The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. In spite of the probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter. 1991 Mathematics...

The Fokker–Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker– Planck equation we describe a construction of exact solution of the Cauchy problem. In the general case, we construct the leadin...

Abstract. In the present paper we are concerned with the Novikov–Veselov equation at negative energy, i.e. with the (2+1)–dimensional analog of the KdV equation integrable by the method of inverse scattering for the two–dimensional Schrödinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non–singular scattering data behaves asymptotically a...

Abstract. Given a system of linear partial differential operators with constant coefficients whose affine algebraic varieties V (℘̌) have dimension 1, we establish in which classes of (small) Gevrey functions the associated Cauchy problem admits at least one solution, looking at the Puiseux series expansions on the branches at infinity of the algebraic curves V (℘̌). We focus, in particular, on t...

In this paper we investigate the numerical solution of non-linear equations in an abstract (Banach space) setting. The main result is that the convergence can be guaranteed by two, directly checkable conditions (namely, by the consistency and the stability). We show that these conditions together are a sufficient, but not necessary condition for the convergence. Our theoretical results are demo...

The goal of this series is to explore various aspects of the inhomogeneous Cauchy– Riemann, or ∂, equation on infinite dimensional complex manifolds. In the first paper in the series we argued for the importance of such an undertaking; we also gave rather complete results when the manifold in question is an infinite dimensional projective space; see [L]. In the present work we turn to the analy...