Some Spectral Properties of the Heun Differential Equation

2002 ◽  
pp. 87-110 ◽  
Author(s):  
P. B. Bailey ◽  
W. N. Everitt ◽  
D. B. Hinton ◽  
A. Zettl
Keyword(s):  
Differential Equation ◽  
Spectral Properties

Additional spectral properties of the fourth-order Bessel-type differential equation

2005 ◽  
Vol 278 (12-13) ◽  
pp. 1538-1549 ◽  
Author(s):  
W. N. Everitt ◽  
H. Kalf ◽  
L. L. Littlejohn ◽  
C. Markett
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Spectral Properties ◽  
Type Differential Equation

IMPLEMENTATIONS OF BOUNDARY–DOMAIN INTEGRO-DIFFERENTIAL EQUATION FOR DIRICHLET BVP WITH VARIABLE COEFFICIENT

2016 ◽  
Vol 78 (6-5) ◽  
Author(s):  
Nurul Akmal Mohamed ◽  
Nur Fadhilah Ibrahim ◽  
Mohd Rozni Md Yusof ◽  
Nurul Farihan Mohamed ◽  
Nurul Huda Mohamed
Keyword(s):  
Differential Equation ◽  
Quadrature Formula ◽  
Spectral Properties ◽  
Variable Coefficient ◽  
Logarithmic Singularity ◽  
Neumann Series ◽  
Numerical Results ◽  
Analytic Method ◽  
Elliptic Type ◽  
Integro Differential Equation

In this paper, we present the numerical results of the Boundary-Domain Integro-Differential Equation (BDIDE) associated to Dirichlet problem for an elliptic type Partial Differential Equation (PDE) with a variable coefficient. The numerical constructions are based on discretizing the boundary of the problem region by utilizing continuous linear iso-parametric elements while the domain of the problem region is meshed by using iso-parametric quadrilateral bilinear domain elements. We also use a semi-analytic method to handle the integration that exhibits logarithmic singularity instead of using Gauss-Laguare quadrature formula. The numerical results that employed the semi-analytic method give better accuracy as compared to those when we use Gauss-Laguerre quadrature formula. The system of equations that obtained by the discretized BDIDE is solved by an iterative method (Neumann series expansion) as well as a direct method (LU decomposition method). From our numerical experiments on all test domains, the relative errors of the solutions when applying semi-analytic method are smaller than when we use Gauss-Laguerre quadrature formula for the integration with logarithmic singularity. Unlike Dirichlet Boundary Integral Equation (BIE), the spectral properties of the Dirichlet BDIDE is not known. The Neumann iterations will converge to the solution if and only if the spectral radius of matrix operator is less than 1. In our numerical experiment on all the test domains, the Neumann series does converge. It gives some conclusions for the spectral properties of the Dirichlet BDIDE even though more experiments on the general Dirichlet problems need to be carried out.

On the spectral properties and positivity of solutionsof a periodic boundary value problemfor a second-order functional differential equation

2020 ◽  
pp. 123-130
Author(s):  
Manuel J. Alves ◽  
Sergey M. Labovskiy
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Functional Differential Equation ◽  
Spectral Properties ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Functional Differential ◽  
The Green Function

For a functional-differential operator Lu = (1/ρ)(-(pu')' + ∫_0^l▒〖u(s)d_s r(x,s)〗) with symmetry, the completeness and orthogonality of the eigenfunctions is shown. Thepositivity conditions of the Green function of the periodic boundary value problem areobtained.

On the study of the spectral properties of differential operators with a smooth weight function

2020 ◽  
pp. 25-47
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Operator ◽  
Spectral Properties ◽  
Differential Operators ◽  
Asymptotic Estimates ◽  
Third Order ◽  
Indicator Diagram ◽  
Estimates Of Solutions ◽  
Smooth Coefficients

In this paper we study the spectral properties of a third-order differential operator with a summable potential with a smooth weight function. The boundary conditions are separated. The method of studying differential operators with summable potential is a development of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The differential equation defining the differential operator is reduced to the solution of the Volterra integral equation by means of the method of variation of constants. The solution of the integral equation is found by the method of successive Picard approximations. Using the study of an integral equation, we obtained asymptotic formulas and estimates for the solutions of a differential equation defining a differential operator. For large values of the spectral parameter, the asymptotics of solutions of the differential equation that defines the differential operator is derived. Asymptotic estimates of solutions of a differential equation are obtained in the same way as asymptotic estimates of solutions of a differential operator with smooth coefficients. The study of boundary conditions leads to the study of the roots of the function, presented in the form of a third-order determinant. To get the roots of this function, the indicator diagram wasstudied. The roots of this equation are in three sectors of an infinitely small size, given by the indicator diagram. The article studies the behavior of the roots of this equation in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the differential operator under study is calculated. The formulas found for the asymptotics of eigenvalues allow us to study the spectral properties of the eigenfunctions of the differential operator under study.

