scholarly journals Fundamental systems and solutions of nonhomogeneous equations for a pair of mixed linear ordinary differential equations

1990 ◽  
Vol 49 (1) ◽  
pp. 161-173
Author(s):  
M. Venkatesulu ◽  
T. Gnana Bhaskar
Keyword(s):  
Differential Equations ◽  
Systematic Study ◽  
Differential Operators ◽  
Linear Ordinary Differential Equations ◽  
Physical Conditions ◽  
Transverse Vibrations ◽  
Ordinary Differential Operators ◽  
Acoustic Waveguides ◽  
Complete Set ◽  
Fundamental Systems

AbstractTwo different ordinary differential operators L1 and L2 (not of the same order) defined on two adjacent intervals I1 and I2, respectively, with certain mixed conditions at the interface are considered. These problems are encountered in the study of ‘acoustic waveguides in ocean’, ‘transverse vibrations in nonhomogeneous strings’, etc. A complete set of physical conditions on the system give rise to three types of (selfadjoint) boundary value problems associated with the pair (L1, L2). In a series of papers, a systematic study of these new classes of problems is being developed. In the present paper, we construct the fundamental systems and exhibit the forms of solutions of nonhomogeneous problems associated with the pair (L1, L2).

scholarly journals Computation of Green's matrices for boundary value problems associated with a pair of mixed linear regular ordinary differential operators

1995 ◽  
Vol 18 (4) ◽  
pp. 789-797 ◽  
Author(s):  
T. Gnana Bhaskar ◽  
M. Venkatesulu
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Transverse Vibrations ◽  
Ordinary Differential Operators ◽  
Acoustic Waveguides

An algorithm for the computation of Green's matrices for boundary value problems associated with a pair of mixed linear regular ordinary differential operators is presented and two examples from the studies of acoustic waveguides in ocean and transverse vibrations in nonhomogeneous strings are discussed.

V.— On the Spectrum of Second Order Partial Differential Operators.

1970 ◽  
Vol 69 (1) ◽  
pp. 77-84
Author(s):  
K. J. Brown ◽  
I. M. Michael
Keyword(s):  
Differential Equations ◽  
Spectral Properties ◽  
Differential Operators ◽  
Second Order ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Ordinary Differential Operators

SynopsisIn a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.

Limit Point Criteria for Differential Equations

1972 ◽  
Vol 24 (2) ◽  
pp. 293-305 ◽  
Author(s):  
Don Hinton
Keyword(s):  
Differential Equations ◽  
Limit Point ◽  
Differential Operators ◽  
Central Importance ◽  
Linearly Independent ◽  
Ordinary Differential Operators ◽  
Image Position

For certain ordinary differential operators L of order 2n, this paper considers the problem of determining the number of linearly independent solutions of class L2[a, ∞) of the equation L(y) = λy. Of central importance is the operator0.1where the coefficients pi are real. For this L, classical results give that the number m of linearly independent L2[a, ∞) solutions of L(y) = λy is the same for all non-real λ, and is at least n [10, Chapter V]. When m = n, the operator L is said to be in the limit-point condition at infinity. We consider here conditions on the coefficients pi of L which imply m = n. These conditions are in the form of limitations on the growth of the coefficients.

scholarly journals The computational approach to commuting ordinary differential operators of orders six and nine

1994 ◽  
Vol 35 (4) ◽  
pp. 399-419 ◽  
Author(s):  
Geoff A. Latham
Keyword(s):  
Differential Equations ◽  
Elliptic Curve ◽  
Ordinary Differential Equations ◽  
Differential Operators ◽  
Computational Approach ◽  
Nonlinear Ordinary Differential Equations ◽  
Ordinary Differential Operators ◽  
Elementary Methods ◽  
Explicit Formulae ◽  
Rational Expressions

AbstractEntirely elementary methods are employed to determine explicit formulae for the coefficients of commuting ordinary differential operators of orders six and nine which correspond to an elliptic curve. These formulae come from solving the nonlinear ordinary differential equations which are equivalent to the commutativity condition. Most solutions turn out to be rational expressions in one or two arbitrary functions and their derivatives. The corresponding Burchnall-Chaundy curves are computed.

scholarly journals The variational theory of higher-order linear differential equations

1984 ◽  
Vol 95 ◽  
pp. 137-161 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Differential Equations ◽  
Compact Riemann Surface ◽  
Differential Operators ◽  
Higher Order ◽  
Variational Theory ◽  
Open Set ◽  
Ordinary Differential Operators ◽  
Linear Differential ◽  
Higher Order Differential Equations ◽  
Projective Coordinate

In his paper [2], [3], D. A. Hejhal investigated the variational theory of linear polynomic functions. In this paper we are concerned with the variational theory of higher-order differential equations. To be more precise, consider a compact Riemann surface having genus g > 1. As is well known, we can choose a projective coordinate covering U = (Ua, za). Fix this coordinate covering of X. We shall be concerned with linear ordinary differential operators of order n defined in each projective coordinate open set Ua

Spectral Methods

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Simple Model ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Algebra ◽  
Spectral Methods ◽  
Differential Operators ◽  
Higher Dimensions ◽  
Standard Linear ◽  
Sommerfeld Equation ◽  
Spectral Discretization

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

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