scholarly journals Uniform estimation of the eigenvalues of Sturm–Liouville problems

1980 ◽  
Vol 21 (3) ◽  
pp. 365-383 ◽  
Author(s):  
John Paine ◽  
Frank de Hoog
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Normal Form ◽  
Error Bound ◽  
Small Perturbation ◽  
Liouville Problem ◽  
Coefficient Function ◽  
Uniform Error ◽  
Sturm Liouville ◽  
Uniformly Bounded

AbstractThe perturbation of the eigenvalues of a regular Sturm–Liouville problem in normal form which results from a small perturbation of the coefficient function is known to be uniformly bounded. For numerical methods based on approximating the coefficients of the differential equation, this result is used to show that a better bound on the error is obtained when the problem is in normal form. A method having a uniform error bound is presented, and an extension of this method for general Sturm–Liouville problems is proposed and examined.

scholarly journals On partial fractional Sturm–Liouville equation and inclusion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zohreh Zeinalabedini Charandabi ◽  
Hakimeh Mohammadi ◽  
Shahram Rezapour ◽  
Hashem Parvaneh Masiha
Keyword(s):  
Differential Equation ◽  
Liouville Equation ◽  
Existence Of Solutions ◽  
Liouville Problem ◽  
Existence Of A Solution ◽  
Contractive Mappings ◽  
Sturm Liouville

AbstractThe Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.

scholarly journals Fermions in the Rindler spacetime

2019 ◽  
Vol 16 (09) ◽  
pp. 1950140 ◽  
Author(s):  
L. C. N. Santos ◽  
C. C. Barros
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Energy Spectrum ◽  
Dirac Equation ◽  
Reference Frame ◽  
Bound States ◽  
Liouville Problem ◽  
External Potential ◽  
Compact Expression ◽  
Sturm Liouville

In this paper, we study the Dirac equation in the Rindler spacetime. The solution of the wave equation in an accelerated reference frame is obtained. The differential equation associated to this wave equation is mapped into a Sturm–Liouville problem of a Schrödinger-like equation. We derive a compact expression for the energy spectrum associated with the Dirac equation in an accelerated reference. It is shown that the noninertial effect of the accelerated reference frame mimics an external potential in the Dirac equation and, moreover, allows the formation of bound states.

X.—The Two Parameter Sturm-Liouville Problem for Ordinary Differential Equations

1971 ◽  
Vol 69 (2) ◽  
pp. 139-148
Author(s):  
B. D. Sleeman
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameter ◽  
General Conditions

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

On a singular Sturm-Liouville problem involving an advanced functional differential equation

2001 ◽  
Vol 12 (6) ◽  
pp. 625-644 ◽  
Author(s):  
B. VAN BRUNT ◽  
G. C. WAKE ◽  
H. K. KI M
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Infinite Number ◽  
Classical Problem ◽  
Liouville Problem ◽  
Advanced Argument ◽  
Isolated Points ◽  
Eigenvalue Parameter ◽  
Sturm Liouville ◽  
Functional Differential

Solutions to a boundary-value problem involving a second-order linear functional differential equation with an advanced argument are investigated in this paper. The boundary conditions imposed on the differential equation are analogous to conditions defining various singular Sturm-Liouville problems, and if an eigenvalue parameter is introduced certain properties of the spectrum can be deduced having analogues with the classical problem. Dirichlet series solutions are constructed for the problem and it is established that the spectrum contains an infinite number of real positive eigenvalues. A Laplace transform analysis of the problem then reveals that the spectrum does not generically consist of isolated points and that there may be an infinite number of eigenfunctions corresponding to a given eigenvalue. In contrast, it is also shown that there is a subset of eigenvalues that correspond to the zeros of an entire function for which the corresponding eigenfunctions are unique.

scholarly journals Solution of Ordinary Differential Equation Using Green's Function & Sturm - Liouville Problem

2019 ◽  
pp. 241-244
Author(s):  
S. Angel Auxzaline Mary ◽  
T. Ramesh
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Initial Value Problem ◽  
Series Representation ◽  
Liouville Problem ◽  
Initial Value ◽  
Sturm Liouville

In this paper, we describe Green's function to determine the importance of this function, i.e. Boundary & Initial Value problem, Sturm-Liouville Problem. Along with the series representation of Green's Function.

scholarly journals A Sturm–Liouville Problem Arising in the Atmospheric Boundary-Layer Dynamics

2020 ◽  
Vol 22 (3) ◽  
Author(s):  
Kateryna Marynets
Keyword(s):  
Differential Equation ◽  
Eddy Viscosity ◽  
Order Differential Equation ◽  
Liouville Problem ◽  
Explicit Solutions ◽  
Governing Equations ◽  
Liouville Type ◽  
First Order ◽  
Sturm Liouville ◽  
Boundary Layer Dynamics

Abstract We present an approach that facilitates the generation of explicit solutions to atmospheric Ekman flows with a height-dependent eddy viscosity. The approach relies on applying to the governing equations, of Sturm–Liouville type, a suitable Liouville substitution and then reducing the outcome to a nonlinear first-order differential equation of Riccati type.

On a Non-Analytic Perturbation Problem

1961 ◽  
Vol 4 (3) ◽  
pp. 243-248 ◽  
Author(s):  
C. A. Swanson
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Closed Interval ◽  
Liouville Problem ◽  
Simple Problem ◽  
Asymptotic Estimates ◽  
Perturbation Problem ◽  
Sturm Liouville ◽  
Characteristic Values ◽  
Asymptotic Perturbation

The purpose here is to study a type of perturbation problem, arising from a differential equation, which is not included in the realm of analytic or asymptotic perturbation theory [4], [6]. Such a problem arises when the domain of the differential operator has been subjected to a variation (rather than the formal operator). We propose to outline one simple problem of this type, concerned with a second order ordinary differential operator. Our purpose is to obtain asymptotic estimates for the characteristic values of a regular Sturm-Liouville problem on a closed interval [a, b] when b is near a singular point of the differential operator. Similar results have been obtained in [3], [8], and [9] by other methods.

On an extension of the theorem of V. A. Ambarzumyan

1988 ◽  
Vol 110 (1-2) ◽  
pp. 79-84 ◽  
Author(s):  
N.K. Chakravarty ◽  
Sudip Kumar Acharyya
Keyword(s):  
Differential Equation ◽  
Liouville Problem ◽  
Second Order ◽  
Differential Systems ◽  
Order Matrix ◽  
Sturm Liouville ◽  
Matrix Differential

SynopsisThe Ambarzumyan theorem connecting the Sturm–Liouville problem and the corresponding problem associated with the Fourier differential equation is extended to a class of second order matrix differential systems.

scholarly journals Non-axisymmetric oscillations of thin twisted magnetic tubes

2007 ◽  
Vol 3 (S247) ◽  
pp. 344-350
Author(s):  
Michael S. Ruderman
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Liouville Problem ◽  
Density Variation ◽  
Tube Axis ◽  
Order Ordinary Differential Equation ◽  
Azimuthal Component ◽  
Kink Mode ◽  
Sturm Liouville ◽  
The Magnetic Field

AbstractIn this paper we study non-axisymmetric oscillations of thin twisted magnetic tubes taking the density variation along the tube into account. We use the approximation of the zero-beta plasma. The magnetic field outside the tube is straight and homogeneous, however it is twisted inside the tube. We assume that the azimuthal component of the magnetic field is proportional to the distance from the tube axis, and that the tube is only weakly twisted, i.e. the ratio of the azimuthal and axial components of the magnetic field is small. Using the asymptotic analysis we show that the eigenmodes and eigenfrequencies of the kink and fluting oscillations are described by a classical Sturm-Liouville problem for a second order ordinary differential equation. The main result is that the twist does not affect the kink mode.

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