Homework 09

• Click here to download this notebook

Exercise 1. Maximum Principle Theorem

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

a.- Show that,

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

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

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

Exercise 2. Solving using Fourier Series

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

b.- Using as initial data $u(x,t=0) = x*(1-x)$ 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 $t>0$ even when it only belonged to $L^2$ at $t=0$. 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 $[0,L]$.

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

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

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

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

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

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

g.- Compute the energy loss formula for the case d. For which values of $a$ 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:

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

where the fundamental solution is given by:

b.- Show that the solution is smooth in both $x$ and $t$ for all $t>0$ and that the solution is infinitely differentiable in both $x$ and $t$ for all $t>0$. 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 $t>0$.

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