Re: symbolic manipulation of operators

*To*: mathgroup at yoda.physics.unc.edu*Subject*: Re: symbolic manipulation of operators*From*: victor*Date*: Sun, 20 Sep 92 13:30:46 CDT

I would like to give an advice how that could work within Mathematica. So, Alex Kasman wants to introduce the new operation D*u = u' + u*D where D is a differential operator and u is a some operator. >From the Mathematica point of view this new operation is a bit incorrect. Because the symbol u' is understanding as a function in Mathematica. Derivative[_] is a functional operator and Derivative[_][u] is a function. Thus, we have operator = function + operator ??? But we can extend the definition of Derivative to work with any operators. Now, let we assume that D and u are operators and Derivative[_] can be applied to any "good" operators too and Derivative[_][u] is an operator. We what to know what D^2*u is. D^2[u] == D[ D[u] ] = D[ Derivative[1][u] + u[D] ] == D[ Derivative[1][u] ] + D[ u[D] ] = (D is an linear operator?) Derivative[1][ Derivative[1][u] ] + Derivative[1][u][ D ] + Derivative[1][ u[D] ] + u[D][ D ] = Derivative[2][u] + Derivative[1][u][D] + Derivative[1][u][D] + u[D][D] = Derivative[2][u] + 2 Derivative[1][u][D] + u[D^2] = u'' + 2 u'*D + u*D^2 that is not the same to: D^2*u = D*(D*u)= D*(u'+u*D) = u'' + 2 u'D + D^2 That's why I am afraid maybe I didn't understand a bit that problem. Anyway, I would continue and maybe my thoughts will be useful to somebody. Here I'll give some routines how to make such operations Clear[u] u /: Head[u] = Operator (* properties of the operator u *) (* u[A'][A''] is u[A'''] *) u[Derivative[n_Integer][b_Symbol]][Derivative[m_Integer][b_]] := u[ Derivative[n+m][b] ]/; Head[ u ] == Operator (* u'[A'][A''] is u'[A'''] *) Derivative[k_Integer][s_/;Head[s] == Operator][ Derivative[n_Integer][b_Symbol]][Derivative[m_Integer][b_]] := Derivative[k][s][ Derivative[n+m][b] ] (* an extension of Derivative[_] *) Derivative[1][ s_[p_] ] := Derivative[1][s][p] /; Head[ s ] == Operator (* the multiplication of operators *) (* the first order There are 4 cases for the expression s2: s2 is u s2 is u' s2 is u[_] s2 is u'[_] *) Derivative[1][s1_Symbol][ s2_ ] := Derivative[1][s2] + s2[Derivative[1][s1]] /; Head[s2] == Operator || ( MatchQ[Head[s2], _Derivative ] && Head[s2[[1]]] == Operator ) || MatchQ[Head[s2], p_/;Head[p] == Operator] || ( MatchQ[Head[Head[s2]], _Derivative] && Head[Head[s2][[1]]] == Operator ) (* derivatives of the high order *) Derivative[n_Integer/;n>1][s1_Symbol][ s2_/;Head[s2] == Operator ] := Derivative[1][s1][ Derivative[n-1][s1][s2] ] (* linearity *) Derivative[n_Integer][s1_Symbol][ s2_Plus ] := Derivative[1][s1][ # ]&/@ s2 Derivative[n_Integer][s1_Symbol][ m_?NumberQ s2_ ] := m Derivative[1][s1][ s2 ] >>>>>>>>>>>>> If we will load that code we will get. In[19]:= S'[u] Out[19]= u[S'] + u' In[20]:= S''[u] Out[20]= u[S''] + u'' + 2 u'[S'] In[21]:= S'''[u]//Expand (3) (3) Out[21]= u[S ] + u + 3 u'[S''] + 3 u''[S'] In[22]:= S''''[u]//Expand (4) (4) (3) (3) Out[22]= u[S ] + u + 4 u'[S ] + 6 u''[S''] + 4 u [S'] Here the operator S' is exactly the above "element called D", S'' is D^2 and so on, i.e. that differential operator D is a composition of the operator Derivative[] and a some operator S. Victor Adamchik