead to appearances of multi-layered spike-bubble pairs, and lots of secondary spikes and bubbles are also generated due to the interactions between internal spikes and bubbles. The present work has potential applications in many research and engineering areas, e.g., in reducing the RTI growth during capsule implosions in inertial confinement fusion.

A Sub-Density Theorem of Sturm-Liouville Eigenvalue Problem with Finitely Many Singularities

We study the distribution of the Sturm-Liouville eigenvalues of a potential with finitely many singularities. There is an asymptotically periodical structure on this class of eigenvalues as described by the entire function theory. We describe the singularities of its potential function explicitly in its eigenvalue asymptotics.

Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order 2 < α < 5/2

Fractional difference operators with discrete-Mittag-Leffler kernels of order α > 1 are defined and their corresponding fractional sum operators are confirmed. We prove existence and uniqueness theorems for the discrete fractional initial value problems in the frame of discrete Caputo (ABC) and Riemann (ABR) operators by using Banach contraction theorem. Then, we prove Lyapunov type inequality for a Riemann type fractional difference boundary value problem of order 2 < α < 5∕2 within discrete Mittag-Leffler kernels, where the limiting case α → 2+ results in the ordinary difference Lyapunov inequality. Examples are given to clarify the applicability of our results and an application about the discrete fractional Sturm-Liouville eigenvalue problem is analyzed.

HiMod Reduction of Advection–Diffusion–Reaction Problems with General Boundary Conditions

We extend the hierarchical model reduction procedure previously introduced in Ern et al. (in: Kunisch, Of, Steinbach (eds) Numerical mathematics and advanced applications, Springer, Berlin, pp 703–710, 2008) and Perotto et al. (Multiscale Model Simul 8(4):1102–1127, 2010) to deal with general boundary conditions, enforcing their prescription in the basis function set. This is achieved by solving a Sturm–Liouville Eigenvalue problem. We analyze this approach and provide a convergence analysis for the associated error in the case of a linear advection–diffusion–reaction problem in rectangles (2D) and slabs (3D). Numerical results confirm the theoretical investigation and the reliability of the proposed approach.

A generalized Lyapunov-type inequality in the frame of conformable derivatives

We prove a generalized Lyapunov-type inequality for a conformable boundary value problem (BVP) of order alpha in (1,2]. Indeed, it is shown that if the boundary value problem \t\t\t(Tαcx)(t)+r(t)x(t)=0,t∈(c,d),x(c)=x(d)=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl(\\textbf{T}_{\\alpha }^{c} x\\bigr) (t)+r(t)x(t)=0,\\quad t \\in (c,d), x(c)=x(d)=0 $$\\end{document} has a nontrivial solution, where r is a real-valued continuous function on [c,d], then \t\t\t1∫cd|r(t)|dt>αα(α−1)α−1(d−c)α−1.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\int_{c}^{d} \\bigl\\vert r(t) \\bigr\\vert \\,dt> \\frac{\\alpha^{\\alpha }}{(\\alpha -1)^{\\alpha -1}(d-c)^{ \\alpha -1}}. $$\\end{document} Moreover, a Lyapunov type inequality of the form \t\t\t2∫cd|r(t)|dt>3α−1(d−c)2α−1(3α−12α−1)2α−1α,12<α≤1,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\int_{c}^{d} \\bigl\\vert r(t) \\bigr\\vert \\,dt> \\frac{3\\alpha -1}{(d-c)^{2\\alpha -1}} \\biggl( \\frac{3 \\alpha -1}{2\\alpha -1} \\biggr) ^{\\frac{2\\alpha -1}{\\alpha }},\\quad \\frac{1}{2}< \\alpha \\leq 1, $$\\end{document} is obtained for a sequential conformable BVP. Some examples are given and an application to conformable Sturm-Liouville eigenvalue problem is analyzed.

Fractional operators with exponential kernels and a Lyapunov type inequality

In this article, we extend fractional calculus with nonsingular exponential kernels, initiated recently by Caputo and Fabrizio, to higher order. The extension is given to both left and right fractional derivatives and integrals. We prove existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial value problems by using Banach contraction theorem. Then we prove Lyapunov type inequality for the Riemann type fractional boundary value problems within the exponential kernels. Illustrative examples are analyzed and an application about Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.

A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel

In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order alphain[0,1] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo (ABC) and Riemann (ABR) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order 2<alphaleq3 in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.

Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications

Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order0&lt;α≤1with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order2&lt;α≤3is proved and the ordinary difference Lyapunov inequality then follows asαtends to2from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.

A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems

This article is devoted to both theoretical and numerical studies of eigenvalues of regular fractional $2\alpha $-order Sturm-Liouville problem where $\frac{1}{2}< \alpha \leq 1$. In this paper, we implement the reproducing kernel method RKM) to approximate the eigenvalues. To find the eigenvalues, we force the approximate solution produced by the RKM satisfy the boundary condition at $x=1$. The fractional derivative is described in the Caputo sense. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the eigenfunctions of the proposed problem. Uniformly convergence of the approximate eigenfunctions produced by the RKM to the exact eigenfunctions is proven.

Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

Variational methods for the solution of fractional discrete/continuous Sturm–Liouville problems

The fractional Sturm-Liouville eigenvalue problem appears in many situations, e.g., while solving anomalous diffusion equations coming from physical and engineering applications. Therefore to obtain solutions or approximation of solutions to this problem is of great importance. Here, we describe how the fractional Sturm-Liouville eigenvalue problem can be formulated as a constrained fractional variational principle and show how such formulation can be used in order to approximate the solutions. Numerical examples are given, to illustrate the method.