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.


1. Do you know such an algorithm to find the action of such an exponential map on a function?
2. Do you have any useful references for this?
3. 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.



Algorithm to find exponential map of differential operators acting on function
A partial answer: What you call the "exponential function" is the so-called flow semigroup, see [Engel-Nagel](http://www.fa.uni-tuebingen.de/research/publications/1999/one-parameter-semigroups-for-linear-evolution-equations/), Section II.3.28.


Another reference on the [Lie derivative](http://en.wikipedia.org/wiki/Lie_derivative) is the monograph by [Chicone and Swanson](http://ejde.math.unt.edu/Monographs/02/abstr.html).


