I need help with b.ii Im not sure how to calculate the perturbation approximatio

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I need help with b.ii Im not sure how to calculate the perturbation approximation. anyhelp would be greatly appreciated

2. (50 points) In the problems below you are asked to analyze a model for weight gain and loss. This w(t) as a function of time t model is based on a differential equation for an individual's body weight w The differential equation results from the input-output balance law and the following assumptions: (t) denotes the rate of weight gain due to food consumption p(t) denotes the rate of weight loss at due to physical activity (e.g. exercise) m(w) denotes the loss of weight due to body metabolism (which is a function of body weight w). Thus, the rate of change of body weight w is given by f(t) p(t) m(n (t)) (a) Assume f and p are positive constants and that m is proportional to the square of body weight w Denote the constant of proportionality by c 0.) (i) Determine the phase line portrait of the resulting differential equation for w. The portrait will depend on the weight loss parameter p (ii) Draw a bifurcation diagram using the weight loss parameter p as the bifurcation parameter. Include in your bifurcation diagram the stability properties of all equilibria. Locate and identify any bifurcations that occur. (iii) Interpret you results in (i) and (ii) with regard to their implications concerning weight loss as a function of the exercise parameter p. periodically (around 1 with amplitude and period 2T). Assume p is a constant satisfying 0 p 1 and take the constant of proportionality c in (w) as specified in (a), to equal 1 (i) Use computer explorations to convince yourself that the individual's weight w(t) approaches a periodic oscillation as t increases. Then with further explorations, answer the following questions. Hand-in graphs of selected examples to back-up your answers. oes the individual's weight w(t) peak at the same time as the food intake rate f(t peaks or does it peak later (or earlier What significant event happens to the person's periodically fluctuating weight w(t) as the exercise parameter p approaches 1 (t ko(t) ki(t)e of the final periodic state ii) Calculate the first order perturbation approximation wi approached by the solutions w( HINT: Take ko(t) to be a constant (calculate it) and find a periodic coefficient k1 (t) using the perturbation method (iii) Use your approximation in (ii) to justify validate (or invalidate) your answers in (i)

Explanation / Answer

Very often, a mathematical problem cannot be solved exactly or, if the exact solution is available, it exhibits such an intricate dependency in the parameters that it is hard to use as such.

It may be the case, however, that a parameter can be identified, say , such that the solution is available and reasonably simple for = 0. Then, one may wonder how this solution is altered for non-zero but small . Perturbation theory gives a systematic answer to this question.

Perturbation theory for algebraic equations. Consider the quadratic equation x2 1 = x. (1) The two roots of this equation are x1 = /2 + q 1 + 2/4, x2 = /2 q 1 + 2/4. (2)

For small , these roots are well approximated by the first few terms of their Taylor series expansion (see figure 1)1 x1 = 1 + /2 + 2 /8 + O(3 ), x2 = 1 + /2 2 /8 + O(3 ). (3)

Can we obtain (3) without prior knowledge of the exact solutions of (1)? Yes, using regular perturbation theory.

The technique involves four steps.

STEP A. Assume that the solution(s) of (1) can be Taylor expanded in .

Then we have x = X0 + X1 + 2X2 + O(3 ), (4) for X0, X1, X2 to be determined.

STEP B. Substitute (4) into (1) written as x2 1 x = 0, and expand the left hand side of the resulting equation in power series of .

Using x2 = X2 0 + 2X0X1 + 2 (X2 1 + 2X0X2) + O(3 ), x = X0 + 2X1 + O(3 ), (5)

1a() = O(b()) as 0, (“a() is big-oh of b()”) if there exists a positive constant M such that |a()| M|b()| whenever is sufficiently close to 0.

The root x1 as a function of (solid line), compared with the approximations by truncation of the Taylor series at O(2), x1 = 1 + /2 (dotted line), and O(3), x1 = 1 + /2 + 2/8 (dashed line).

Notice that even though the approximations are a priori valid in the range 1 only, the approximation x1 = 1 + /2 + x2/8 is fairly good even up to = 2.

this gives X2 0 1 + (2X0X1 X0) + 2 (X2 1 + 2X0X2 X1) + O(3 ) = 0. (6)

STEP C. Equate to zero the successive terms of the series in the left hand side of (6): O(0) : X2 0 1 = 0, O(1): 2X0X1 X0 = 0, O(2) : X2 1 + 2X0X2 X1 = 0, O(3) : ··· (7)

STEP D. Successively solve the sequence of equations obtained in (7). Since X2 0 1 = 0 has two roots, X0 = ±1, one obtains X0 = 1, X1 = 1/2, X2 = 1/8, X0 = 1, X1 = 1/2, X2 = 1/8. (8)

It can be checked that substituting (8) into (4) one recovers (3). From the previous example it might not be clear what the advantage of regular perturbation theory is, since one can obtain (3) more directly by Taylor expansion of the roots in (2). To see the strength of regular perturbation theory, consider the following equation x2 1 = ex.

However, the Taylor series expansion of these solutions can be obtained by perturbation theory

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