RE: Differential operators, Help

• To: mathgroup at smc.vnet.net
• Subject: [mg25366] RE: [mg25332] Differential operators, Help
• From: "David Park" <djmp at earthlink.net>
• Date: Sun, 24 Sep 2000 03:01:33 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Bill,

op = (D[#1, {#2, 2}] + #2 D[#1, #2]) &;

op[f[z], z]
op[%, z]
z*Derivative[1][f][z] + Derivative[2][f][z]
2*Derivative[2][f][z] + z*Derivative[3][f][z] +
z*(Derivative[1][f][z] + z*Derivative[2][f][z] +
Derivative[3][f][z]) + Derivative[4][f][z]

Which is the same as

D[Derivative[2][f][z] + z*Derivative[1][f][z],
{z, 2}] + z*D[Derivative[2][f][z] +
z*Derivative[1][f][z], {z, 1}]

David Park

> -----Original Message-----
> From: Bill Bertram [mailto:wkb at ansto.gov.au]
To: mathgroup at smc.vnet.net
> Hi group,
>
> 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]
>
> Any suggestion will be greatly appreciated.
>
> Bill
>
>
>
>
>
>

```

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