8 Feb 2016 Exam

Pagina 1

ADV ANCED MA THEMA TICAL ANAL YSIS F eb 8, 2016 Surname: Name: Code: LM — Chemical Engineering E1 E2 E3 E4 Q Lab T ot These problems and relative solutions are protected by copyright, therefore they cannot be exploited for any commercial purpose or editorial publication. Any abuse or infringement of copyright will be prosecuted to the full extent of the law.c⃝ • W rite solutions below the corresponding text in the blank spaces of these sheets and, if needed, on the reverse side. Drafts will not be acknowledged. • Provide motivations for your answers. Exercise 1. (7 points) Solve the transport problem with damping { ut + 2ux + 2uy +u = 0 ( x,y )∈ R2, t> 0 u(x,y, 0) =e−(|x|+|y|) (x,y )∈ R2 . Answer: Letting v =etu, v solves the following transport problem (without damping) { vt + 2vx + 2vy = 0 ( x,y )∈ R2,t> 0 v(x,y, 0) =e−(|x|+|y|) (x,y )∈ R2 after setting g(x,y ) =v(x,y, 0) =e−(|x|+|y|) , we get v(x,y,t ) =g(x− 2t,y− 3t) , thus v(x,y,t ) =e− ( |x−2t|+|y−2t| ) , therefore u(x,y,t ) =e− ( t+|x−2t|+|y−2t|2 ) . Exercise 2. (10 points) Solve1 the following boundary value problem for Laplace equation:    ∆u = 0 x2 +y2< 4,y >0 u(x, 0) = 0 −2≤x≤ 2 u(x,y ) = 4y3 x2 +y2 = 4,y >0 where we look for a solution C2 in the interior of the domain and continuous up to the boundary. Answer: We switch to polar coordinates: if we let u =u(ϱ,ϑ ), then u solves    ∆u =uϱϱ + 1 ϱuϱ + 1 ϱ2uϑϑ = 0 ϱ< 2, 0<ϑ<π (PDE ) u(ϱ, 0) =u(ϱ,π ) = 0 0 ≤ϱ≤ 2 (BC1) u(2,ϑ ) = 32 sin3ϑ 0≤ϑ≤π (BC2) the equation ∆u = 0, jointly with boundary condition (BC1), is solved by all the functions un(ϱ,ϑ ) =Rn(ϱ)Tn(ϑ) =ϱn sin(nϑ), 1It may be useful to note that sin 3 ϑ = 3 4 sin ϑ − 1 4 sin(3ϑ). where the terms ϱ−n sin(nϑ) and lnϱ are discarded to avoid singularity at the origin. Then the general solution of…