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MA3AM/MA4AM Asymptotic Methods

创建日期：2024-09-02 15:31:33

所属分类：数学mathematics

标签： MA4AM

Asymptotic Methods

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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 .

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.)

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.

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