Sturm-liouville problem for a differential equation of second order with discontinuous coefficients

2000 ◽  
Vol 73 (4) ◽  
pp. 735-740 ◽  
Author(s):  
B. V. Averin ◽  
D. I. Kolotilkin ◽  
V. A. Kudinov
Keyword(s):  
Differential Equation ◽  
Liouville Problem ◽  
Discontinuous Coefficients ◽  
Second Order ◽  
Sturm Liouville

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 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.

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 Completeness of the system of eigenfunctions of the Sturm-Liouville problem with the singularity

2020 ◽  
Vol 30 (1) ◽  
pp. 59-63
Author(s):  
V.P. Tanana
Keyword(s):  
Thermal Conductivity ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Liouville Problem ◽  
Discontinuous Coefficients ◽  
Order Differential Operator ◽  
Temperature Conductivity ◽  
Inverse Boundary ◽  
Sturm Liouville ◽  
Inverse Boundary Value Problems

Mathematical modeling of composite materials plays an important role in modern technology, and the solution and study of inverse boundary value problems of heat transfer is impossible without the use of systems of eigenfunctions of the Sturm-Liouville problem for the differential equation with discontinuous coefficients. One of the most important properties of such systems is their completeness in the corresponding spaces. This property of systems allows to prove theorems of existence and uniqueness of both direct problems and inverse boundary value problems of thermal conductivity, and also to prove numerical methods of solving such problems. In this paper, we prove the completeness of the Sturm-Liouville problem in the space $L_2[r_0,r_2]$ for a second-order differential operator with a discontinuous coefficient. This problem arises when investigating and solving the inverse boundary problem of thermal conductivity for a hollow ball consisting of two balls with different temperature conductivity coefficients. Self-conjugacy, injectivity, and positive definiteness of this operator are proved.

Oscillation criteria of Nehari-type for Sturm–Liouville operators and elliptic differential operators of second order and the lower spectrum

1988 ◽  
Vol 109 (1-2) ◽  
pp. 127-144 ◽  
Author(s):  
F. Fiedler
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Oscillation Criteria ◽  
Elliptic Differential Equations ◽  
Ordinary Differential Operators ◽  
Elliptic Differential Operators ◽  
Sturm Liouville ◽  
Lower Spectrum ◽  
Liouville Operators

SynopsisSufficient oscillation criteria of Nehari-type are established for the differential equation −uʺ(t) + q(t)u(t) = 0, 0<t<∞, with and without sign restrictions on q(t), respectively. These results are extended to Sturm-Liouville equations and elliptic differential equations of second order.In Section 7 we present conclusions for the lower spectrum of elliptic differential operators and also for the discreteness of the spectrum of certain ordinary differential operators of second order.

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.

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