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In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem.

I've never seen integral equations outside of functional analysis, but apparently they are useful for ordinary/partial differential equations. If someone familiar with integral equation methods could give some motivation, I would really appreciate it.

Also, are there any good textbooks that discuss integral equation methods for PDE's?

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Already integral equations are used in the proof of the existence and uniqueness theorem for ODE. For a book, DiBenedetto's 'Partial Differential Equations' has a discussion of integral equations (he treats somewhat explicitly the double layer potential method for the Laplacian). – Jose27 Dec 8 '12 at 20:53
Thanks for the recommendation! I also found a nice chapter on integral equations in Partial Differential Equations by David L. Colton. – B0112358 Dec 8 '12 at 22:34
There is also a chapter on this in Handbook of Integral Equations Second Edition Andrei D . Polyanin and Alexander V . Manzhirov Chapman and Hall/CRC 2008 – kelu Mar 24 '14 at 6:56
An excellent place to start is Courant-Hilbert, Volume II. This work is written as the level of Advanced Calculus. The chapter on Potential Theory is particularly well-done and deals with integral equations and methods of solution based on operator expansions. This material is presented in an elegant and classical way. It should be very accessible to someone with your background, and should prove useful in understanding classical applications. – TrialAndError Jan 17 at 17:06

Contractible mapping principles and Banach fixed point theorems have their direct application to the systems of linear equation, partial differential equation and INTEGRAL equation. Their applications are interwoven, since every contraction mapping has a fixed point.

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