How to replace derivatives of a function with symbolic symbols and numbers with a replacement rule? (I.e. U'' maps to u2)

Question

Suppose I have the expression:

expr = U''[x] + U'[x]

I would like to be able to write an abstract replacement rule so that I can transform the above expression to:

u2 + u1

I was hoping that the replacement rule:

U''[x] + U'[x] /. Derivative[n_][U][x]->StringJoin["u",ToString[n]]

would work. However, this disappointingly uses a literal n, and not a variable n:

U''[x] + U'[x] /. Derivative[n_][U][x]->StringJoin["u",ToString[n]] (* 2 un *)

I suspect I'm close with my solution and I need to use some kind of # and & method but I still do not fully understand those functions.

Any help would be appreciated.

(dexpr = U''[x] + U'[x]) // FullForm and try: dexpr /. Derivative[x_][f_][v_] :> ToLowerCase@ToString[f] <> ToString[x] — Syed
– Syed, Commented May 4, 2022 at 17:53
@Syed ah yes that works. I always forget about the delayed replace. Using :> in my example also makes the replacement works as desired. — akozi
– akozi, Commented May 4, 2022 at 17:56

Bob Hanlon · Accepted Answer · 2022-05-04 18:01:11Z

It is generally easier to deal with indexed variables and to format them for display in any desired manner.

expr = U''[x] + U'[x] + U[x]; Format[u[n_]] := Row[{u, n}] expr2 = expr /. {Derivative[n_][U][_] :> u[n], U[_] :> u[0]}

Or,

Format[u[n_]] := Subscript[u, n] expr2

Reversing the replacements

expr2 /. u[n_] :> Derivative[n][U][x]

Whoops, sorry I have no idea why I did not accept this nearly a year ago, but this was the perfect solution to my problem. — akozi
– akozi, Commented Sep 13, 2023 at 14:03

1 Answer 1