A vibrating system is governed by the differential equationd2udt2−(u2dudt+u)+ 4u= 0, u(0) = 0,dudt(0) =v,0< 1,wherev >0is a constant parameter.
1) Determine the first two non-zero terms in the regular perturbation expansion foruas→0, and state whether the expansion is uniform int.[20 marks]
2) By introducing the two timescales,T0=tandT1=tobtain the leading-order term inthe expansion ofuas→0, which is uniformly valid fort=O(1/).[40 marks]
3) Numerical results.
a) (i) Obtain a numerical solution of the differential equation for= 0.5andv= 0.5using 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 therange0≤t≤15.[20 marks]
(ii) Compare the accuracy and the range of validity of both approximate solutions.[5 marks]
b) (i) Use the multiple-scales solution to plot a phase portrait of ̇uagainstufor= 0.5andv= 0.5and for0≤t≤100.[10 marks]
(ii) Comment on the qualitative behaviour of the phase trajectory and justify it using the constructed multiple-scales solution.[5 marks
]Your mathematical work may be handwritten or typed in LATEX and marks will be given forclear presentation. You must include printouts of the graphs but it is not essential to includethe Maple (or other used software) codes.
