MATH 543: METHODS OF APPLIED MATHEMATICS I

Final Exam: 10 January 2001 Wednesday 09.30-12.30 SAZ:19

QUESTIONS: Solve at least five questions.

(1). [25] Find the large eigenvalues and the corresponding eigenfunctions of the problem (λ >> 1).

\frac{d^{2} u(x)}{dx^{2}} + λ (π + x)^{4} u(x) = 0,

u(0) = u(π) = 0.

(Hint: Use the WKB approximation method)

(2). [25] Verify the following approximations for large λ

i. ∫_{0}^{∞} e^{−λ t} ln(1 + t^{2})dt = \frac{2}{λ^{3}} − \frac{1}{2} \frac{4!}{λ^{5}} + O(\frac{1}{λ^{7}} )
ii.  ∫_{0}^{π} e^{λ sin t} dt = e^{λ}\sqrt{\frac{2π}{λ}} + ⋯

(3). [25] Find uniform asymptotic approximation to the solutions of the following problem on 0 ≤ t ≤ 1 with 0 < ε ≤ 1

ε u′′ + (x^{2} + 1) u′ − x^{3} u = 0,
u(0) = u(1) = 1.

(4). [25] Expand the Dirac delta function (distribution) δ(x − y) interms of the Legendre polynomials.

(5). [25] Find the solution initial value problem (w is a real constant and f(x) is an integrable function)

\frac{d^{2}u(x)}{dx^{2}} + w^{2}u(x) = f(x), x ∈ [0, 1]

u(0) = a, \frac{du}{dx}|_{x=0} = b.

(6). [25] Let P_{n}(x) denote the Legendre polynomials and

\frac{1}{\sqrt{1 − 2tx + t^{2}}} = ∑_{n=0}^{∞} t^{n} P_{n}(x).

Prove the following

i. ∫_{−1}^{1} P_{n}(x) P_{m}(x)dx = \frac{2}{2n + 1} δ_{nm},

ii. P′_{n−1} − x P′_{n} + n P_{n} = 0,

iii. (1 − x^{2}) P′_{n} + nxP_{n} − nP_{n−1} = 0,

iv. Using ii-iii find the second order differential equation satisfied by P_{n}(x). (7). [25] Find the most general solution of the equation

2x u′′ + (1 − 2x) u′ − u = 0.

(Hint: Use the series expansion and the Frobenius method)