Asymptotic Differential Equation Research Articles

Complements to the Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions

The purpose of this paper is to add some complements to the general theory of higher-order types of asymptotic variation developed in two previous papers so as to complete our elementary (but not too much!) theory in view of applications to the theory of finite asymptotic expansions in the real domain, the asymptotic study of ordinary differential equations and the like. The main results concern: 1) a detailed study of the types of asymptotic variation of an infinite series so extending the results known for the sole power series; 2) the type of asymptotic variation of a Wronskian completing the many already-published results on the asymptotic behaviors of Wronskians; 3) a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation; 4) a discussion about the simple concept of logarithmic variation making explicit and completing the results which, in the literature, are hidden in a quite-complicated general theory.

On the solution of the asymptotic differential equations in quantum mechanics

The solution of the asymptotic quantum mechanical differential equations in the neighbourhood of singular points is suggested to overcome some complexities occurring with the technique of the indicial equations. After separation of the variables, the differential equations are solved in their asymptotic regions, the general regular solutions in the whole interval being then found by replacing integration constants by functions of the variables and finding the differential equations determining these functions. The resulting differential equations are now free from singularities and can be solved by the usual series expansion, giving simple polynomial solutions for the harmonic oscillator and the hydrogen atom and more complicated solutions for H 2 + .

The dynamics of boundary spikes for a nonlocal reaction-diffusion model

An asymptotic reduction of the Gierer–Meinhardt activator-inhibitor system in the limit of large inhibitor diffusivity and small activator diffusivity ε leads to a singularly perturbed nonlocal reaction-diffusion equation for the activator concentration. In the limit ε → 0, this nonlocal problem for the activator concentration has localized spike-type solutions. In this limit, we analyze the motion of a spike that is confined to the smooth boundary of a two or three-dimensional domain. By deriving asymptotic differential equations for the spike motion, it is shown that the spike moves towards a local maximum of the curvature in two dimensions and a local maximum of the mean curvature in three dimensions. The motion of a spike on a flat segment of a two-dimensional domain is also analyzed, and this motion is found to be metastable. The critical feature that allows for the slow boundary spike motion is the presence of the nonlocal term in the underlying reaction-diffusion equation.

An analysis of thickness-extensional trapped energy resonant device structures with rectangular electrodes in the piezoelectric thin film on silicon configuration

It has recently been shown that highly uniform thin layers can be etched in a small well-defined region of a silicon wafer and that good quality thin piezoelectric films such as zinc-oxide can be deposited along with the electrodes to form high frequency trapped energy resonant device structures. The accompanying analytical work has been for pure thickness vibrations only. In this work an analysis of essentially thickness-extensional trapped energy modes in the thin piezoelectric film on silicon composite structure is performed. It is shown that for small wavenumbers along the plate the dispersion equation is isotropic in the plane of the plate even though the silicon is severely anisotropic in that plane. From the resulting dispersion relation an asymptotic differential equation describing the mode shape along the surface of the composite plate vibrating in the vicinity of a thickness-extensional resonance is obtained along with the associated edge conditions. Since the mode is essentially thickness-extensional, trapping does not usually occur in the electroded regions for the fundamental mode of the flat composite plate. However, by appropriately thickening the silicon outside the electroded regions trapping in the electroded regions can be made to occur for the fundamental mode. Furthermore, in the case of zinc-oxide on silicon, trapping in the electroded regions can be made to occur for the fundamental mode of a flat plate simply by making the zinc-oxide sufficiently thicker than the etched silicon. The aforementioned asymptotic differential equation and edge conditions are applied in the analyses of the steady-state vibrations of trapped energy resonators and acoustically coupled two-pole resonator filters, both with rectangular electrodes. Lumped parameter representations of the admittance in the case of the single-pole device and admittance matrix in the case of the two-pole device, which are valid in the vicinity of resonances, are obtained. The analyses hold for the first few thickness-extensional modes and their accompanying lateral overtones.

An asymptotic solution for the large deflection of a circular plate with a central hole

A linear asymptotic differential equation is derived in this work which closely approximates the deflection of a circular plate with a central hole for large values of the load parameter. This equation, therefore, replaces the exact non-linear von-Karman differential equation which results from the vertical force summation. The solution to the above equation may be used as an initial iterate to the numerical solution technique (hereafter referred to as the ‘exact’ solution) or may be used as a good engineering approximation in the useful load range. For a particular geometric configuration, an overlap of the linear (small deflection) solution and the asymptotic (large deflection) solution can result. This will eliminate the necessity of employing the exact solution. Numerical results are presented which compare the exact deflections with those obtained from the asymptotic solution.