INTEGRAL EQUATIONS SM-4325
1. Find the solution, for any $\lambda \in \mathbb{R}$, of the integral equation:
$\phi(s) = 1 + \lambda \int_{-1}^{1} (3s + t) \phi(t) dt$.
(1)
2. Solve (1), for $\lambda \neq \pm 1/2$, using Fredholm's iteration scheme.
Hint:
$\Gamma(s, t; \lambda) = \frac{\sum_{p=0}^{\infty} \frac{(-\lambda)^p}{p!} C_p(s, t)}{\sum_{p=0}^{\infty} \frac{(-\lambda)^p}{p!} C_p}$,
where $c_0 = 1$, $C_0(s, t) = K(s, t)$, $C_p = \int_a^b C_{p-1}(s, s) ds$ and $C_p(s, t) = c_p K(s, t) - p \int_a^b K(s, x) C_{p-1}(x, t) dx$.
3. Prove that the integral equation:
$2\phi(s) = 1 + \int_0^1 (\phi^2(s) + \phi(t)) dt$.
has no real solution.
4. Find the integral equation that is equivalent to the ordinary differ-
ential equation:
$\phi''(s) + s\phi'(s) + s\phi(s) = 2$, $\phi(0) = 0$, $\phi'(0) = 1$.
Propose a method for solving the integral equation.
