Problems and Techniques

Abstract

The three papers in this issue are concerned with asymptotic expansions of integrals, calculus of variations, and singular perturbation techniques. 1. Among integral transforms, the Fourier transform is arguably the best known, but there are many others, including Laplace, Stieltjes, Mellin, Hankel, and Poisson transforms. They can all be represented as $I(x)\equiv\int_0^{\infty}{f(t)h(xt)dt}$. If x is an asymptotic parameter, which means that x is close to zero or very large, then one may be able to approximate $I(x)$ by an asymptotic expansion. In his paper “Asymptotic Expansions of Mellin Convolution Integrals,” José López presents a general and simple method to generate asymptotic expansions for $I(x)$ that encompasses many existing methods as special cases. 2. Analysis of “slope stability” is an important aspect of geology and soil mechanics: How likely is a sloped terrain (an embankment, a dam) to succumb to erosion and turn into a landslide? And how should the slope profile be modified to prevent further sliding? Analysis of slope stability is one of the applications envisioned by Enrique Castillo, Antonio Conejo, and Ernesto Aranda in their paper “Sensitivity Analysis in Calculus of Variations. Some Applications.” They propose to perform slope stability by means of a sensitivity analysis in the calculus of variations. In the context of slope stability, for instance, one might want to determine how sensitive the slope safety factor is to changes in the slope profile or to changes in soil strength. The mathematical problem comes down to this: How sensitive to changes in parameters are the following quantities: objective function values, primal solutions, and dual solutions? The authors show how to express the sensitivities in terms of partial derivatives, and how to compute them numerically by solving a boundary value problem. 3. In their paper “High-Frequency Oscillations of a Sphere in a Viscous Fluid near a Rigid Plane,” Richard Chadwick and Zhijie Liao model the operation of atomic force microscopes. An atomic force microscope (AFM) is a high resolution scanning device for imaging at nanoscales. Unlike conventional microscopes, which use light, an AFM scans a specimen by “feeling” the surface with a mechanical probe, which in this case consists of a sphere attached to a cantilever. When the specimens are delicate biological samples, such as tissues of the inner ear, they are scanned in a fluid, and the AFM is employed in “tapping mode.” In tapping mode, the sphere is forced to oscillate up and down. When it approaches the specimen, its oscillations are changed by hydrodynamic forces. This provides information about properties of the specimen. To understand the hydrodynamic interactions taking place between sphere and wall, one can model the situation as a fluid that contains a sphere oscillating close to a rigid wall. The problem becomes singular when the sphere approaches the wall. The authors revive singular perturbation techniques developed in the 1960s to derive asymptotic expansions for the forces acting on the sphere.