Recently, I managed to find a digital copy of "Handbook of Integral Equations" by Andrei D. Polyanin and Alexander V. Manzhirov (1998 edition).
Linear integral equation of the first kind have the form:
$$ \int_a^x K(x,t) y(t) dt = f(x) $$ where $K(x,t)$ and $f(x)$ are known functions and you solve for $y(t)$.
I think $K$ is called the Kernel function, I don't know why though.
The first result presented is:
$$ \int_a^x y(t) dt = f(x)$$
And the solution is $y(x) = f'(x)$. This result makes sense because if you plug it back in, you get $f(x) - f(a) = f(x)$ and with the convention that $f(a) = f'(a) = 0$ in the preliminary remark, it works out.
However, my understanding of how the kernel integral results gets stopped with the second and third results:
At first, I thought 2 was a result of integration by parts, but I didn't managed to make it; while the 3 is still a mystery for me.
I believe it has to do with the hypotheses presented in introduction that I might have missed, here they are:
Please note how remark 2 was used for the result 1.
Finally, when you scroll a bit deeper in the document, you find in chapters 8+ methods to solve those problems.
It starts with solving the Volterra Integral Equations of the First Kind $ \int_a^x K(x,t) y(t) dt = f(x)$, which is exactly my problem. The Kernel is degenerate because it has the form $K(x,t) = \sum_i g_i (x) h_i (t)$.
The book presents a rewriting of the Volterra equations as the following with the introduction of the sequence of function $w_m (x) = \int_a^x h_m (t) y(t) dt$: $$\sum_{m=1}^{n} g_m (x) w_m (x) = f(x)$$
And then the author says:
On differentiating formulas (10) and eliminating y(x) from the resulting equations, we arrive at the following linear differential equations for the functions $w_m = w_m(x)$: $$h_1(x) w_m = h_m (x) w_1$$ $m = 2, ... , n$
I didn't quite understood how this result was reached.
Then,
Any solution of system (11), (12) determines a solution of the original integral equation (9) by each of the expressions $y(x) = \frac{w_m(x)}{h_m(x)}$, $m = 1, ... , n$, which can be obtained by differentiating formula (10)
In particular, this result is given for $n=2$:
Equation (3) is said to be obtained by integration by parts of $g_2 (x) \int_a^x h_2 (t) y(t) dt$, but I don't see which part should be integrated and which one should be differentiated to get the result.
Then, once the rewriting of (3) into equation (5) seems to lead to (6) and (7) but I don't see how either.
My questions are:
- When the Kernel is a sum of terms like in the example 3, why isn't it a span of functions spaces (3 in the case of the example 3)?
- How is equation (3) achieved?
- How are equations (6), (7) and (8) found?
- What are the applications of Kernel integral equations on mathematics and on physics?
- Are the Kernel integral equations taught in university (Master?)? And if so, at which stage of the education?