1
vote
1answer
114 views

Understanding a periodic discontinuous function

This is from an example in my PDE text. It's something I should probably know, but maybe I am just reading the wording wrong. The text (Asmar as it happens, section 2.1) gives the following example ...
2
votes
1answer
118 views

How to show $C_p^k([-\ell, \ell])$ is not a Banach space?

I need to show the space $$C_p^k([-\ell, \ell])=\{f\in C^k(\mathbb R; \mathbb C); f(x+2\ell)=f(x), \forall x\in\mathbb R\},$$ is not a banach space with the norms ...
0
votes
1answer
26 views

Convergence of a series of a given metric..

I'm trouble with a metric defined over a given set: Consider $\mathcal{P}=C_p^\infty([-\pi, \pi])$, that is, $\mathcal{P}$ is the set of all infinitely differentiable functions $f:\mathbb R\rightarrow ...
1
vote
2answers
160 views

Asymptotic Expansion of a Multiscale Partial Differential Equation

I'm trying to understand how to solve $$-\nabla\cdot(K(\frac{x}{\epsilon})\nabla u(x,\frac{x}{\epsilon})=f \text{ in } \Omega$$ $$u(x,\frac{x}{\epsilon})=u_D \text{ in } \partial \Omega$$ where ...
2
votes
1answer
652 views

FFT for solving Poisson equation in 3 dimensions with periodic boundary conditions

In electrostatics, the laplacian of the electrostatic potential $\Delta V(\mathbf{r})$ created by a charge distribution $\rho(\mathbf{r})$ is $$ \Delta ...