Singular Ordinary Differential Equations Research Articles

Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball

In this paper we show that a radially symmetric superlinear Dirichlet problem in a ball has infinitely many solutions. This result is obtained even in cases of rapidly growing nonlinearities, that is, when the growth of the nonlinearity surpasses the critical exponent of the Sobolev embedding theorem. Our methods rely on the energy analysis and the phase-plane angle analysis of the solutions for the associated singular ordinary differential equation.

A new procedure for determining the diffusion coefficients of electron swarms according to the modern transport theory

In this paper we are concerned with the solution of the kinetic equations which are met when expanding the electron velocity distribution function with respect to the gradient of the electron density in the Boltzmann equation. The approximation of the corresponding zero- and first-order coefficients by an expansion in Legendre, or associated Legendre, polynomials up to a given number (2l)_ of terms, which transform the kinetic equations into three coupled hierarchies for the corresponding expansion coefficients, is assumed. The solutions of these hierarchies permit to determine the drift speed, as well as the longitudinal and transverse diffusion coefficients of an electron swarm. Based on an analysis of the mathematical structure of these singular ordinary differential equation systems (with additional difference terms due to the inelastic,i.e. energy loss, collision processes) a new procedure is developed which isolates and constructs (for arbitrary even approximation order 2l) the nonsingular part of the general solutions (NSPGS) of each hierarchy. The NSPGS is firstly obtained in the region of small electron energies (up to an appropriate connection point) and secondly in the region of large energies (down to the connection point) starting from the two singular points of each hierarchy,i.e. from very small and sufficiently large energies, respectively. The continuous connection of the two NSPGSs, obtained in the two different energy regions, together with the normalization condition, yield the physically relevant solution for each one of the three hierarchies and make it possible to calculate the diffusion coefficients we are particularly interested in. Our new technique, even if much more complex, is a logical generalization of the conventional backward prolongation technique used for the well-known two-term approximation of the stationary and spatially homogeneous Boltzmann equation. The present paper also reports a first application of this new procedure to a model plasma. The diffusion coefficients are calculated with increasing approximation order 2l up to their converged values, under different parameter conditions of the model and are compared with corresponding values found by accurate Monte Carlo simulations as well as with results of the conventional transport theory for the same model. The diffusion coefficients obtained for different (even) order of approximation well illustrate the power of our new procedure. In particular, the converged values of the diffusion coefficients are found to be in very good agreement with the corresponding Monte Carlo values.

Integrable-square solutions of a singular ordinary differential equation

SynopsisAbsolutely square integrable solutions are determined for the equation= λywhere the ζn−r(x) are holomorphic in a sector of the complex plane and have asymptotic expansions asxapproaches infinity. It is shown that the number of such solutions depends upon the roots of the characteristic equation and their multiplicity, and upon the sign of the derivative of the characteristic polynomial. Application is made to formally symmetric ordinary differential operators.

On the eigenfunction expansions of singular ordinary differential equations

On the eigenfunction expansions of singular ordinary differential equations

Weyl’s theory applied to predissociation by rotation. I. Mercury hydride

Weyl’s theory for singular ordinary second order differential equations is presented as a numerical method for analyzing the spectral properties of the Schrödinger operator associated with predissociation by rotation in diatomic molecules. It is pointed out that the poles of Weyl–Titchmarsh’s m function characterize both the discrete eigenvalues and the resonances in the continuous spectrum, thus allowing a unified treatment of the entire spectrum. The conventional hard core treatment of the origin is compared to two alternative approaches, both applicable to more general singular behavior of the potential. A detailed discussion of the other singularity, infinity, is given and Kodaira’s theorem is generalized to an expression involving Wronskian’s between the regular and the asymptotic solutions. The calculation of the Weyl–Titchmarsh m function is based on numerical Runge–Kutta integrations of both independent solutions, combined with asymptotic information derived from a Riccati expansion. Im(m) is shown to have its physical significance as flux of predissociating particles. The resonances in the continuous spectrum are obtained by numerical analytical continuation of Im(m) across the real energy axis. The method is applied to Stwalley’s isotopically combined (IC1) potential for HgH. All resonance energies and lifetimes are reported and compared to Stwalley’s phase shift results. Excellent agreement is found for sharp resonances whereas considerable discrepancies occur for the broader ones. This latter fact is attributed to the failure of the Breit–Wigner fit to account for asymmetric non-Lorentzian shapes.

Decomposition theorems for the generalized metaharmonic equation in several independent variables

In this paper solutions of the generalized metaharmonic equation in several independent variables where λ > 0 are uniquely decomposed into the sum of a solution regular in the entire space and one satisfying a generalized Sommerfeld radiation condition. Due to the singular nature of the partial differential equation under investigation it is shown that the radiation condition in general must hold uniformly in a domain lying in the space of several complex variables. This result indicates that function theoretic methods are not only the correct and natural avenue of approach in the study of singular ordinary differential equations, but are basic in the investigation of singular partial differential equations as well.

Sur quelques problèmes de perturbation singulière en théorie des équations différentielles—i

This study considers a singular ordinary differential equation with almost periodic coefficients. An almost periodic solution is obtained in a unique manner through a uniformly convergent series. The limit of this solution, when the singularity parameter ε tends to zero is a known almost periodic solution of the reduced system obtained by setting ε = 0.

Existence of exact solutions of singular ordinary differential equations near their approximate solutions.

Existence of exact solutions of singular ordinary differential equations near their approximate solutions.

The Expansion Theorem For Singular Self-Adjoint Linear Differential Operators

1. There has recently appeared three papers giving independently and almost simultaneously a much simplified approach to the expansion theorem for selfadjoint singular ordinary differential equations. Levitan [7] treats by means of a simple lemma the differential operator of arbitrary order but does not deal with the problem of uniqueness of the expansion. Yosida [11] and Levinson [5, 6] considered only the second order case but with uniqueness proved in the limit point case. All three accounts make important use of the Helly selection theorem for a set of functions of bounded variation. Here it will be shown that using the lemma of Levitan [7] and a treatment rather similar to that used by Levinson [6] for the inverse transform theorem, it is possible to obtain a simple self-contained proof of uniqueness under broad conditions. In particular none of the apparatus of Hilbert space or the theory of singular integral equations will be required. One of the results that follow, Theorem II, has been demonstrated by Coddington [2] with the use of Levitan's lemma and theorems from Hilbert space theory. Moreover Theorem III is implicit in Coddington's results because of the remarks which follow his Theorem 3 [2, p. 735]. The methods of the present paper can easily be carried over to singular selfadjoint systems using the formulations given by Bliss [1] for the non-singular