Suppose there is a partial differential equation (say second order, two independent variables and nonlinear) and I have a method to solve differential equations of this form to very good accuracy numerically. For sake of simplicity, let me refer to this equation as $E$.
Now my question- Is there a way to solve other arbitrary differential equations (I'll call them required equations $R$ from now) by reposing the problem as a cascading set of $E$s?
Here is a relatively more clear version of my question-
Say I have a set of initial conditions well defined. My task is to obtain the output of $R$ with these initial conditions. I find that $R$ is painful to solve even numerically. But, I do have a magically simple way to solve $E$. So what I want to do is to build a set of $E$s, say $E_1$, $E_2$ and $E_3$. Now I use the original initial conditions and obtain a solution using $E_1$. I use the solution of $E_1$ as the initial conditions for $E_2$, and so on. The solution of $E_3$, or in general, $E_n$ must be an approximate solution of $R$ subject to the original initial conditions.
I want to know if this is possible at all. What differential equations can be solved this way for a given structure of $E$?
A relatively more mathematical (but by no means rigorous) representation-
Given an $U(0,x)$ as my initial conditions, equation $R$ is used to obtain $U(y,x)$.
I want to solve it in the following way-
Use $U(0,x)$ as initial conditions for $E_1$ which gives $U(y_1,x)$.
$U(y_1,x)$ is used as initial conditions for $E_2$ to obtain $U(y_2,x)$.
.
.
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$U(y_{n-1},x)$ is used as initial conditions for $E_n$ and the result is $U(y,x)$ which is the same as what would have been obtained by solving $R$ directly using $U(0,x)$ as initial conditions.
Clarification: $E_m$ is different from $E_n$ in its choice of coefficients.
Another clarification: I am assuming I have found a way to solve $E$ which has precisely zero computation cost. I understand this is probably not possible, but I find my question very intriguing even otherwise.