# Volterra integral equation solver

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Michal on 31 Jan 2023
Commented: John D'Errico on 3 Feb 2023
I am looking for good (fast and accurate, with recent state of art methods) numerical (not symbolic) Volterra Integral Equation (1st kind especially) solver. A few solvers available on FEX are only based on very naive and ineffective methods.
Thanks in advance for any help.
Motivation note:
I would like to numerically solve the following inverse (Linear Dynamic System) problem:
x'(t) = Ax(t) + Bu(t)
y(t) = Cx(t); x(0) = x0
where A,B,C are known constant matrices and y(t) is known (measured) noisy output discrete signal. The goal is estimation of input signal u.
There exist an analytical form of solution y(t):
y(t) = Integral_0^t (C*exp(A*(t-s))*B*u(s) ds = Integral_0^t (K(s,t)*u(s) ds) [*]
where kernel is K(s,t) = C*exp(A*(t-s))*B, so the eq [*] has the form of Volterra intergral equatioon of first kind.
Question: Is there any other possibility how to effectivelly and robustly estimate uknown input signal u(t)?
Torsten on 31 Jan 2023
I don't have experience with integral equations, but aren't there more listed here:
?
John D'Errico on 3 Feb 2023
I'm also not an expert at all on the subject of first kind integral equations, having not touched them for nearly 40 years. But my memory does include the words ill-posed. In turn, that suggests finding an off the shelf solver that will be robust, efficient, etc., will be a difficult thing to hope for. I would contact someone with some expertise in the area.

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