Matrix differential operator

Question

i'm new in MATHEMATICA. I want to create an operator $D^{(f)}=\partial_x+f'-\partial^2_x$ and $D^{(g)}=\partial_x+g'-\partial^2_x$ and put it into a matrix element, then multiplied by a vector whose components are functions on $x$, say $u(x), v(x)$. For example $$\begin{pmatrix}D^{(f)} & D^{(g)}\\ D^{(g)} & D^{(f)} \end{pmatrix} \begin{pmatrix} u(x) \\ v(x) \end{pmatrix}$$

I saw the posts: Computing with matrix differential operators, How to do matrix operation if the first matrix is an operator?; but i'm very newbie on this.

My problem lives on the fact that I have problems with the matrices products because I have operators

Thanks!

EDIT: $f=f(x,y,z)$ and $g=g(x,y,z)$, both of them are function of a vector $(x,y,z)$, but we can neglect the components $y,z$ and consider only the $x$ part

Answer 1

We can start by defining our two differential operators and putting them into a matrix, like this:

ClearAll["Global`*"]

df[w_] := D[w, x] + f'[x] w - D[w, x, x]
dg[w_] := D[w, x] + g'[x] w - D[w, x, x]
m = {{df, dg}, {dg, df}};

Now we need a way for the matrix operator to act on the vector. This is not plain multiplication. One way to get the matrix operator to act on a vector is to define a function to do it, like this:

operate[matrix_, column_] := 
 Table[Inner[#1[#2] &, μ, column, Plus], {μ, matrix}]

The above function is basically what those other posts were saying. Now we can use the function to apply the matrix operator to our vector.

result = operate[m, {u[x], v[x]}];
result // MatrixForm

(*
  {u[x] f'[x] + v[x] g'[x] + u'[x] + v'[x] - u''[x] - v''[x], 
   v[x] f'[x] + u[x] g'[x] + u'[x] + v'[x] - u''[x] - v''[x]}  *)

We could have defined our functions f[x] and g[x] at any point. For example, we can do it now, as

f[x] := Sin[x]
result
(*
  {Cos[x] u[x] + v[x] g'[x] + u'[x] + v'[x] - u''[x] - v''[x], 
   Cos[x] v[x] + u[x] g'[x] + u'[x] + v'[x] - u''[x] - v''[x]}
*)

Answer 2

You can make use of my DifferentialOperator paclet to do this. Install with:

PacletInstall["https://github.com/carlwoll/DifferentialOperator/releases/download/0.1/DifferentialOperator-0.0.2.paclet"]

and load with:

<<DifferentialOperator`

Here is an animation:

Also, notice how I used two different variations of the second order differential operator.

Here is the final answer so that it can be compared with other answers:

{
u[x] f'[x] + v[x] g'[x] + u'[x] + v'[x] - u''[x] - v''[x], 
v[x] f'[x] + u[x] g'[x] + u'[x] + v'[x] - u''[x] - v''[x]
}