I'm an engineer so please forgive my lack of knowledge in exact mathematical terminology:
For linear differential equations, it is possible to use the principle of superposition or convolution techniques to extend the solutions from simple BCs to more complicated cases. On the other hand, numerical methods sometimes utilize operator splitting techniques to break a problem into simpler problems. My question is about something in between these two cases.
Let's say I have a nonlinear initial value problem in the form of first order ODE, for which I found the analytical solution for constant values of boundary condition. Now, I can use this solution in iterations and assuming a piecewise linear approximation of any BC, obtain a solution for a general case. However, I don't know what this technique is called and what to search for on the web to find material on error estimation, stability, and proof of convergence for such approaches. I would appreciate it if anyone can point me in the right direction, telling me what these methods are based on or even better, suggest references.