I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator.
Examples:
$\exp(\varepsilon \partial_x) f(x) = f(x + \varepsilon)$
$\exp(\varepsilon x \partial_x) f(x) = f(x \exp(\varepsilon) )$
$\exp(\varepsilon x^2 \partial_x) f(x) = f\left( \frac{x}{1-\varepsilon x}\right)$
$\exp\Big(\varepsilon (x \partial_y + y \partial_x) \Big) f(x, y) = f\Big(\cosh(\varepsilon) x + \sinh(\varepsilon) y, \sinh(\varepsilon) x + \cosh(\varepsilon) y\Big) $
All these equalities are varifiable by Taylor expanding the variable $\varepsilon$ around zero.
My ideas:
-
Taylor expand the exponential of differential and try to guess the function yielding the same expansion (think this requires going through differential equations, which my program cannot do).
-
Recognize Lie algebras equivalent to the differential form, contruct a Lie algebra matrix and exponentiate.
I'm stuck because I do not know precisely how to proceed, except for particular cases in which the Lie algebra identification is obvious.
I also tried to look for papers both on Google and Google Scholar to get something out of it, but I didn't manage to find an explanation for such an algorithm.
- Do you know such an algorithm to find the action of such an exponential map on a function?
- Do you have any useful references for this?
- Is the Lie-algebra/Lie-group approach correct and valid for all types of differential operators?
EDIT:
I identified the case of linear operators, such as
$ \exp\Big( \epsilon ( \sum_{i,j} a_{ij} x_i \partial_{x_j} ) \Big ) f(x_1, \ldots, x_N) $
which would be easily solved by constructing a matrix
$M_{ij} = a_{ij}$
and then exponentiating it, using well known algorithms for matrix exponentiation, then make the matrix act on the function parameters $x_1, \ldots, x_N$ and substitute the resulting vector as new parameters.
e.g.:
the hyperbolic rotation mentioned earlier:
$\exp\Big(\varepsilon (x \partial_y + y \partial_x) \Big) f(x, y) = f\Big(\cosh(\varepsilon) x + \sinh(\varepsilon) y, \sinh(\varepsilon) x + \cosh(\varepsilon) y\Big) $
has matrix:
$ \begin{pmatrix} 0 & \varepsilon \end{pmatrix}$
$\begin{pmatrix} \varepsilon & 0 \end{pmatrix} $
which results by exponentiation in:
$ \begin{pmatrix} \cosh(\varepsilon) & \sinh(\varepsilon) \end{pmatrix} $
$ \begin{pmatrix} \sinh(\varepsilon) & \cosh(\varepsilon) \end{pmatrix} $
The problem concerns general differential operators, such as
$\exp\Big(\varepsilon x^n \partial_{x}\Big) f(x) $
or maybe even multivariable non-homogeneous differential operators, such as:
$\exp\Big(\varepsilon ( x^2 y^3 \partial_x + x y^5 \partial_y ) \Big) f(x, y) $
How do I find a non-infinite formula for the action on $f(x, y)$?
4) Is the matrix algebra approach a good way?
5) Is an analysis of the structure of the Taylor expansion a good way?
SOLUTION
Given a differential operator $D$, the exponential action $\exp(t \, D) f(x_1,\ldots)$ is given by the partial differential equation:
$\partial_t g(t, x_1, \ldots, x_n) = D [g(t, x_1, \ldots, x_n)] $
$ g(0, x_1,\ldots,x_n) = f(x_1, \ldots, x_n)$
Then $g(1, x_1, \ldots, x_n)$ is the result of the exponential action.
Best Answer
A partial answer: What you call the "exponential function" is the so-called flow semigroup, see Engel-Nagel, Section II.3.28.
Another reference on the Lie derivative is the monograph by Chicone and Swanson.