代做MA3AM/MA4AM Asymptotic Methods Problems 7代写C/C++程序

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 + 23 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

x(t ; μ)  =  cost + μ  − 

+ μ2   sint −   )+ . . . and deduce that the region of non-uniformity is    t = O(1μ .

(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 x  = cosT + μ − 

+ μ2  −   ) + . . . where   t = T(12  + . . .) .

(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

x(t ; μ)  =  cos(

where  ξ = μt ,  η= t(1+ O(μ )),  and   C

[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 ) -  (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.