Regular Singular Points Research Articles

Comments on "A Simple Numerical Method for the Cutoff Frequency of a Single-Mode Fiber with an Arbitrary Index-Profile"

In the above paper, Sharma and Ghatak have proposed a numerical method for calculating the cutoff frequency of single mode operation in optical fibers with an arbitrary index profile. We want to comment on the validity of their results using their notations. The referenced paper requires in particular the knowl-edge of the boundary conditions on the modal field Psi and its derivatives at R=0. To this aim, Sharma and Ghatak have used a series solution method. Their series expansion [1, eq. (A. 1)] is valid only in the vicinity of an ordinary point but the point R=0 is a regular singular point for the scalar wave differential equation [1, eq. (A.3)].

Growth of solutions of linear differential equations at a logarithmic singularity

We consider differential equations Y ′ = A Y Y’ = AY with a regular singular point at the origin, where A A is an n × n n \times n matrix whose entries are p p -adic meromorphic functions. If the solution matrix at the origin is of the form Y = P exp ⁡ ( θ log ⁡ x ) Y = P\exp (\theta \log x) , where P P is an n × n n \times n matrix of meromorphic functions and θ \theta is an n × n n \times n constant matrix whose Jordan normal form consists of a single block, then we prove that the entries of P P have logarithmic growth of order n − 1 n - 1 .

Arithmetic Properties of Power Series Solutions of Algebraic Differential Equations

in some non-trivial disk. We considered such a problem at a regular singular point and also at an irregular singular point. The problem at a regular singular point was more difficult. To deal with this difficulty, we utilized a Newton iteration procedure. This procedure was originally used by J. Moser and V. I. Arnol'd to deal with small divisors in celestial mechanics (cf. S. Sternberg 1141). In the present work, we improve our previous results so that they imply the following Main Theorem (cf. Chapter 4): Let Q be the algebraic closure of the rational numbers. Let y

Regularity of Singular Two-Point Boundary Value Problems

Two-point boundary value problems like $(p(x)x^\sigma u')' = f(x)$ on $(0,1)$ have a regular singular point at the boundary, and solutions that are not smooth but behave like $x^{1 - \sigma } $ near 0. Bounds on weighted Sobolev norms of their solutions, essential to the theory of finite element approximations for such problems, are obtained here.

The Connection Problem for Two Neighboring Regular Singular Points of General Linear Complex Ordinary Differential Equations

A two point connection problem for local solutions at two regular singular points of a general linear ordinary differential equation is studied. Generalizing the results of a recent paper of D. Schmidt and the author [SIAM J. Math. Anal. 11(1980), pp. 848–862] explicit formulas for the connection matrix of the fundamental solutions are derived without restrictive assumptions. Applications to the hypergeometric equation in a Banach algebra yield new formulas for their connection factors.

Invariant polynomials and zonal spherical functions

In the second and third sections we derive again some results of Nolan Wallach (with some modest simplifications). In the second section we give his proof of a simple lemma (lemma 10) which is the basis for the inversion of the Abel transform. To add a little spice, we cast the lemma in the setting of Euclidean motion groups; this gives us the Paley-Wiener theorem for certain Bessel functions (Jn/2, n ~ 7/). The third sect~n gives Wallach's proof of the inversion formula for the Abel transform and of the zonal Paley-Wiener theorem for rank one semi-simple groups [H2, G]. I have included these because they are not readily available. The fourth section is devoted to finding the Plancherel measure in rank one, class one. The techniques of this paper are quite elementary; the most difficult facts used are Mather's theorem and the parameterization of zonal spherical function [H, ch 10]. (Actually one can give a proof of Mather's theorem in the case used here which is considerably simpler). Aside from this we use only some standard structure theory for semi-simple groups, regular singular point theory for ordinary differential equations, and a contour integral.

Monodromy- and spectrum-preserving deformations I

A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformations and with classical and recent work on monodromy preserving deformations are discussed. Specific new results include the reduction of the general initial value problem for the Painlevé equations of the second type and a special case of the third type to a system of linear singular integral equations. Several classes of solutions are discussed, and in particular the general expression for rational solutions for the second Painlevé equation family is shown to be −d/dx ln(Δ+/Δ−), where Δ+ and Δ− are determinants. We also demonstrate that each of these equations is an exactly integrable Hamiltonian system. The basic ideas presented here are applicable to a broad class of ordinary and partial differential equations; additional results will be presented in a sequence of future papers.

