Assignment 4

# Assignment 4 - ε = 0(b Find the system of IVPs satisﬁed by the 3 ﬁrst terms(to order ε 2 of the Lindstedt-Poincar´ e perturbation series

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AM351 Spring 08 Due Wednesday, July 30th 2008 Assignment 4 Instructions: write your solutions clearly, explain your steps carefully and refer to the- orems and lemmas when needed. The assignment is due either in class or in the as- signments drop box (4th ﬂoor) by 2pm. 1. ( 8 marks ) Consider the following initial value problem: y 0 = y + εxy, y (0) = 1 . (a) Calculate a regular perturbation series solution to the third order in ε . (b) From the pattern observed in (a), ﬁnd y k ( x ) for all k 1. (c) Show the validity of your result by solving the IVP exactly. 2. ( 10 marks ) Consider the Van der Pol equation for an oscillator with non-linear damping in dimensionless form: y 00 ( t ) + ε ( y 2 ( t ) - 1) y 0 ( t ) + y ( t ) = 0 , y (0) = 2 ,y 0 (0) = 0 . You can check that regular perturbation theory leads to secular terms. However it can be shown that the equation has a periodic solution. Thus you are asked to use the Lindstedt-Poincar´ e technique to approximate this solution: (a) What is the natural frequency of the unperturbed ODE? (Take

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Unformatted text preview: ε = 0) (b) Find the system of IVPs satisﬁed by the 3 ﬁrst terms (to order ε 2 ) of the Lindstedt-Poincar´ e perturbation series solution. (c) Solve those IVPs and choose the appropriate parameters so that no secular term arise from the 3 ﬁrst terms of the Lindstedt-Poincar´ e perturbation series solution. 3. ( 7 marks ) Consider the centrifuge problem : u ”( t )-ωu ( t ) = 0 , u (0) = 0 , u (2) = 20 . (1) a) Solve the centrifuge BVP by using an analytical version of the shooting method. 1 b) Suppose we use Heun’s 2 stage together with the shooting method to solve (1). What is the error between the exact shooting speed s and the approx-imate shooting speed s h . (Do not only apply the formula given in class, rederive it). If we want this error to be of at least 10-4 , what should be the order of Δ t ? (i.e. ﬁnd p such that Δ t = 10 p ). 2...
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## This note was uploaded on 07/28/2009 for the course AMATH 351 taught by Professor Sivabalsivaloganathan during the Spring '08 term at Waterloo.

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Assignment 4 - ε = 0(b Find the system of IVPs satisﬁed by the 3 ﬁrst terms(to order ε 2 of the Lindstedt-Poincar´ e perturbation series

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