# Kernel evaluations of and order approximations of 2nd order Volterra integral equation

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?