20MAC251 - Vibrations And Waves - Science Assignment Help

Assignment Task

A vibrating system is governed by the differential equation where 0 < a < 2 is a constant parameter.
 

1) Determine the first two non-zero terms in the regular perturbation expansion for u as  → 0, and state whether the expansion is uniform in t.
 

2) By introducing the two timescales, T0 = t and T1 =  t obtain the leading-order term in the expansion of u as  → 0, which is uniformly valid for t = O(1/).
 

3) Numerical results.
 

a) (i) Obtain a numerical solution of the differential equation for  = 0.2 and a = 1 using Maple (or any other suitable software) and plot graphs which compare the numerical solution with the regular perturbation expansion and the multiple-scales solution, over the range 0 ≤ t ≤ 25.

(ii) Compare the accuracy and the range of validity of both approximate solutions.

b) (i) Use the multiple-scales solution to plot a phase portrait of u ? against u for  = 0.2 and a = 1 and for 0 ≤ t ≤ 25. 
 

(ii) Comment on the qualitative behaviour of the phase trajectory and justify it using the constructed multiple-scales solution. 
 

Your mathematical work may be handwritten or typed in LATEX and marks will be given for clear presentation and explanations. You must include the graphs but it is not essential to include the Maple (or other used software) codes.

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