Differential operators, Help

*To*: mathgroup at smc.vnet.net*Subject*: [mg25332] Differential operators, Help*From*: "Bill Bertram" <wkb at ansto.gov.au>*Date*: Sat, 23 Sep 2000 03:35:55 -0400 (EDT)*Organization*: Australian Nuclear Science and Technology Organisation*Sender*: owner-wri-mathgroup at wolfram.com

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