I'm interested in finding numerical approaches to solving functional equations such as f(xy)=f(x)+f(y), where the equations had no derivatives or integrals, and contains arguments involving x and y .
Reason I ask is that when searching the web/literature, I can find information on some methods for functional differential/integral equations (e.g. Laplace transform and method of steps for time-variant problems, etc.). But I have not been able to find numerical methods (if there are any) for the type of functional equations like shown above.
My idea for general approach (which has been around a long time):
Search for smooth solutions by repeatedly differentiating functional equation wrt x or y , solving resulting system to eliminate one variable, obtaining ODE, solving ODE numerically given value of function at single point. This method can be used to solve d'Alembert's equation, for example. Of course, requiring differentiability greatly limits possible numerical solutions that could be searched for.
I'm looking for areas of research and curious what information is out there on this topic. Any guidance is appreciated.
Intuitively, ideal method would involve being able to evaluate f at points which can be iteratively generated, do not form a cycle, and are closely spaced in desired interval on the real line.
