in the present article, we focus on the numerical approximation of stochastic partial differential equations of itˆo type with space-time white noise process, in particular, parabolic equations. for each case of additive andmultiplicative noise, the numerical solution of stochastic diffusion equations is approximated using two stochastic finite difference schemes and the stability and consisten...

We present a model hierarchy multilevel optimisation approach to solve an optimal boundary control problem in glass manufacturing. The process is modelled by radiative heat transfer and formulated as an optimal control problem restricted by partial differential algebraic equations (PDAE) and additional control constraints. We consider a sequence of model approximations given by space-time depen...

The purpose of this article is to explain how some apparently simple problems from numerical linear algebra are in fact extremely difficult, so that we cannot hope to solve them effectively in general. However, if we build and analyze algorithms to solve them in special cases of interest for the numerical analysis of partial differential equations, we find that the theory needed to validate the...

If T⃗ denotes a vector tangent to C at t,x,u then the direction numbers of T⃗ must be a,b, f. But then (1.2) implies that T⃗ n⃗, which is to say, T⃗ lies in the tangent plane to the surface S. But if T⃗ lies in the tangent plane, then C must lie in S. Evidently, solution curves of (1.2) lie in the solution surface S associated with (1.2). Such curves are called characteristic curves for (1.2). W...

In the rst part of this article, we have shown how time-dependent optimal control for partial di erential equations can be realized in a modern high-level modeling and simulation package. In this second part we extend our approach to (state) constrained problems. Pure state constraints in a function space setting lead to non-regular Lagrange multipliers (if they exist), i.e. the Lagrange multip...

2 Separation of variables and the complete integral 5 2.1 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The envelope of a family of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 The complete integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Determining the characteristic strips from t...

The fundamental solution of the Laplace equation in two dimensions is not contained in the Sobolev space H1(Ω) such that finite element error estimates are non-standard and quasi-uniform meshes are inappropriate. By using graded meshes L2-error estimates of almost optimal order are shown. As a by-product, we show for the Poisson equation with a right-hand side in L2 that appropriate mesh refine...

This is a survey on Chaos in Partial Differential Equations. First we classify soliton equations into three categories: 1. (1+1)-dimensional soliton equations, 2. soliton lattices, 3. (1+n)-dimensional soliton equations (n ≥ 2). A systematic program has been established by the author and collaborators, for proving the existence of chaos in soliton equations under perturbations. For each categor...

Mathematicians often speak of the unity of their subject. Such a unity from the 19th century origins in the history of linear ordinary differential equations, especially those closely connected with elliptic and modular functions. It is concerned with the impact of group theoretical and geometrical ideas upon the problem of understanding the nature of the solutions to a differential equation (s...