# Replacement rules for partial derivatives

I have a system of PDEs, and it contains terms of the form $$A^{(n,m)}(x,t)$$, which is shorthand for $$\frac{\partial^n}{\partial x^n}\frac{\partial^m}{\partial t^m}A(x,t),$$ (ie Derivative[n,m][A][x,t]) and want to replace output of the form $$A^{(n,m)}(x,t),$$ with $$\partial^n_t(-i\partial_t^2-2\partial_t)^mA(x,t).$$

I think I can take care of the $$\partial^n_t$$ part by defining a rule for the derivative, i.e. ruleDerF = Derivative[n_, m_][f][x_, t_] -> D[ f[x, t], {t, n}], and then applying this to output in question.

However, I don't know how to take care of the expansion of the next operator. For example, I can define the operator $$-i\partial_t^2-2\partial_t,$$ by, say L_1[x_,t_]=(-I*D[#,{t,2}]-2*D[#,{t,1}] and then applying L_1[x,t][A[x,t]]. I am not sure how to then apply this $$m$$ times, nor how to set up these replacement rules for this operator.

Any suggestions are appreciated.

## 1 Answer

Is this it?

Derivative[4, 2][a][x, t]
% /. Derivative[n_, m_][f_][x_, t_] :> D[Nest[-I D[#, {t, 2}] - 2 D[#, t] &, f[x, t], m], {t, n}]