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I am looking for numerical packages (ideally python) to solve second kind Volterra integral equations, such as

$$u(t)=g(t)+\int_0^tK(t,s)u(s) ds$$

or Volterra-Fredholm integral equations

$$u(x,t)=g(t,x)+c\int_0^t\int_\Omega K(t,s,x,\xi)u(s,\xi) d\xi ds$$

Are there any callable functions in python to solve such equations? if not, are there any standard algorithm to solve such equations? Thanks.

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You can't go wrong if you follow Numerical Recipes. Chapter 18.2 has the code for the Volterra integral equation of the second kind. Here is the book itself, there may also be downloadable code online.

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    $\begingroup$ One should take care when using Numerical Recipes, particularly if you are considering distributing the resulting code at some point. numerical.recipes/licenses $\endgroup$ – J.J. Green Apr 16 '18 at 6:57
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    $\begingroup$ @J.J.Green. Indeed. With the current ideas of copyright I'll soon be put in jail for making scrambled eggs for breakfast without purchasing a proper license from the cookbook publishers. The code for VE in NR is not bad, but nothing really surprising or far superior to any homemade contraption of the type "just break the eggshells and pour what's inside on a moderately hot pan". $\endgroup$ – fedja May 16 '18 at 10:25

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