Deriving a Fredholm integral equation of the second kind from ODE

Question

I have been giving the following BVP:

$$\begin{cases} -u''(t) + a(t)u(t) = f(t)\\ u(0) = u(1) = 0\end{cases}$$

and I am asked to turn it into a Fredholm integral equation of the second kind.

Following the answer on this question, I started by writing

$$\phi(t) = u''(t)$$

and then I wrote

$$u'(s) = u'(0) + \int_0^s u''(w) dw = u'(0) + \int_0^s \phi(w)dw$$

which then led to

$$u(t) = u(0) + \int_0^t u'(s)ds = \cdots = \int_0^t (t-w)\phi(w)dw$$

So I was able to rewrite the differential equation as

$$\phi(t) = -f(t) + \int_0^t a(t)(t-w)\phi(w)dw$$

but this doesn't look like a F.I.E. of the second kind because the upper bound of the integral isn't fixed... I am right in assuming that the upper bound of the integral must be fixed? If it must be, how should I do it? I only need a pointer in the right direction as I have never seen this done, but I do believe I can finish the calculations alone.

Thanks for your time.

Answer 1

The problem is that if you want to convert a given boundary value problem to a Fredholm integral equation, then the kernel mostly would be piecewise-defined; this is somewhat the reason why the "Volterra approach" doesn't seem to work. The standard procedure is as follows:

1. Integrate the given differential equation twice with respect to $t$. You are going to get a double integral when you integrate the second time, so you will need to use an identity that converts double integral into single integral over the triangular domain. You will also need to use $u(0)$ here. The end result here should be a Volterra integral equation and there will be a constant floating around.
2. Evaluate the integral equation from Step 1 at $t=1$ and use $u(1)$ to determine the constant; this constant should involve a definite integral from $t=0$ to $t=1$.
3. Together, Step 1 and Step 2 yield an integral equation involving an integral from $0$ to $t$ and a definite integral (coming from the constant). I will let you figure out how to combine these two integrals into a single definite integral.