Generalized solutions of parabolic initial value problems with discontinuous nonlinearities

Discontinuous Parabolic Problems with a Nonlocal Initial Condition

Let Ω be a an open bounded domain in R , N ≥ 2, with a smooth boundary ∂Ω. Let QT Ω × 0, T and ΓT ∂Ω × 0, T where T is a positive real number. Then ΓT is smooth and any point on ΓT satisfies the inside and outside strong sphere property see 1 . For u : QT → R we denote its partial derivatives in the distributional sense when they exist byDtu ∂u/∂t, Diu ∂u/∂xi,DiDju ∂u/∂xi∂xj , i, j 1, . . . ,N....

A Generalized Maximum Principle for Boundary Value Problems for Degenerate Parabolic Operators with Discontinuous Coefficients

In [14] M.G.Platone Garroni has extended the classical generalized maximum principle (see, for instance, [15]), when the coefficients of the operator are discontinuous, to subsolutions of elliptic linear second order equations with mixed type boundary unilateral conditions, that is, on a portion of the boundary ∂Ω of Ω, the values of the solution are assigned, while on the other part a unilater...

Degenerate Parabolic Initial-Boundary Value Problems*

in Hilbert space and their realizations in function spaces as initial-boundary value problems for partial differential equations which may contain degenerate or singular coefficients. The Cauchy problem consists of solving (1.1) subject to the initial condition Jdu(0) = h. We are concerned with the case where the solution is given by an analytic semigroup; it is this sense in which the Canchy p...

Initial - Boundary Value Problems for Parabolic Equations

Computing Exact Solutions of Initial Value Problems

Finding the exact solution of initial value problems is the main topic of this thesis. Because differential equations in practice can be hard to solve, we require algorithms to solve them on a computer. Since the solution of the equation needs to be stored on a computer, we first discuss the representation of real functions. We show that approximating a function by polynomials gives a suitable ...