With Charles Matthews comments in perspective, these are some notes I made sometime ago on this topic. I dont have the books in front of me so I can't look up the details right now.
1) In Zabreyko's book Integral equations (902860393X), there is the method based on Green's functions in Appendix A.
2) Bellman in Perturbation techniques Sec 10 points out that the other way (ODE to integral equation) is actually better
Conversion of differential eqn to integral equation
is one of the powerful devices in
approximation theory. Its potency is
due to the fact that integration is a
smoothing op, while differentiation
accentuates small variations. If u(t)
and v(t) are close together, then
∫u(s)ds and ∫v(s)ds will be comparable
in value, but du/dt and dv/dt may be
arbitrarily far apart. Consequently,
when carrying out successive
approximations, we prefer integral
operators to differential operators.
On the other hand, in numerical
solutions, we prefer differential
operators to integral operators.
3) You can also look up Handbook of Integral Equations by Polyanin.
Sec 8.4.5, Sec 9.7 and sec 9.3.3 are three situations where the method reduces a specific integral equation to an ODE
No comments:
Post a Comment