2 - Parabolic Equations

Abstract

This chapter discusses the properties of parabolic equations. Problems considered are usually assumed to be properly posed from the outset. Existence and uniqueness are usually ensured under physically reasonable assumptions such as those of piecewise differentiability of coefficients and initial data. Parabolic partial differential equations arising in scientific and engineering problems are often of the form u1 = L, where L is a second-order elliptic partial differential operator that may be linear or nonlinear. Diffusion in an isotropic medium, heat conduction in an isotropic medium, fluid flow through porous media, boundary layer flow over a flat plate, persistence of solar prominences, and the wake growth behind a submerged object have been modeled by the parabolic equation. An explicit formula provides for a noniterative marching process for obtaining the solution at each present point in terms of known preceding and boundary values. As parabolic equations characteristically have open integration domains, explicit methods are applicable to these problems.