The integral equation $u:[a,b]\to \mathbb{R}$

$$u(t) = f(t) + \int\limits_a^t K(t,s)u(s)ds$$

defined on the interval $[a,b]$, with $f:[a,b]\to \mathbb{R}$ and $K: [a,b]^2 \to \mathbb{R}$ some known functions, can be approximated by the Trapezoid rule for integrals:

$$u_m = f_m + \frac{h}2 \sum\limits_{i=1}^m K_{m,i} u_i$$

So if we follow the same procedure for $u_m$ as in the Trapezoidal rule for finding the order of approximation, we will get $O(h^2)$, that is order 2.

Now if $K(t,s) = K(t-s)$ then our numerical formula for $K$ becomes

$$K_{m,i} = K_{m-i}$$

so the number of evaluations of $K$ is $m$ in both cases. Is this correct?


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