scholarly journals Nonlocal multipoint problem for a differential equation of $2n$-th order with operator coefficients

2021 ◽  
Vol 13 (2) ◽  
pp. 501-514
Author(s):  
Ya.O. Baranetskij ◽  
I.I. Demkiv ◽  
A.V. Solomko ◽  
O.M. Sus'
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Multipoint Problem ◽  
Root Functions ◽  
Multiple Eigenvalues ◽  
Finite Set ◽  
Uniqueness Of The Solution ◽  
Differential Operator Equation ◽  
Transmutation Operators

In the article, the spectral properties of a multipoint problem for a differential operator equation of order $2n$ are studied. The operator of the problem has an infinite number of multiple eigenvalues. Each multiple eigenvalue corresponds to a finite set of root functions. A commutative group of transmutation operators is constructed. Each element of the group corresponds to the isospectral perturbation of the problem operator with antiperiodic conditions. The conditions for the existence and uniqueness of the solution are established for the selected family of multipoint problems, and this solution is constructed too.

A general version of the Hardy–Littlewood–Polya–Everitt (HELP) inequality

1984 ◽  
Vol 97 ◽  
pp. 9-20 ◽  
Author(s):  
Christer Bennewitz
Keyword(s):  
Differential Equation ◽  
Spectral Properties ◽  
Positive Function ◽  
Best Constants ◽  
The Other ◽  
Valid Inequality ◽  
General Version ◽  
Best Constant ◽  
Sturm Liouville

SynopsisThe inequality (0·1) below is naturally associated with the equation −(pu′)′ + qu = λu. By assuming that one end-point of the interval (a, b) is regular and the other limit-point for this equation, Everitt characterized the best constant K in tems of spectral properties of the equation. This paper sketches a theory for more general inequalities (0·2), (0·3) similarly related to the equation Su = λTu. Here S and T are ordinary, symmetric differential expressions. A characterization of the best constants in (0·2), (0·3) is given which generalises that of Everitt.For the case when S is of order 1 and T is multiplication by a positive function, all possible inequalities are given together with the best constants and cases of equality. Furthermore, an example is given of a valid inequality (0·1) on an interval with both end-points regular for the corresponding differential equation. This contradicts a conjecture by Everitt and Evans. Finally, the general theory for the left-definite inequality (0·3) is specialised to the case when S is a Sturm-Liouville expression. A family of examples is given for which the best constants can be explicitly calculated.

VII.—On the Spectrum of Ordinary Second Order Differential Operators.

1969 ◽  
Vol 68 (2) ◽  
pp. 95-119 ◽  
Author(s):  
Jyoti Chaudhuri ◽  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Analytic Function ◽  
Spectral Theory ◽  
Spectral Properties ◽  
Differential Operators ◽  
Second Order ◽  
Space Theory ◽  
Alternative Approach

SynopsisThis paper considers properties of the spectrum of differential operators derived from differential expressions of the second order. The object is to link the spectral properties of these differential operators with the analytic, function-theoretic properties of the solutions of the differential equation. This provides an alternative approach to the spectral theory of these differential operators but one which is consistent with the standard definitions used in Hilbert space theory. In this way the approach may be of interest to applied mathematicians and theoretical physicists.

scholarly journals The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

2020 ◽  
Vol 54 (1) ◽  
pp. 64-78 ◽  
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M. I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Spectral Properties ◽  
Nonlocal Problem ◽  
Sufficient Conditions ◽  
Adjoint Problem ◽  
Multipoint Problem ◽  
Even Order

The spectral properties of the nonself-adjoint problem with multipoint perturbations of the Dirichlet conditions for differential operator of order $2n$ with involution are investigated. The system of eigenfunctions of a multipoint problem is constructed. Sufficient conditions have been established, under which this system is complete and, under some additional assumptions, forms the Riesz basis. The research is structured as follows. In section 2 we investigate the properties of the Sturm-type conditions and nonlocal problem with self-adjoint boundary conditions for the equation $$(-1)^ny^{(2n)}(x)+ a_{0}y^{(2n-1)}(x)+ a_{1}y^{(2n-1)}(1-x)=f(x),\,x\in (0,1).$$ In section 3 we study the spectral properties for nonlocal problem with nonself-adjoint boundary conditions for this equation. In sections 4 we construct a commutative group of transformation operators. Using spectral properties of multipoint problem and conditions for completeness the basis properties of the systems of eigenfunctions are established in section 5. In section 6 some analogous results are obtained for multipoint problems generated by differential equations with an involution and are proved the main theorems.

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