[SciPy-user] Linear Integral Equation second kind

Anne Archibald peridot.faceted@gmail....
Tue Apr 3 16:17:36 CDT 2007

On 03/04/07, Bobo South Africa <wizzard028wise@gmail.com> wrote:
> Hi all,
> I would want to compute the numerical solution of the following integral
> equation
>  ( Volterra integral equation for the second kind) :
>  f(t) = \int_{a}^{ b} f(t-x)g(x)dx + h(t)                      (latex
> symbol)
> where the unknown function is f(t).

As far as I know, scipy does not implement solving these equations.
But it does implement some tools which should allow you to solve them
quite efficiently (at least in terms of your time, possibly not in
terms of CPU time).

I recommend looking at Numerical Recipes: http://www.nrbook.com/b/bookcpdf.php

See if the algorithms they describe can take advantage of the tools in
scipy - in particular, there are both smart adaptive quadrature rules
and splines, if you'd like to work with a spline representation of
your function. Alternatively, if you want to compute your function on
a grid and use numerical integration adapted to the grid, scipy
implements not only trapezoidal and Simpson integration on arbitrary
grids, in scipy.special you will find various families of orthogonal
polynomials along with all the coefficients to do Gaussian integration
in terms of those. Finally, even if you implement the calculation
mostly by hand, be aware that there are a host of linear algebra tools
to help.


> Thank for your help
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