Homework 09

Exercise 1. Maximum Principle Theorem

Let uC1[0,T]×C2(Ω)u \in C^1[0,T] \times C^2(Ω), satisfying tΔu>0\partial_t - \Delta u > 0.

a.- Show that,

maxu[0,T]×Ω=maxu([0,T]×Ω). \max{u}_{[0,T] \times \Omega} = \max{u}_{\partial([0,T] \times \Omega)}.

b.- Extend to the case tΔu>0\partial_t - \Delta u > 0.

c.- What does it say physically for the case of the heat equation?

d.- Solve the heat equation in the interval [0,L][0,L] with homogeneous Dirichlet boundary conditions and initial data u0(x)=x(xL)u_0(x) = x * (x-L), find the maximum for each time and check the validity of the above theorem. What happens if you now solve for boundary conditions u0(0)=0,u0(L)=L2/8u_0(0) = 0, u_0(L) = L^2/8.

Exercise 2. Solving using Fourier Series

a.- Solve the heat equation in the unit circle using Fourier Transform.

b.- Using as initial data u(x,t=0)=x(1x)u(x,t=0) = x*(1-x) find the maximum and its location for each time and check the validity of the maximum principle theorem of Exercise 1.

Exercise 4. Sources

a.- Using a base of eigenfunctions solve for arbitrary sources the heat equation on the unit circle.

b.- Check that the solution belongs to arbitrary Sobolev spaces in space for any t>0t>0 even when it only belonged to L2L^2 at t=0t=0. What does that say about the differentiability of the solution in space? And in time?

Exercise 4. Boundary Conditions

Solve the heat equation in an interval [0,L][0,L].

a.- With homogeneous Dirichlet boundary conditions u(0,t)=u(L,t)=0u(0,t)=u(L,t)=0.

b.- With homogeneous Neumann boundary conditions ux(0,t)=ux(L,t)=0u_x(0,t)=u_x(L,t)=0.

c.- With mixed boundary conditions u(0,t)=0u(0,t)=0 and ux(L,t)=0u_x(L,t)=0.

d.- With Robin boundary conditions u(0,t)=0u(0,t)=0 and ux(L,t)+au(L,t)=0u_x(L,t) + au(L,t)=0.

e.- With periodic boundary conditions u(0,t)=u(L,t)u(0,t)=u(L,t) and ux(0,t)=ux(L,t)u_x(0,t)=u_x(L,t).

f.- With non-homogeneous Dirichlet boundary conditions u(0,t)=0u(0,t)=0 and u(L,t)=Su(L,t)=S, where SS is a constant.

g.- Compute the energy loss formula for the case d. For which values of aa the solution is stable? What does it mean physically?

Exercise 5. Behaviour of solutions, smoothness and infinite propagation speed of the heat equation. Consider the heat equation in one space dimension:

ut=α2ux2,u(x,0)=u0(x). \frac{\partial u}{\partial t} = \alpha\frac{\partial^2 u}{\partial x^2}, \quad u(x,0) = u_0(x).

a.- Show that the solution to the heat equation with initial data u0(x)u_0(x) is given by the convolution of the initial data with the fundamental solution:

u(x,t)=K(xy,t)u0(y)dy, u(x,t) = \int_{-\infty}^{\infty} K(x-y,t)u_0(y)dy,

where the fundamental solution is given by:

K(x,t)=14παtex24αt. K(x,t) = \frac{1}{\sqrt{4\pi \alpha t}}e^{-\frac{x^2}{4\alpha t}}.

b.- Show that the solution is smooth in both xx and tt for all t>0t>0 and that the solution is infinitely differentiable in both xx and tt for all t>0t>0. Check that the same holds for the solutions found on the previous exercise.

c.- Argue that even when the initial data is of compact support the solution extends to infinity for any t>0t>0.

d.- The solution decays exponentially fast as x|x| \to ∞.

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