Perturbation Methods

Abstract

Perturbation methods are one of the fundamental tools used by all applied mathematicians and theoretical physicists. In this book, the author has managed to present the theory and techniques underlying such methods in a manner which will give the text wide appeal to students from a broad range of disciplines. Asymptotic expansions, strained coordinates and multiple scales are illustrated by copious use of examples drawn from all areas of applied mathematics and theoretical physics. The philosophy adopted is that there is no single or best method for such problems, but that one may exploit the small parameter given some experience and understanding of similar perturbation problems. The author does not look to perturbation methods to give quantitative answers but rather to give a physical understanding of the subtle balances in a complex problem.

Singular SPDEs in domains with boundaries

“…To the best of our knowledge, this is the first observed instance of "boundary renormalisation" for stochastic PDEs. On the other hand, it is somewhat similar to the effects one observes in the analysis of (deterministic) singularly perturbed problems in the presence of boundary layers, see for example [15,16]. The remainder of the article is structured as follows.…”

Spreading speeds for stage structured plant populations in fragmented landscapes

“…We can now reconsider (A.7) (F app s,0 − ρ app s,0 ) φ app s,n + F s,1 φ app s,n−1 − n j=1 ρ app s,j φ app s,n−j = 0 (A.11) which defines φ app s,n and ρ app s,n in terms of lower order terms. We find ρ app s,n by taking its inner product with This is analogous to standard results from matrix perturbation theory [52,53], which for n = 1 are derived rigorously in [54]. For ρ app s,1 , we use the definitions of φ app s,0 andφ …”

Integral constraints in multiple-scales problems

“…The equations for linearised motion about a periodic array of bubbles with centres (i, j) are , and γ is the ratio of specific heats in the gas, and ρ g and ρ l are the undisturbed densities of the gas and liquid respectively 1 If we eliminate u as in [2] we find 5) in the liquid region x : |x − x j | > δ, and…”

“…The technique of asymptotic homogenisation via multiple scales is widely used for a variety of problems involving a (locally) periodic microstructure for which bulk or effective equations are required [5][6][7]. Typical applications include the derivation of Darcy flow from the Stokes equations, homogenisation of a rapidly varying porosity/conductivity/diffusion coefficient, effective elastic properties of composite materials, and the transmission of acoustic waves through bubbly fluids.…”