Hw5 - Dr. Yuan GAO PDF

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Dr. Yuan GAO...

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Math5351 Fall2018 Homework 5 Due to Nov. 28 2018 Updated on Nov. 21 2018, completed

1.1 Consider the shallow water system ht + uhx + hux = 0 ut + uux + hx = 0 with a simple derivative jump at x = 0 t = 0. Denote the wavefront ξ(x, t) = x − X(t) = 0 and assume u ≡ u0 , h ≡ h0 ahead of the front. After the power series expansion near ξ = 0, (1) Write down the approximated gradient jump [hx ], [u √x ] across ξ under the assumption the wavefront is upstream wave i.e. X ′ (t) = u0 − h0 ; (2) Assume further the initial jump [hx ] is positive, what’s the breaking time tb for [hx ]? 1.2 Assume the stretching transformation Tε is defined as x1 = εax, t1 = εb , u1 = εc u. When the PDE ut − (ux1/3)x = 0,

x > 0, t > 0

with initial data and boundary condition u(x, 0) = 0, x > 0;

ux (0, t) = −1, u(∞, t) = 0, t > 0

is invariant? Find out the similarity transformation to reduce the PDE to an ODE. 1.3 Associated with an Ornstein-Uhlenbeck process in stochastic dynamics is the PDE 1 ut = uxx − xux , 2 where u(x, t) is the probability that a particle starting at x in (−a, a) at time t = 0 stays within [−a, a] for all s with 0 ≤ s ≤ t. Assume u(−a, t) = u(a, t) = 0 for all t. In the long time limit show that the particle will escape the domain [−a, a] with probability 1. Hint: Assume when t → ∞, the solution goes to the steady state ut = 0. 1

1.4 Consider the nonlinear Reaction-Diffusion equation ut − (u2 )xx = u(1 − u). Investigate the existence of wavefront-type traveling wave solution U (z), i.e. the traveling wave solution tends to constants as z → ±∞. 1.5 Consider the nonlinear diffusion problem (K (u)ux )x − ut = 0,

0 < x < 1, 0 < t < T,

u(x, 0) = 1 + x(1 − x), u(0, t) = u(1, t) = 1,

0 < x < 1, 0 < t < T,

where the diffusion coefficient K=K(u) is a continuously differentiable function and K(u) > 0. Show that if u is a solution to this problem, then 1 ≤ u(x, t) ≤ 45.

2...