Download PDF by Etienne Emmrich, Petra Wittbold: Analytical and numerical aspects of partial differential

By Etienne Emmrich, Petra Wittbold

ISBN-10: 1282296477

ISBN-13: 9781282296473

ISBN-10: 3110204479

ISBN-13: 9783110204476

ISBN-10: 3110212102

ISBN-13: 9783110212105

This article features a sequence of self-contained studies at the cutting-edge in several components of partial differential equations, awarded via French mathematicians. issues comprise qualitative homes of reaction-diffusion equations, multiscale equipment coupling atomistic and continuum mechanics, adaptive semi-Lagrangian schemes for the Vlasov-Poisson equation, and coupling of scalar conservation laws.

Show description

Read Online or Download Analytical and numerical aspects of partial differential equations : notes of a lecture series PDF

Similar differential equations books

Download PDF by Dennis G. Zill: A First Course in Differential Equations with Modeling

A primary direction IN DIFFERENTIAL EQUATIONS WITH MODELING functions, tenth variation moves a stability among the analytical, qualitative, and quantitative ways to the examine of differential equations. This confirmed and obtainable booklet speaks to starting engineering and math scholars via a wealth of pedagogical aids, together with an abundance of examples, factors, "Remarks" containers, definitions, and crew initiatives.

Download PDF by Peter H. Baxendale, Sergey V. Lototsky: Stochastic Differential Equations: Theory and Applications,

This quantity includes 15 articles written through specialists in stochastic research. the 1st paper within the quantity, Stochastic Evolution Equations through N V Krylov and B L Rozovskii, was once initially released in Russian in 1979. After greater than a quarter-century, this paper continues to be a regular reference within the box of stochastic partial differential equations (SPDEs) and keeps to draw the eye of mathematicians of all generations.

Get Recent trends in differential equations PDF

This sequence goals at reporting new advancements of a excessive mathematical typical and of present curiosity. every one quantity within the sequence will be dedicated to mathematical research that has been utilized, or almost certainly acceptable to the suggestions of medical, engineering, and social difficulties. the 1st quantity of WSSIAA comprises forty two learn articles on differential equations by way of best mathematicians from world wide.

Extra info for Analytical and numerical aspects of partial differential equations : notes of a lecture series

Sample text

Nonetheless, as we have demonstrated in the previous section, so defined generalized solutions of the Cauchy problem may fail to be unique (even for the case u0 (x) ≡ 0). It is clear that the non-uniqueness stems from the fact that the “wrong” solutions uδ , δ = 0, have discontinuities. One could guess that not all the discontinuities are admissible; but how can we find the appropriate restrictions on the discontinuities? 1 Admissibility condition on discontinuities: the case of a convex flux function Let us make the additional assumption f ′′ 0, f ∈ C 3 ( R) , u 0 ∈ C 2 ( R) .

11. 22), holds. Proof. Fix a point (t0 , x0 ) ∈ Γ, x0 = x(t0 ), on the discontinuity curve Γ. As usual, denote by u± (t0 , x0 ) the one-sided limits of u(t0 , x) on Γ as x approaches x0 . To be specific, assume that u− (t0 , x0 ) < u+ (t0 , x0 ). 35) u(t, x) > k for (t, x) ∈ O+ ≡ {(t, x) ∈ O | x > x(t)}. 36) This is always possible since we consider a piecewise smooth solution. Moreover, without loss of generality, we can assume that u is smooth in each of the subdomains O+ and O− . 30) it follows that for any test function ϕ ∈ C0∞ (O), ϕ(t, x) holds |u − k|ϕt + sign(u − k ) (f (u) − f (k )) ϕx dx dt 0, there 0.

10) with respect to t, we obtain dx dx = vt (t, x(t)) + vx (t, x(t)) · dt dt Here and in the sequel, ux , vx , ut , vt denote the corresponding limits of the derivatives as the point (t, x) tends to the weak discontinuity curve Γ. 2), we have ut (t, x(t)) + ux (t, x(t)) · dx dx − f ′ (u(t, x(t)))ux = vx (t, x(t)) · − f ′ (v (t, x(t)))vx. 10), we obtain ux (t, x(t)) · ux (t, x(t)) − vx (t, x(t)) dx − f ′ (u(t, x(t)) = 0. 9) follows. 26 Gregory A. Chechkin and Andrey Yu. 1. 2) in the strip ΠT (remind that ΠT = {−∞ < x < +∞, 0 < t < T }), for (i) f (u) = u2 /2, u(t, x) = 0 1 for x < t, for x > t; (ii) f (u) = u2 /2, u(t, x) = 0 2 for x < t, for x > t; (iii) f (u) = u2 /2, u(t, x) = 2 0 for x < t, for x > t; (iv) f (u) = −u2 , u(t, x) = 1 −1 for x < 0, for x > 0; (v) f (u) = −u2 , u(t, x) = −1 1 for x < 0, for x > 0; (vi) f (u) = u3 , u(t, x) = 1 −1 for x < 0, for x > 0; (vii) f (u) = u3 , u(t, x) = −1 1 for x < t, for x > t; (viii) f (u) = u3 , u(t, x) = 1 −1 for x < t, for x > t?

Download PDF sample

Analytical and numerical aspects of partial differential equations : notes of a lecture series by Etienne Emmrich, Petra Wittbold


by Joseph
4.2

Rated 4.99 of 5 – based on 4 votes