ASSIGNED EXERCISES OF MATH544: IntegraL Equations set 8 April, 2001

THE INTEGRAL EQUATIONS AND THE GREEN’S FUNC-TION Ronald B Guenther and John W Lee, Partial Differential Equations of Mathematical Physics and Integral Equations See also Ch.4 of the lecture notes.

1. Show that the Green’s function is unique, if it exists. Hint. If both G(x, y) and H(x, y) provides the same u(x) given by

u(x) = ∫_{a}^{b} G(x, y)f(y)dy,  (1)
u(x) = ∫_{a}^{b} H(x, y)f(y)dy,  (2)

then

∫_{a}^{b}[G(x, y) − H(x, y)]f(y)dy = 0

.

2. Find Green’s function of the following problem

−k u′′ + l u = 0, u(0) − u′(0) = 0, u(1) = 0.

3. Discuss the existence and and the construction of the Green’s function of the following problem

−[p(x)u′]′ + q(x)u = f(x), 0 < x < 1,  (3)
u(0) = u(1), u′(0) = u′(1)  (4)

4. Find Green’s function for the initial value problem

Lu = f(x), u(0) = 0, u′(0) = 0.

Show that the solution has the form

u(x) = ∫_{0}^{x} G(x, y)f(y)dy.

5. Let u′′ − u = f(x). Find Green’s function for the boundary conditions u(x) → 0 ax |x| → ∞.