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