symbolic manipulation of operators

• To: mathgroup at yoda.physics.unc.edu
• Subject: symbolic manipulation of operators
• From: deutsch at bu-pub.bu.edu
• Date: Fri, 18 Sep 92 08:52:36 -0400

```Hello Mathgroup,

This question was sent to me from Alex Kasman, Dept. of
Mathematics, Boston University. If anyone has experience creating
the type of operator described below, we would appreciate hearing from you.

Thank you,

David Deutsch
Information Technology
Boston University
deutsch at bu-it.bu.edu

------------------------------------------------------------------

I need to have multiplicative differential operators.  They MOSTLY act
just like multiplication, the only difference is that there is also an
element called D which, when multiplied by u behaves like this:

D*u = u' + u*D

You will note that this cannot be a commutative multiplication since
D*u NOT= u*D.

I would like D^2*u = D*(D*u)= D*(u'+u*D) = u'' + 2 u'D + D^2
and so on for higher powers of D (although I will probably never get
past D^4 in my applications.)

The way that I TRIED to do this was to turn off the commutativity of
Mathematica's "Times" function and teach it how to treat D.  This
worked fine...EXCEPT, matrix multiplication, Simplify, Expand, simple
subtraction(!) and many other operations still assume that
multiplication is commutative.  Consequently, whenever I try to do
anything, I get recursion overflow errors (since the D keeps moving
back and forth) and loads of mathematical errors (which look like
a+b-a=2a+b !!) etc.

For example, I can correctly calculuate D*u and u*D.  However, it
won't do D*u-u*D because something in the subtraction process tries to
simplify things in a way that assumes multiplication to be
commutative.

It is necessary that the action of D be done via multiplication
because the eventual goal is to multiply and add matrices made up of
differential operators.

difficult one; I will do my best to explain it to you in greater
detail if you think you can help.

Thanks
Alex

P.S. Here is what I did in attempting to get this all to work:

Unprotect[Times]
Attributes[Times]={OneIdentity,Flat,Listable}

Times[d,u_]=u'+u*d
Times[d,0]=0
Times[d,d,u_]=u''+2u'*d+u*d^2
Times[d,d,d,u]=u'''+2u''*d+2u'*d^2+u'''*d^3
Times[dd,u_]=u''+2u'*d+u*d^2
Times[ddd,u_]=u'''+3u''*d+3u'*d^2+u'''*d^3
Times[(d+u_),v_]=d*v+u*v
Times[v_,(d+u_)]=v*d+v*u
Times[(dd+u_),v_]=dd*v+u*v
Times[v_,(dd+u_)]=v*dd+v*u
Times[(ddd+u_),v_]=ddd*v+u*v
Times[v_,(ddd+u_)]=v*ddd+v*u
Times[d,d]=dd
Times[dd,d]=ddd
Times[d,dd]=ddd
Times[d,d,d]=ddd
Protect[Times]

Unprotect[Power]
Power[d,2]=dd
Power[d,3]=ddd

Protect[Times]

d'=0

SimpleLax[exp_]:=Collect[exp/.{d->D,dd->D^2,ddd->D^3},{D,I}]

LaxMul[x_,y_]:=SimpleLax[x*y]

Lax[v_,w_]:=MatrixForm[{{LaxMul[v[[1,1]],w[[1,1]]]+LaxMul[v[[2,1]],w[[1,2]]]
-LaxMul[w[[1,1]],v[[1,1]]]-LaxMul[w[[2,1]],v[[1,2]]],
LaxMul[v[[1,1]],w[[2,1]]]+LaxMul[v[[2,1]],w[[2,2]]]
-LaxMul[w[[1,1]],v[[2,1]]]-LaxMul[w[[2,1]],v[[2,2]]]},
{LaxMul[v[[1,2]],w[[1,1]]]+LaxMul[v[[2,2]],w[[1,2]]]
-LaxMul[w[[1,2]],v[[1,1]]]-LaxMul[w[[2,2]],v[[1,2]]],
LaxMul[v[[1,2]],w[[2,1]]]+LaxMul[v[[2,2]],w[[2,2]]]
-LaxMul[w[[1,2]],v[[2,1]]]-LaxMul[w[[2,2]],v[[2,2]]]}}]

```

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