ASSIGNED EXERCISES OF MATH544: Integral Equations set 9 April, 2001

THE INTEGRAL EQUATIONS, GREEN’S FUNCTION AND THE NEUMANN SERIES

1. Ronald B Guenther and John W Lee, 2. Hildebrandt See also Ch.4 of the lecture notes.

1. An integral equation of the form

y(x) = f(x) + λ ∫_{a}^{x} g(x, s) y(s)ds

is called a Volterra equation.

a. Show that , if y(x) satisfies the differential equation

y′′ + xy = 1

and the conditions y(0) = y′(0) = 0 , then y satisfies the Volterra equation

y(x) = ∫_{0}^{x} (s − x) s y(s)ds + \frac{x^{2}}{2}

b. Prove that the converse of the preceding statement is also true.

2. Transform the problem

y′′ + xy = 1, y(0) = y(1) = 0

to the integral equation

y(x) = −\frac{1}{2}s(1 − x) + ∫_{0}^{1} G(x, s)s y(s)ds

where G(x, s) = x(1 − s) when x < s and G(x, s) = s(1 − x) when x > s.

3. Consider the integral equation

y(x) = f(x) + λ ∫_{a}^{b} e^{x−s} y(s)ds.

(i) Find the iterated kernels g_{n}(x, s) defined in the lecture (ii) the resolvent kernel γ(x, s) (iii) Find the solution (iv) find the cases (restrictions on λ where there is no solution and (v) the solution is not unique (vi) find the solution when f(x) = x^{2}.

4. a. Consider the integral equation

y(x) = f(x) + λ ∫_{a}^{b} q(x) q(s) y(s)

where q(x) with x ∈ [a, b] is a continuous function. Answer all questions (i)-(v) as in Pr.3. Consider also b. let q(x) = x discuss the cases (i)-(v) and c. let f(x) = sin x the consider part b.

5. Consider the integral equation

y(x) = f(x) + λ ∫_{0}^{1} (1 − k x s)y(s)ds

where f(x) is a continuous function in [0, 1] , k is a constant and λ is the parameter of the integral equation. Follow the cases (i)-(v) in Problem 3 and answer them. Discuss all possible cases.

6. Solve the following equation

y(x) = λ ∫_{0}^{2π} sin(x + s) y(s)ds.

Discuss all possibilities

7. Obtain the most general solution of the integral equation

y(x) = f(x) + λ ∫_{0}^{2π} sin(x + s) y(s)ds.

when f(x) = x and f(x) = 1. Discuss all cases.

8. Prove that the equation

y(x) = f(x) + \frac{1}{π} ∫_{0}^{2π} sin(x + s) y(s) ds

has no solution when f(x) = x, but that it possesses infinitely many solutions when f(x) = 1. Determine all such solutions