In this paper we derive the sharp long time stability and error estimates of nite element approximations for parabolic integro di erential equations First the exponential decay of the solution as t is studied and then the semi discrete and fully discrete approximations are considered using the Ritz Volterra projection Other related problems are studied as well The main feature of our analysis i...

This paper deals with development and analysis of a fully discrete nite element method for a nonlinear di erential system for describing an air water system in groundwater hydrol ogy The nonlinear system is written in a fractional ow formulation i e in terms of a saturation and a global pressure The saturation equation is approximated by a nite element method while the pressure equation is trea...

We classify all partial di¤erential equations with polynomial coe¢ cients in x and y of the form A(x)uxx + 2B(x; y)uxy + C(y)uyy +D(x)ux + E(y)uy = nu; which has weak orthogonal polynomials as solutions and show that partial derivatives of all order are orthogonal. Also, we construct orthogonal polynomials in d-variables satisfying second order partial di¤erential equations in d-variables.

An important question about a ne skeletons is the existence of di erential equations that are related to the A ne Distance and the Area Distance (hence to a ne skeletons) in the same way the Eikonal equation is related to the Euclidean Distance (and the medial axis). We show a surprisingly simple nonlinear second order PDE of Monge-Ampère type that relates to the a ne skeletons (and extends the...

In many applications, large systems of ordinary di erential equations (ODEs) have to be solved numerically that have both sti and nonsti parts. A popular approach in such cases is to integrate the sti parts implicitly and the nonsti parts explicitly. In this paper we study a class of implicit-explicit (IMEX) linear multistep methods intended for such applications. The paper focuses on the linea...

— We present necessary and su‰cient conditions to guarantee that at least one solution of an infinite dimensional stochastic di¤erential equation, which starts from a regular closed subset K of an Hilbert space, remains in K for all times.

We reformulate the one-dimensional complex Ginzburg-Landau equation as a fourth order ordinary differential equation in order to find stationary spatiallyperiodic solutions. Using this formalism, we prove the existence and stability of stationary modulated-amplitude wave solutions. Approximate analytic expressions and a comparison with numerics are given.

In this paper we study the regularity of the solution of an elliptic partial di erential equation with discontinuous coe cients. This result is used to establish the continuous Fr echet di erentiability of a shape functional arising in the optimal shape design of an electromagnet.