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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

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