Re: Differential operators, Help
- To: mathgroup at smc.vnet.net
- Subject: [mg25364] Re: [mg25332] Differential operators, Help
- From: BobHanlon at aol.com
- Date: Sun, 24 Sep 2000 03:01:32 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
In a message dated 9/23/2000 4:03:07 AM, wkb at ansto.gov.au writes:
>This should be relatively easy, but after several tries I have not been
>able
>to do it.
>
>I want define a differential operator in the following way. Let Dx and
>Dx2
>denote the operators for first and second order differentiation with respect
>to x. I want P to be an operator which depends on x, Dx and Dx2.
>
>For example, with
>
> P = Dx2 + x Dx
>
>I want P[f[z],z] = f''[z] + z f'[z].
>
>This much I can do, but I cannot find a method which also gives the
>following result,
>
>P[ P[f[z],z], z] -> (f''[z] + z f'[z])''+ z (f''[z] + z f'[z])'
>
>Or more generally, if I have two such operators P and Q I want the correct
>result from expressions such as
>
> P[ Q[f[x],x], x]
>
P[expr_, x_Symbol:x] := D[expr, {x, 2}] + x*D[expr, x];
Q[expr_, x_Symbol:x] := D[expr, {x, 2}] - 2x^2*D[expr, x];
P[f[z], z]
z*Derivative[1][f][z] + Derivative[2][f][z]
P[P[f[z], z], z] // Simplify
z*Derivative[1][f][z] + (2 + z^2)*Derivative[2][f][z] +
2*z*Derivative[3][f][z] + Derivative[4][f][z]
D[f''[z] + z f'[z], {z, 2}] + z *D[f''[z] + z f'[z], z] == % // Simplify
True
P[ Q[f[x]]]
-4*Derivative[1][f][x] - 8*x*Derivative[2][f][x] -
2*x^2*Derivative[3][f][x] +
x*(-4*x*Derivative[1][f][x] - 2*x^2*Derivative[2][f][x] +
Derivative[3][f][x]) + Derivative[4][f][x]
Bob Hanlon