Equivalence of a differential operator of n-Th order with a regular singular point and a Euler operator in the space A(G)

Equivalence of a differential operator of n-Th order with a regular singular point and a Euler operator in the space A(G)

Behaviour of boundary conditions transferred to the neighbourhood of a regular singular point

Behaviour of boundary conditions transferred to the neighbourhood of a regular singular point

Stokes multipliers for a class of ordinary differential equations

A new method is presented for calculating the Stokes multipliers for a class of linear second-order ordinary differential equations. The Stokes multipliers allow the asymptotic solutions of these equations to be continued across the Stokes lines on which they are dominant. The differential equations, of the class considered here, have an irregular singular point at infinity and a singular point at the origin, which may be either regular or irregular. The Stokes multipliers, as functions of the coefficients in the differential equation, are obtained in the form of convergent infinite series, whose terms must be obtained from the solution of recursion relations, which are derived. In the case of Whittaker’s equation (when the origin is a regular singular point), the known results are obtained analytically. When the origin is an irregular singular point, numerical evaluation of the series is necessary, but the method seems to be quite efficient for use with digital computers. In the special case of an equation, with two irregular singular points, which can be transformed to Mathieu’s equation, the numerical results for the Stokes multiplier show good agreement with available known results for the characteristic exponents of Mathieu’s equation.

Stability of Shock Waves of Arbitrary Strength With Viscosity and Heat Conduction

The problem of the stability of shock waves with viscosity and heat conduction has been previously formulated as an eigenvalue problem involving a set of linear ordinary differential equations in a finite domain with what are shown to be regular singular points at the ends of the domain. By means of a computer-aided Frobenius type of analysis, it is shown that the (continuous) eigenvalue spectrum is such that the shock waves will be stable for all values of the shock-strength parameter. Some actual solutions of the disturbance equations are shown.

Equivalence of third-order differential operators with a regular singular point

Equivalence of third-order differential operators with a regular singular point

A reduction theorem for a linear Pfaffian system with regular singular points

A reduction theorem for a linear Pfaffian system with regular singular points

ON THE FUNDAMENTAL MATRIX OF A PFAFFIAN SYSTEM OF FUCHSIAN TYPE

A necessary and sufficient condition is given in order that a completely integrable Pfaffian system with regular singular point, given on a polydisk Dn, be of Fuchsian type. It is proved that the system is Fuchsian if and only if the space of solutions is weakly singular. In the one-dimensional case, this problem was studied by A. H. M. Levelt (MR 25 ##1301-1304). For higher dimensions, a similar attempt was made by R. Gerard (MR 39 #4433), but his paper contains an error.Bibliography: 4 titles.

A Numerical Method for Singular Two Point Boundary Value Problems

The numerical solution of boundary value problems for linear systems of first order equations with a regular singular point at one endpoint is considered. The standard procedure of expanding about the singularity to get a nonsingular problem over a reduced interval is justified in some detail. Quite general boundary conditions are included which permit unbounded solutions. Error estimates are given and some numerical calculations are presented to check the theory.

PFAFFIAN SYSTEMS OF FUCHS TYPE ON A COMPLEX ANALYTIC MANIFOLD

In this paper we give necessary and sufficient conditions for a completely integrable Pfaffian system with regular singular points on to be a Fuchsian system, where is a divisor with normal crossings in a compact Kahler manifold . We prove that the condition of being a Fuchsian system is equivalent to the solvability of some first Cousin problem on . This condition appears particularly simple when is complex projective space.Bibliography: 12 titles.

Necessary and sufficient conditions for equivalence of differential operators with regular singular points

Necessary and sufficient conditions for equivalence of differential operators with regular singular points

Global properties of radial wave functions in Schwarzschild's space-time

Two solutions ℛ5(x, xs) and ℛ6(x, xs) related to the irregular singular point atx=+∞ of the radial wave equation in Schwarzschild's space-time are studied as functions of the independent variablex and the parameterxs. Analytic continuations of ℛ5 and ℛ6 are derived and their relation to the flat-space case solutions is established. Explicit expressions for ℛ3(x, xs) and ℛ4(x, xs) (the solutions about the regular singular point atx=xs) are given. From these expressions and the analytic continuations of ℛ5 and ℛ6 the coefficients relating linearly ℛ5 and ℛ6 with ℛi(i=1, 2, 3, 4) are calculated.

Nonlinear perturbation of a linear system of differential equations with a regular singular point

Nonlinear perturbation of a linear system of differential equations with a regular singular point

Asymptotic solutions of second-order linear equations with three transition points

A uniformly valid asymptotic expansion is obtained for the regular solution of a class of second-order linear differential equations with three transition points-a turning point and two regular singular points. The solution is found by matching three different solutions obtained using the Langer Transformation. The matching yields the eigenvalues and the eigenfunctions.