Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation
$$\int_a^xf(s)g(s,x)~ds=h(x)$$
where $h$ and $g$ are known. and we want to obtain function $f(x)$.
As I know, It does not have analytic solution except special cases. so numerical solution can be considered. the most basic approach is as follows:
$$\sum_j w_jf(t_j)g(s_i,t_j)=h(s_i), i=1,...,n$$
by discretizing variables. Then $n$ equation solves $n$ values of $w_j$. Consequently, we can obtain $f(t_i)$ of dicreted version.
What I am wondering is that this numerical solution converges to $f(x)$ as $n$ increases to $\infty$??
If so, what condition is needed?
Please let me know releavanet paper or books
Thanks in advances
The Classical Theory of Integral Equations of S. M. Zemyan can be helpful. I guess, boundedness of g(x,s) (on a sufficiently large region around a) is needed. Most of the time, Volterra integral equations are well posed problems, so there will be a unique solution to the problem.