CODEWITHSUNDEEP
wolfram-mathematica
Qiang Li
Qiang Li

Reputation: 10865

How to expand the arithmetics of differential operators in mathematica

For example, I want mma to expand out the following differential operator

(1+d/dx+x*d2/dy2)^2*(1+y*d/dy)^2

I found Nest is not good enough to do this sort of things.

Upvotes: 2

Views: 1826

Answers (1)

Daniel Lichtblau
Daniel Lichtblau

Reputation: 6894

A bit dated, but see

http://library.wolfram.com/infocenter/Conferences/325/

The section "Some noncommutative algebraic manipulation" gives a few ways to go about this. The first example, defining a function called differentialOperate, is probably best suited for your purposes.

---edit, reedited---

Here is the code I use. Probably it is (still) missing a few refinements. It is taken from a couple of examples in the notebook mentioned above.

I will define and use an auxiliary predicate, scalarQ. This gives the flexibility of declaring entities other than explicit numerical values to be scalar.

I define a noncommutative multiplication, called ncTimes. Ideally I would just use NonCommutativeMultiply, but Iwas not able to get the pattern matching to behave the way I wanted with respect to zero or one argument forms, or pulling out scalars. (Less technical explanation: it's mojo was more powerful than mine.)

scalarQ[a_?NumericQ] := True scalarQ[_] := False

ncTimes[] := 1 ncTimes[a_] := a ncTimes[a_, ncTimes[b_, c], d_] := ncTimes[a, b, c, d] ncTimes[a_, x_ + y_, b_] := ncTimes[a, x, b] + ncTimes[a, y, b] ncTimes[a_, i_?scalarQ*c_, b_] := i*ncTimes[a, c, b] ncTimes[a_, i_?scalarQ, b___] := i*ncTimes[a, b]

differentialOperate[a_, expr_] /; FreeQ[a, D] := a*expr differentialOperate[L1_ + L2_, expr_] := differentialOperate[L1, expr] + differentialOperate[L2, expr] differentialOperate[a_*L_, expr_] /; FreeQ[a, D] := a*differentialOperate[L, expr] differentialOperate[a : HoldPattern[D[] &], expr_] := a[expr] differentialOperate[ncTimes[L1, L2_], expr_] := Expand[differentialOperate[L1, differentialOperate[L2, expr]]] differentialOperate[L1_^n_Integer, expr_] /; n > 1 := Nest[Expand[differentialOperate[L1, #]] &, expr, n]

In[15]:= ddvar[x_, n_: 1] := D[#, {x, n}] &

Here are some of your examples, both from post and comments.

In[17]:= diffop = ncTimes[(1 + ddvar[x] + ncTimes[x, ddvar[y, 2]])^2, (1 + ncTimes[y, ddvar[y]])^2]

Out[17]= ncTimes[(1 + (D[#1, {x, 1}] & ) + ncTimes[x, D[#1, {y, 2}] & ])^2, (1 + ncTimes[y, D[#1, {y, 1}] & ])^2]

Apply this operator to f[x,y].

In[25]:= differentialOperate[diffop, f[x, y]]

Out[25]= f[x, y] + 3*y*Derivative[0, 1][f][x, y] + 9*Derivative[0, 2][f][x, y] + 18*x*Derivative[0, 2][f][x, y] + y^2*Derivative[0, 2][f][x, y] + 7*y*Derivative[0, 3][f][x, y] + 14*x*y*Derivative[0, 3][f][x, y] + 25*x^2*Derivative[0, 4][f][x, y] + y^2*Derivative[0, 4][f][x, y] + 2*x*y^2*Derivative[0, 4][f][x, y] + 11*x^2*y*Derivative[0, 5][f][x, y] + x^2*y^2*Derivative[0, 6][f][x, y] + 2*Derivative[1, 0][f][x, y] + 6*y*Derivative[1, 1][f][x, y] + 18*x*Derivative[1, 2][f][x, y] + 2*y^2*Derivative[1, 2][f][x, y] + 14*x*y*Derivative[1, 3][f][x, y] + 2*x*y^2*Derivative[1, 4][f][x, y] + Derivative[2, 0][f][x, y] + 3*y*Derivative[2, 1][f][x, y] + y^2*Derivative[2, 2][f][x, y]

Those edge cases.

In[26]:= differentialOperate[ncTimes[1, 1], f[t]]

Out[26]= f[t]

We can declare a symbol to be scalar.

In[28]:= scalarQ[a] ^= True;

Now it will get pulled out as a simple multiplier.

In[29]:= differentialOperate[ncTimes[a, b], f[t]]

Out[29]= a b f[t]

---end edit---

Daniel Lichtblau Wolfram Research

Upvotes: 2

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