3
$\begingroup$

The name of the operator $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]$ is one-parameter semigroup.

For example one writes $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]f$, where $f$ is a function of $\overrightarrow{v}$.

What is the meaning of this operator?

Does $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]f$ mean $EXP[\frac{ \theta }{2}\frac{\partial^2 f}{\partial \overrightarrow{v}^2}]$ ?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

This is a symbol which reminds that the solution of linear system of ODEs is represented as an exponential function of a matrix. This symbol represents the solution family of the heat equation, and indeed called a one-parameter semigroup (or semi-dynamical system).

Great introduction in the subject are the books of Engel and Nagel, the original one and a short version.

So $ EXP[t\frac{\partial^2}{\partial \overrightarrow{v}^2}]f$ is the solution of the partial differential equation.

$$\partial_t u(\overrightarrow{v},t)= \frac{\partial^2}{\partial \overrightarrow{v}^2}u(\overrightarrow{v},t)$$ with initial condition $$u(\overrightarrow{v},0) = f(\overrightarrow{v})$$ and appropriate boundary conditions depending on the domain and the problem at hand.

Improve this answer
$\endgroup$
2
  • $\begingroup$ thanks for your answer! can you tell me how to compute it? $\endgroup$
    – Zhang He
    Commented Dec 4, 2013 at 7:45
  • $\begingroup$ It is a great help,thank you very much！ $\endgroup$
    – Zhang He
    Commented Dec 4, 2013 at 8:54

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .