[Q} MMA: General Linear Differential Operator

• Subject: [mg3118] [Q} MMA: General Linear Differential Operator
• From: paddy at sun4.bham.ac.uk (Patrick Jemmer)
• Date: 3 Feb 1996 01:03:30 -0600
• Approved: usenet@wri.com
• Distribution: local
• Newsgroups: wri.mathgroup
• Organization: The University of Birmingham, UK.
• Sender: daemon at wri.com

```---------------------------------7006122884703
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Hello All!

I need to be able to define a "second-derivative"
type operator (ie like del-squared on scalars).

I feel I am being very inelegant in having to make
all the following declarations for the properties(* define general delsquared
*)

L[a_+b_,q_]:=L[a,q]+L[b,q]
L[a_*b_,q_]:=a*L[b,q]+b*L[a,q]+2*D[a,q]*D[b,q]
L[m_Integer*a_,q_]:=m*L[a,q]
L[a_[q_]*b_[q_],q_]:=a[q]*L[b[q],q]+b[q]*L[a[q],q]+2*D[a[q],q]*D[b[q],q]
L[a_[q_]^n_,q_]:=n*a[q]^(n-2)*(a[q]*L[a[q],q]+(n-1)*D[a[q],q]^2)
L[a_[q1]*b_[q2],q1]:=b[q2]*L[a[q1],q1]
L[a_[q1]*b_[q2],q2]:=a[q1]*L[b[q2],q2]
L[a_[q1]^n_,q2]:=0
L[a_[q2]^n_,q1]:=0
L[a_[q1],q2]:=0
L[a_[q2],q1]:=0
L[a_Real,q_]:=0
L[a_Rational]:=0
---------

Is there any systematic way around this ?

I feel I should be using the built-in properties of
D, Dt or things of that kind...

What I want to calculate is the following:

-------------
rho1[q_]:=2*chi1[q]^2
rho2[q_]:=2*chi2[q]^2
rho12[q1_,q2_]:=2*(chi1[q1]*chi1[q2]+chi2[q1]*chi2[q2])
p2[q1_,q2_]:=rho[q1]*rho[q2]-(1/2)*rho12[q1,q2]^2//Expand

d2r[fn_,q1_,q2_]:=L[fn,q1]/4+L[fn,q2]/4-D[fn,q1,q2]/2

d2p2[q1_,q2_]:=d2r[p2[q1,q2],q1,q2]
d2p2a=d2p2[q1,q2]

d2p2=d2p2a /. {q1->r,q2->r}

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

Thanks... Patrick Jemmer

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

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```

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