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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.

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These are just formal solutions of first order partial differential equations... –  H. Arponen May 18 '13 at 18:40
    
The formula for the action on $f(x,y)$ is given in detail in the Engel-Nagel reference below. Does it help? –  András Bátkai May 19 '13 at 17:49
    
I had a look at Engel-Nagel, it works! It's just a first order differential equation to solve to find the flow of the vector field. Thanks! –  user34129 May 19 '13 at 23:21
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1 Answer

up vote 2 down vote accepted

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.

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