代做MA3AM/MA4AM Asymptotic Methods Problems 7代做Statistics统计

Department of Mathematics

MA3AM/MA4AM Asymptotic Methods

Problems 7

1.    Consider the problem

x.. + x + μx2  = 0 ,     0 < μ << 1 ,

x(0; μ) = 1, x.(0; μ) = 0 .

(a)   Show that x.2  = (1− x) (1+ x + 2/3 μ(1+ x + x2 )) .

Hence sketch the trajectory in the phase plane and deduce that the motion is periodic.

(b)  Show that the straightforward expansion for x is

and deduce that the region of non-uniformity is

(c)  Using the method of renormalisation with

t = T(1+ f1μ+ f2μ2 + ...)

show that the  s.e. in (a) can be rendered uniform, and show that a uniform expansion is

where t = T(1+12/5 u2  + . . .) .

(d)  Apply Linstedt’s method to this problem with

x(t ; μ) = X(T ; μ)     , T = t(1+ w1 μ+ w2 μ2 + ...)  to deduce the uniformly valid

expansion in (c).

(You will show that w1  = 0 , w2  = − 512   .)

2.    (FOR STUDENTS TAKING MA4AM. NOT FOR STUDENTS TAKING MA3AM.) Apply the method of multiple scales to Duffing’s equation

x.. + x + μx3 = 0 ,     0 < μ<< 1 , x(0; μ) = 1 , x.(0; μ) = 0

with slow variable/scale  ξ = μt ,  fast variable/scale  η= t(1+ w2 μ2  + ...)   to deduce the uniformly valid expansion

where ξ = μt , η= t(1+ O(μ )),  and C (0) = − 1 32 , φ1 (0) = 0 .

[Note:  Write the solution for x0     in the form. C0 (ξ) cos(η+ φ0 (ξ)) and deduce that

C0 (0) = 1, φ0 (0) = 0 .   Then show that C0 (ξ) 三 1,  φ0 (ξ) = 38 ξ.   Deduce the conditions above on C1,φ1 ,  but do not attempt to find C1 (ξ) , φ1 (ξ)  or w2.]

3.    Consider the problem

x.. + x = μ(1- x2 )x. ,     0 < μ<< 1 .

(a)  Show that the straightforward expansion only gives a uniformly valid solution for the limit cycle solution

x = 2 cos(t + α0 ) + μ(c1 cos(t + α1 ) - 4/1sin[3(t + a0)])+O(u2)

(α0 ,α1 , c1   constants).

Hint: Review Problem Sheet 1: Question 2 & its Solution.

(FOR STUDENTS TAKING MA4AM. NOT FOR STUDENTS TAKING MA3AM.) (b)  Use the method of multiple scales to deduce the solution

where K, φ0    are constants, ξ = μt , η= t (1+ O(μ2 )) .

(Note:  Write the solution for X0    as in Question 2, and deduce that  φ0 (ξ) = constant .) (FOR STUDENTS TAKING MA4AM. NOT FOR STUDENTS TAKING MA3AM.)

(c)   Show that if x(0; μ) = 0 , x.(0; μ) = 6  then K = , φ0  = - .

4.    Consider the problem

x.. + x + x3  = 0

where x is ‘small’ .  By letting the value of x at t = 0  be μ,  where  0 < μ<< 1, and x. = 0   at t = 0,  i.e. x(0) = μ, x.(0) = 0 ,  and expanding: x(t) = μx1 (t) + μ2x2 (t) + μ3x3 (t) + ...  find

(a) the straightforward expansion,  (b) a uniform expansion by Linstedt’smethod,  (c) a uniform. expansion by rendering the expansion in (a) uniform. using renormalisation.