代写ASYMPTOTIC METHODS (MA3AM)代写Web开发

MA3AM 2023/4 A 800

ASYMPTOTIC METHODS (MA3AM)

April/May 2024

SECTION A

1.   (a) Show that

cot(E) = O(E—1)

as E → 0.

[4 marks]

(b) Show that

as E → 0.

[4 marks]

(c) Show that

as E → 0.

(You may use e —ttm dt = m! form = 0, 1, 2, · · · .)

[12 marks]

2. Show by construction that the roots of

x cot(x) = 1

for large x are given by

where nis a large integer

[16 marks]

and determine the next term in the series.

[4 marks]

(You may use the expansion tan(z) = z + z3 /3 + O(z5 ), and the identity )

3. Show, by integrating by parts three times, that if μ > 0

as μ → 0+ .

[12 marks]

Show further that

I(μ) = μ + o(μ2 ) ,

as μ → 0+ .

[8 marks]

SECTION B

4. Using an asymptotic expansion of the formu = u0  + ∈ u1  + · · · , where u0 ; u1 = O(1) are to be determined, show that two of the roots of the     equation

u4 — u2 + ∈ = 0

as ∈ → 0 are given by

u = 土1 干 ∈ + · · ·  .

[8 marks]

Similarly, determine the leading TWO non-zero terms in the asymptotic expansions for each of the other two roots as ∈ → 0.

[12 marks]

5.   (a) Use integration by parts to show that

as x → ∞ .

[17 marks]

(b) When using integration by parts as in part (a) for the asymptotic expansion of

what issue arises, and what alternative technique could you use instead?

[3 marks]

6. Consider the following initial value problem

Use Linstedt’s method to show that an asymptotic expansion of the

solution of (1) is given by

x(t; μ) = sinT +32/1μ (sin 3T - 7 sinT) + O(μ2 )

where T = t (1 + 8/1μ + O(μ2)).

[Note: sin 3Q = 3 sin Q - 4 sin3 Q.]

[20 marks]

7. Consider the boundary value problem

Show that the one-term outer expansion of the solution of (2) is given by xouter (t) = 2et-1  - 1 + O(∈) .

Use the method of matched asymptotic expansions to show that the corresponding one-term inner expansion is given by

xinner (t; ∈) = (1 - 2e-1)(e-t/ E  - 1) + O(∈) ,

and determine a one-term composite expansion for the solution of (2) that is valid throughout [0, 1].

[20 marks]