Charles G. Lange

SINGULAR PERTURBATION ANALYSIS OF BOUNDARY-VALUE PROBLEMS FOR DIFFERENTIAL-DIFFERENCE EQUATIONS*

A study is made of a class of boundary-value problems for linear second-order differential-difference equations in which the highest-order derivative is multiplied by a small parameter. Consideration of the exact solution of simple model equations provides insight into the appropriate use of singular perturbation techniques for more general problems. The resulting analysis leads to several novel features which are not present in problems without differences.

A Stability Criterion for Envelope Equations

The statistical initial value problem for the envelope equation \[ \frac{{\partial w}}{{\partial t}} - \gamma \frac{{\partial ^2 w}}{{\partial x^2 }} = \chi w - \beta w^2 w^ * ,\quad \gamma ,\beta \,{\text{complex}} \], is discussed. It is shown that there exists an instability (stability) criterion \[ \beta _r \gamma _r + \beta _i \gamma _i \gtreqless 0,\quad \beta _r > 0 \], which determines whether the system underlying the above equation achieves or does not achieve a monochromatic state.

Singular perturbation analysis of boundary-value problems for differential-difference equations. VI.: small shifts with rapid oscillations

This paper continues the authors’ study of boundary-value problems for singularly perturbed linear second-order differential-difference equations with small shifts. This study was initiated in the companion paper [SIAM J. Appl. Math., 54 (1994), pp. 249–272]. In this paper, the study is extended to problems that have solutions that exhibit rapid oscillations. Restrictions on the sizes of the shifts in terms of the small parameter are found such that, generally, the shifted terms cannot be replaced with truncated Taylor series. In particular, it is shown that, even when the shifts are small relative to the width of an oscillation, they can affect the solution to leading order. The conclusion is that oscillatory solutions are more sensitive to small delays than are layer solutions. It is shown that a suitably modified version of the standard WKB method can be used to obtain leading-order oscillatory solutions of these differential-difference equations. These preliminary studies of differential-difference eq...

Singular Perturbation Analysis of Boundary Value Problems for Differential-Difference Equations III. Turning Point Problems

This paper continues a study of a class of boundary-value problems for linear second-order differential-difference equations in which the second-order derivative is multiplied by a small parameter (SIAM J. Appl. Math., 42 (1982), pp. 502–531; 45 (1985), pp. 687–707). The previous papers focused on problems involving boundary and interior layer phenomena, rapid oscillations, and resonance behavior. The problems studied here have solutions which exhibit turning point behavior, i.e., transition regions between rapid oscillations and exponential behavior. The presence of the shift terms can induce large amplitudes and multiphase behavior over parts of the interval. A combination of exact solutions, singular perturbation methods, and numerical computations are used in these studies.

Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Quations II. Rapid Oscillations and Resonances

This paper continues a study of a class of boundary-value problems for linear second-order differential-difference equations in which the second-order derivative is multiplied by a small parameter (SIAM J. Appl. Math., 42 (1982), pp. 502–531). The previous paper focused on problems involving boundary and interior layer phenomena. Here the problems studied have solutions exhibiting rapid oscillations. The presence of the shift terms can induce large amplitudes, multiphase behavior, and resonance phenomena.n particular, we study two types of resonance phenomena, namely “global” and “local” resonance. A combination of exact solutions, singular perturbation methods, and numerical computations are used in these studies. In a companion paper (SIAM J. Appl. Math., 45 (1985), pp. 708–734) we study problems with solutions which exhibit turning point behavior.

On large axisymmetrical deflection states of spherical shells

A boundary layer analysis is carried out to determine the possible form of large axisymmetrical deflection states for thin elastic spherical shells subjected to uniform external pressure. For the case of complete spheres it is shown that the governing equations admit boundary layer solutions corresponding to large deflections provided the pressure is sufficiently small. However, such solutions are found to exist for nonshallow clamped spherical caps for a much wider range of pressure. Numerical results are presented for the latter case.

Spherical Shells Like Hexagons: Cylinders Prefer Diamonds—Part 1

Abstract : The purpose of the paper is to describe the initial postbuckling behavior of cylindrical shells under axial compression. Multiple scale techniques are employed to investigate the dynamic interaction and evolution of competing diamond patterns to investigate the dynamic interaction and evolution of competing diamon patterns and the propagation of spatial inhomogeneities. It is shown that if no substanital bias is present in either the initial imperfections or initial conditions a natural selection mechanism exists which favors the square diamond configuration. Numerical results are included which support the findings of the analytical investigation. (Author)

Branching from Closely Spaced Eigenvalues with Application to a Model Biochemical Reaction

Branching solutions for a class of bifurcation problems involving closely spaced eigenvalues are constructed. The mathematical features are illustrated by consideration of a model biochemical network involving reaction and diffusion proposed by I. Prigogine. For the case of weak diffusion the severe restriction on the region of validity of the standard constructive procedures is established and a perturbation method is utilized to extend the solution branches.

Particular solutions of forced generalized Airy equations

Differential equations with delayed forcing terms arise naturally in the explicit solution of differential-delay equations [ 11. In our present studies of singular perturbation analyses of boundary-value problems for secondorder differential-difference equations with a turning point (see [2] for earlier results), we have had to deal with forcing terms consisting of shifted Airy and Lommel functions and their derivatives. One example of the type of equation which arises is the inhomogeneous Airy equation (see the Appendix for a derivation) y”-xy=c,Ai(x-a)+c,Bi(x-a)+c,L(x-a) + d, Ai’(x a) + d, Bi’(x a) + d3 L’(x a), (1) where a is a constant (0 < a < co); ci, di, i = 1, 2, 3, are constants; Ai, Bi are Airy functions; and L is a Lommel function (see [ 31) satisfying L”-xL= 1. (2)

BRANCHING FROM LARGE EIGENVALUES OF NONLINEAR STURM-LIOUVILLE SYSTEMS*

The behavior of solutions branching from large eigenvalues of certain nonlinear Sturm–Liouville systems is studied. A perturbation method is presented which permits the solution branches to be extended beyond the region of validity of standard perturbation and iteration techniques. Numerical calculations support the asymptotic results.