Approximation of differential equations

Question

Can someone provide me a good reference about approximation techniques in continous domain (not piecewise nor numerical methods) for differential equations?

Answer 1

There really would be different types of methods for this, particularly dependent on how and what your differential equation is and also what you want to achieve.

I did some time approximations by studying systems of differential equations (also non-linear) by corresponding stochastic methods such as the master equations, Fokker Planck equations (vs. Langevin equations). For reference to both types of approach you just need to start on wikki to get primary information and then forward.

However this is one way. What is often interesting as well is to apply Fourier techniques. This is quite often used by electrical engineers (systems and signals) and in physics. The references there are really massive you need just goolgle the terms. Rather difficult to sieve what is the best technique for your case of equations.

Hope this helps well.

Answer 2

As a reference, check out Advanced Mathematical Methods for Scientists and Engineers by Carl Bender and Steven Orszag.

It handles approximate solutions of both linear and nonlinear differential equations. Among other things, Perturbation Theory is discussed thouroughly. It involves turning a difficult problem (a hard to crack ODE for example) into an infinite amount of easy problems leading to an asymptotic solution.

Also, check out the excellent lecture series on the subject by the above author on youtube.