We discuss the use of the WKB ansatz in a variety of parabolic problems involving a small parameter. We analyse the Stefan problem for small latent heat, the Black–Scholes problem for an American put option, and some nonlinear diffusion equations, in each case constructing an asymptotic solution by the use of ray methods.

We consider the Cauchy problem for a system of fully nonlinear parabolic equations. In this paper, we shall show existence global-in-time solutions to problem. Our condition ensure global is specific system.

The technique of space invariant embedding, applied to a second order boundary value problem defined in a cylindrical domain, gives rise to a system of uncoupled first order initial value problems that includes a nonlinear Riccati equation on a unbounded functional operator. We present a method to justify this equation that uses a parabolic regularization of the original problem.

The martingale representation theorem in a Brownian filtration represents any square integrable r.v. ξ as a stochastic integral with respect to the Brownian motion. This is the simplest Backward SDE with nul generator and final data ξ, which can be seen as the non-Markov counterpart of the Cauchy problem in second order parabolic PDEs. Similarly, the notion of Second order BSDEs is the non-Mark...

Abstract. In this paper, the author propose a numerical method to compute the solution of the Cauchy problem: wt − (w m wx)x = w , the initial condition is a nonnegative function with compact support, m > 0, 1 < p < m + 1. The problem is split in two parts: A hyperbolic term solved by using the Hopf and Lax formula and a parabolic term solved by a backward linearized Euler method in time and a ...

We study a discretization technique for the parabolic fractional obstacle problem in bounded domains. The fractional Laplacian is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic equation posed on a semi-infinite cylinder, which recasts our problem as a quasi-stationary elliptic variational inequality with a dynamic boundary condition. The rapid decay of the solution suggest...

Numerical approximations to the solution of a linear singularly perturbed parabolic problem are generated using a backward Euler method in time and an upwinded finite difference operator in space on a piecewise-uniform Shishkin mesh for a convectiondiffusion problem. A proof is given to show first order convergence of these numerical approximations in appropriately weighted C-norm. Numerical re...