Analytical and Numerical Aspects of Partial Differential by Etienne Emmrich, Petra Wittbold

By Etienne Emmrich, Petra Wittbold

This article features a sequence of self-contained reports at the state-of-the-art in numerous components of partial differential equations, offered via French mathematicians. issues comprise qualitative homes of reaction-diffusion equations, multiscale tools coupling atomistic and continuum mechanics, adaptive semi-Lagrangian schemes for the Vlasov-Poisson equation, and coupling of scalar conservation legislation.

Show description

Read or Download Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series (De Gruyter Proceedings in Mathematics) PDF

Best differential equations books

Principles of Discontinuous Dynamical Systems

Discontinuous dynamical structures have performed an incredible position in either thought and purposes over the past numerous many years. this is often nonetheless a space of energetic learn and methods to make the functions better are an ongoing subject of curiosity. rules of Discontinuous Dynamical structures is dedicated to the speculation of differential equations with variable moments of impulses.

Fixed Point Theorems for Plane Continua with Applications (Memoirs of the American Mathematical Society)

During this memoir the authors current proofs of easy effects, together with these built to this point through Harold Bell, for the aircraft mounted aspect challenge: Does each map of a non-separating airplane continuum have a hard and fast aspect? a few of these effects have been introduced a lot previous through Bell yet with no available proofs.

Extra resources for Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series (De Gruyter Proceedings in Mathematics)

Sample text

Let us provide a “physical” explanation of the admissibility condition obtained for the case where the monotonicity of f ′ is strict. At any point of an admissible discontinuity curve x = x(t), consider the slopes f ′ (u+ ) and f ′ (u− ) of the characteristics 33 The Kruzhkov lectures x = f ′ (u± )t + C which impinge at this point from the two sides of the discontinuf (u+ )−f (u− ) ity. Consider also the slope ω = dx of the discontinuity curve (more dt = u+ −u− exactly, the slope of its tangent line); notice that ω is equal to the value f ′ (u˜ ) at some point u˜ which lies strictly between u+ and u− .

In addition to the integrals over the domains O− and O+ , also integrals over their boundaries will arise, that is, we will get integrals over ∂O and over Γ ∩ O. As ϕ is compactly supported in O, the integral over ∂O is zero. Consequently, we obtain ut + (f (u))x ϕ dx dt O− − Γ∩O (u− − k ) cos(ν, t) + (f (u−) − f (k )) cos(ν, x) ϕ dS ut + (f (u))x ϕ dx dt − O+ − Γ∩O (u+ − k ) cos(ν, t) + (f (u+) − f (k )) cos(ν, x) ϕ dS 0. , the outward normal vector to the boundary of O− and, at the same time, the interior normal vector for O+ ).

2), which is the problem of evolution from a simplest piecewise constant initial datum. 1) where u− and u+ are two arbitrary constant states. The solutions we want to construct will be piecewise smooth in ΠT . 5) and the entropy increase condition on each curve of jump discontinuity. These solutions will converge to the function u0 as t → +0 at all points, except for the point x = 0. 1) can be found in [27, Lectures 4–6]; its existence is demonstrated below with an explicit construction. First of all, let us notice that the equation we consider is invariant under the change x → kx, t → kt; moreover, the initial datum also remains unchanged under the action of homotheties x → kx, k > 0.

Download PDF sample

Rated 4.54 of 5 – based on 19 votes