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suppose the kernel has a lot of good properties

is there any stable and fast method for solving integral equations?

it is tempting to solve it just by converting it into a matrix equation

but is this naive method stable?

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All depends on the integral equation you want to solve. (Linear or not, singular or not, first or second kind etc.) To make the question meaningful, you write your integral equation). –  Alexandre Eremenko Mar 27 '13 at 13:02
If your equation is Fredholm of the first kind and your equation is ill-posed you may look into "The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind" by Chuck Groetsch. –  Dirk Mar 27 '13 at 15:20
For other types, you may consult Hackbusch's "Integral equations: Theory and numerical treatment". –  Dirk Mar 27 '13 at 15:22

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