<i>Lectures on Ordinary Differential Equations</i>

Hurewicz, Witold; Davis, Philip J.

doi:10.1063/1.3062332

Cited by 158 publications

(94 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The SLME expressed in (8) has three important properties: 1) In the case of an homogeneous layer the linearly independent (LI) solutions of the differential equations system (8) can be expressed by means of exponentials [18,19]:…”

Section: Slme Analysismentioning

confidence: 99%

See 1 more Smart Citation

Sturm-Liouville Matrix Equation for the Study of Electromagnetic-Waves Propagation in Layered Anisotropic Media

Pernas-Salomón¹,

Pérez‐Álvarez²

2014

PIER M

View full text Add to dashboard Cite

Abstract-We obtain a Sturm-Lioville matrix equation of motion (SLME) for the study of electromagnetic wave propagation in layered anisotropic structures. Conducting media were taken into account so that ohmic loss is considered. This equation can be treated using a 4 × 4 associated transfer matrix (T) in layered anisotropic structures, where the tensors: permittivity, permeability and the electric conductivity have a piecewise dependence on the coordinate perpendicular to the layered structure. We use the SLME eigenfunctions and eigenvalues to analyze qualitatively the numerical instability (Ωd problem) which potentially affects practical applications of the transfer matrix method. By means of the SLME coefficients we show analytically that T determinant value can be used to keep a check on the numerical accuracy of calculations. We derive equations to analyze wave propagation in linear layered isotropic structures. The SLME approach is applied on two typical layered structures to verify theoretical predictions and experimental results.

show abstract

Section: Slme Analysismentioning

confidence: 99%

“…This matrix, denoted here by Hy can be defined by changing the arrangement of the E T (z), A(z), E T (z 0 ) and A(z 0 ) vectors in (18):…”

Section: Electromagnetic-wave Propagation In Linear Layered Isotropicmentioning

confidence: 99%

Sturm-Liouville Matrix Equation for the Study of Electromagnetic-Waves Propagation in Layered Anisotropic Media

Pernas-Salomón¹,

Pérez‐Álvarez²

2014

PIER M

View full text Add to dashboard Cite

show abstract

“…Since the fundamental mode and the first few harmonics contribute most to the flux distribution the series expansion for the Laplace transformed flux will be truncated a s follows: Now the following inequalities must all be satisfied for stability of the perturbed flux a s given by equation (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24).…”

Section: (5-22)mentioning

confidence: 99%

“…Substitution of equations (6-15) a r Furthermore, since it was assumed that wo := e , then wl becomes, Thus, imposing the boundary conditions found in equations (6-21) upon equations (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) and (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) leads to the following matrix formulation .…”

mentioning

confidence: 99%

Xenon-135 Induced Instabilities of the Neutron Flux in Nuclear Reactors

Sandquist¹

1964

View full text Add to dashboard Cite

“…They can be regarded as a finite analogue of the logistic differential equation In this case u n+ , -«" has the same sign as N' for a given population size and, in particular, the equilibrium values are the same. Indeed, equation (1.1) is the Cauchy-Euler approximation to (1.2) and the solutions behave in the same way for h small (Hurewicz [4]). However, the solutions of the difference equation show a much richer variety of possible behaviour for larger values of h. In a review article [8], May has drawn attention to these possibilities and suggested that the difference equation provides a more appropriate model than the differential equation in a number of practical applications.…”

Section: Introductionmentioning

confidence: 97%

Equations for periodic solutions of a logistic difference equation

Brown

1981

J. Aust. Math. Soc. Series B, Appl. Math

View full text Add to dashboard Cite

The paper is concerned with periodic solutions of the difference equation u n + l -2au n -bu*, where a and b are constants, with a > \ and b > 0. A new method is developed for dealing with this problem and, for period lengths up to 6, polynomial equations are given which allow the periodic solutions to be determined in a precise and practical manner. These equations apply whether the periodic solutions are stable or unstable and the elements of the cycle can be determined with an accuracy which is not affected by instability of the cycle.A simple transformation puts the equation into the form w n+i => w 1 -A, where A = a 2 -a, and the detailed discussion is based on this simpler form. The discussion includes details such as the number of cyclic solutions for a given value of A, the pattern of the cycles and their stability. For practical purposes, it is enough to consider a restricted range of values of A, namely -j < A < 2, although the equations obtained are valid for A > 2.

show abstract

Lectures on Ordinary Differential Equations

Cited by 158 publications

References 0 publications

Sturm-Liouville Matrix Equation for the Study of Electromagnetic-Waves Propagation in Layered Anisotropic Media

Sturm-Liouville Matrix Equation for the Study of Electromagnetic-Waves Propagation in Layered Anisotropic Media

Xenon-135 Induced Instabilities of the Neutron Flux in Nuclear Reactors

Equations for periodic solutions of a logistic difference equation

Contact Info

Product

Resources

About