I am new to integral equations. In this field, people study the Fredholm equation

$$\phi(x) + \int_0^1 K(x, y) \phi(y) dy = f(x). $$

I am a bit surprised to see the first term on the left hand side. In linear algebra, we have the equation

$$ \sum_j A_{ij} x_j = b_i . $$

Here $A$ is the counterpart of $K$, and $b$ the counterpart of $f$. Therefore, the simplest and most natural integral equation for me would be

$$ \int_0^1 K(x, y) \phi(y) dy = f(x). $$

So why did people not start from this one, but the strange one above? This might be a very naive question, but I did not find any discussion in the literature.


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