Contains papers presented by leading international experts at the well-established weekly seminar held at the College de France.

Nonlinear Partial Differential Equations & Their Applications: College de France Seminar Vol. 1 (Research Notes in Mathematics)

Science & Math

Lions†, J.-L.

Brezis, H.

Biological Sciences

Pitman Publishing (UK)

Li Nonlinear Partial

Mathematics

Finite Difference Methods Solutions of Equations in R HNA 1 (Volume 1) (Handbook of Numerical Analysis, Volume 1)

Ciarlet, P.G.

Elsevier Science

This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence methods. Before this book, multiple scale methods were primarily used for non-linear oscillation problems in the applied mathematics community, not for analyzing spatial oscillations as in homogenization.  In the current printing a number of minor corrections have been made, and the bibliography was significantly expanded to include some of the most important recent references. This book gives systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate. The book continues to be interesting and useful to readers of different backgrounds, both from pure and applied mathematics, because of its informal style of introducing the multiple scale methodology and the detailed proofs.

Asymptotic Analysis for Periodic Structures (Volume 5) (Studies in Mathematics and its Applications, Volume 5)

Bensoussan, A.

Papanicolau, G.

North-Holland

The applications of the theory of Optimal Control of distributed parameters is an extremely wide field and, although a large number of questions remain open, the whole subject continues to expand very rapidly. The author does not attempt to cover the field but does discuss a number of the more interesting areas of application.

Some Aspects of the Optimal Control of Distributed Parameter Systems (CBMS-NSF Regional Conference Series in Applied Mathematics, Series Number 6)

Society for Industrial and Applied Mathematics

<p>These lecture notes by very authoritative scientists survey recent advances of mathematics driven by industrial application showing not only how mathematics is applied to industry but also how mathematics has drawn benefit from interaction with real-word problems.<br>The famous David Report underlines that innovative high technology depends crucially for its development on innovation in mathematics. The speakers include three recent presidents of ECMI, one of ECCOMAS (in Europe) and the president of SIAM.</p>

Computational Mathematics Driven by Industrial Problems: Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) ... 1999 (Lecture Notes in Mathematics, 1739)

Burkard, R.

Deuflhard, P.

Jameson, A.

Strang, G.

Reference

Springer

1. We begin by giving a simple example of a partial differential inequality that occurs in an elementary physics problem. We consider a fluid with pressure u(x, t) at the point x at the instant t that 3 occupies a region Q oflR bounded by a membrane r of negligible thickness that, however, is semi-permeable, i. e., a membrane that permits the fluid to enter Q freely but that prevents all outflow of fluid. One can prove then (cf. the details in Chapter 1, Section 2.2.1) that au (aZu azu aZu) (1) in Q, t>o, -a - du = g du = -a z + -a z + -a z t Xl X X3 z l g a given function, with boundary conditions in the form of inequalities u(X,t»o => au(x,t)/an=O, XEr, (2) u(x,t)=o => au(x,t)/an?:O, XEr, to which is added the initial condition (3) u(x,O)=uo(x). We note that conditions (2) are non linear; they imply that, at each fixed instant t, there exist on r two regions r~ and n where u(x, t) =0 and au (x, t)/an = 0, respectively. These regions are not prescribed; thus we deal with a "free boundary" problem.

Inequalities in Mechanics and Physics (Grundlehren der mathematischen Wissenschaften)

Duvant, G.

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