In the rectangle D = (0, a) × (0, b) with the boundary Γ the Dirichlet problem ∂4u ∂x2∂y2 = p(x, y)u + q(x, y), u(x, y) = 0 for (x, y) ∈ Γ is considered, where p and q : D → R are locally summable functions and may have nonintegrable singularities on Γ. The effective conditions guaranteeing the unique solvability of this problem and the stability of its solution with respect to small perturbati...

Let A denote a square complex matrix and let E be a perturbation matrix. The purpose of this paper is to investigate the perturbation of the Drazin inverse when B = A + E satisfies the rank conditions rankA = rankB = rankAB, where r and s denote the indices of A and B, respectively. We will derive an explicit representation of B as a function of A and B −A , for certain positive integers j, k. ...

We re-examine the dynamical stability of the nakedly singular, static, spherically symmetric solutions of the Einstein-Klein Gordon system. We correct an earlier proof of the instability of these solutions, and demonstrate that there are solutions to the massive Klein-Gordon system that are perturbatively stable.

The KdV and modified KdV integrable hierarchies are shown to be different descriptions of the same 2D gravitational system – open-closed string theory. Nonperturbative solutions of the multi-critical unitary matrix models map to nonsingular solutions of the ‘renormalisation group’ equation for the string susceptibility, [P̃ , Q] =Q . We also demonstrate that the large N solutions of unitary matr...

We analyze asymptotically a differential-difference equation, that arises in a Markovmodulated fluid model. We use singular perturbation methods to analyze the problem with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used.

The numerical solution of boundary value problems for certain stiff ordinary differential equations is studied. The methods developed use singular perturbation theory to construct approximate numerical solutions which are valid asymptotically; hence, they have the desirable feature of becoming more accurate as the equations become stiffer. Several numerical examples are presented which demonstr...

We investigate pinning regions and unpinning asymptotics in nonlocal equations. We show that phenomena are related to but different from pinning in discrete and inhomogeneous media. We establish unpinning asymptotics using geometric singular perturbation theory in several examples. We also present numerical evidence for the dependence of unpinning asymptotics on regularity of the nonlocal convo...

We show the existence of travelling wave solutions for a lubrication model of surfactant-driven flow of a thin liquid film down an inclined plane, in various parameter regimes. Our arguments use geometric singular perturbation theory. Mathematics Subject Classification: 76D08, 34B16, 34B40, 34C37 (Some figures in this article are in colour only in the electronic version)

We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with finite-range, non-local coupling. Our approach replaces methods from geometric singular perturbation theory, that had been crucial in previous existence proofs, by a P...

We consider the problem of partial hedging of an European derivative under the assumption that the volatility is stochastic, driven by two diffusion processes, one fast mean reverting and the other varying slowly. For options with long maturities typically beyond 90 days, the singular perturbation analysis in [Partial Hedging in a Stochastic Volatility Environment, M. Jonsson and K.R. Sircar, M...