Tagged Questions
1
vote
1answer
135 views
PDE - (homogeneous) Heat equation - Solution?
today I have a question in PDE. It concerns the heat equation:
Formulate the (homogeneous) heat equation for functions $f:(0,\infty)\times\mathbb{R^n} \longrightarrow\mathbb{C}$. Derive an equation ...
2
votes
1answer
77 views
Evolution operator
We call a function that assigns a starting value of a time-dependent differentialfunction to a solution of a later timevalue as the evolution operator $E(t)$.
Look at the thermal equation
$$
...
2
votes
2answers
172 views
Find derivative of convolution with gaussian
Let $A(\sigma)$, $\sigma > 0$ be an operator that acts on bounded continuous functions $f$ on $\mathbb{R}$ by the rule
$$
(A(t)f)(x) = \int\limits_{\mathbb{R}} f(y)\frac{1}{\sqrt{2 \pi ...
0
votes
1answer
117 views
Verify this distribution convolution: $E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$
In our class notes we are asked to verify the following equality:
$$E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$$
where ...
