# Math 128A Spring 2003 Week 12 Solutions

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1 Math 128A Spring 2003 Week 12 Solutions Burden & Faires 5.9: 1b, 2b, 3, 5, 6, 7 Burden & Faires 5.10: 4, 5, 8 Burden & Faires 5.11: 1c, 2, 5, 6, 8 Burden & Faires 5.9. Higher-Order Equations and Systems of Differential Equations 1. Use the Runge-Kutta method for systems to approximate the solution of the following system of first-order differential equations, and compare the result to the actual solution. b. u 1 = 4u 1 2u 2 + cos t + 4 sin t, 0 t 2, u 1 (0) = 0; u 2 = 3u 1 + u 2 3 sin t, 0 t 2, u 2 (0) = 1; h = 0.1; actual solutions u 1 (t) = 2e t 2e 2t + sin t and u 2 (t) = 3e t + 2e 2t. Solution. b. Applying the Runge-Kutta method, we obtain the following results. t i w i, 1 u 1 (t i ) w i, 2 u 2 (t i ) Use the Runge-Kutta for Systems Algorithm to approximate the solution of the following higherorder differential equation, and compare the result to the actual solution. b. t 2 y 2ty + 2y = t 3 ln t, 1 t 2, y(1) = 1, y (1) = 0, with h = 0.1; actual solution y(t) = 7 4 t t3 ln t 3 4 t3. 1

2 Solution. b. Solving this higher-order differential equation is equivalent to solving the following system of differential equations. u 1 = u 2, 1 t 2, u 1 (1) = 1; u 2 = 2 t u 2 2 t 2 u 1 + t ln t, 1 t 2, u 2 (1) = 0; where y(t) = u 1 (t) and y (t) = u 2 (t). following. Applying Runge-Kutta to this system we get the t i w i, 1 y(t i ) w i, Change the Adams Fourth-Order Predictor-Corrector Algorithm to obtain approximate solutions to systems of first-order equations. Solution. The changed algorithm is as follows. Adams Fourth-Order Predictor Corrector for Systems To approximate the solution of the mth-order system of first-order initial-value problems u j = f j (t, u 1, u 2,..., u m ) a t b, with u j (a) = α j for j = 1, 2,..., m at (N + 1) equally spaced numbers in the interval [a, b]: INPUT endpoints a, b; number of equations m; integer N; initial conditions α 1,..., α m. OUTPUT approximations of w j to u j (t) at the (N + 1) values of t. Step 1 Set h = (b a)/n; t 0 = a. Step 2 For j = 1, 2,..., m set w j = α j. Step 3 OUTPUT (t, w 1, w 2,..., w m ). Step 4 For i = 1, 2, 3 do steps Step 5 For j = 1, 2,..., m set k 1, j = hf j (t, w 1, w 2,..., w m ). Step 6 For j = 1, 2,..., m set k 2, j = hf j (t + h 2, w k 1, 1, w k 1, 2,..., w m k 1, m). Step 7 For j = 1, 2,..., m set k 3, j = hf j (t + h 2, w k 2, 1, w k 2, 2,..., w m k 2, m). Step 8 For j = 1, 2,..., m set k 4, j = hf j (t + h, w 1 + k 3, 1, w 2 + k 3, 2,..., w m + k 3, m ). Step 9 For j = 1, 2,..., m set w i, j = w i 1, j + (k 1, j + 2k 2, j + 2k 3, j + k 4, j )/6. Step 10 Set t i = a + ih. Step 11 OUTPUT (t, w 1, w 2,..., w m ). Step 12 For i = 4,..., N do Steps Step 13 Set t i = a + ih. Step 14 For j = 1, 2,..., m set 2

3 Step 15 w i, j = w i 1, j + h[55f j (t i 1, w i 1, 1, w i 1, 2,..., w i 1, m ) 59f j (t i 2, w i 2, 1, w i 2, 2,..., w i 2, m ) + 37f j (t i 3, w i 3, 1, w i 3, 2,..., w i 3, m ) 9f j (t i 4, w i 4, 1, w i 4, 2,..., w i 4, m )]/24 For j = 1, 2,..., m set w i, j = w i 1, j + h[9f j (t i, w i, 1, w i, 2,..., w i, m ) + 19f j (t i 1, w i 1, 1, w i 1, 2,..., w i 1, m ) 5f j (t i 2, w i 2, 1, w i 2, 2,..., w i 2, m ) + f j (t i 3, w i 3, 1, w i 3, 2,..., w i 3, m )]/24 Step 16 OUTPUT (t, w i, 1, w i, 2,..., w i, m ). Step 17 STOP 5. Repeat Exercise 2 using the algorithm developed in Exercise 3. Solution. b. Applying the algorithm, we arrive at the following results: t i w i, 1 w i, 2 y(t i ) Suppose the swinging pendulum described in the lead example of this chapter is 2 ft long and that g = ft/s 2. With h = 0.1 s, compare the angle θ obtained for the following two initial-value problems: a. b. at t = 0, 1, and 2 s. d 2 θ dt 2 + g L sin θ = 0, θ(0) = π 6, θ (0) = 0. d 2 θ dt 2 + g L θ = 0, θ(0) = π 6, θ (0) = 0, Solution. a. The solution for this initial-value problem using the Runge-Kutta method for systems is as follows. 3

4 t i w i, 1 w i, b. The solution for this initial-value problem using the Runge-Kutta method for systems is as follows. t i w i, 1 w i, Thus we see that at 0, 1 and 2 s, we have , , and radians respectively for the first initial-value problem and , , and radians respectively for the second initial-value problem. 7. The study of mathematical models for predicting the population dynamics of competing species has its origin in independent works published in the early part of this century by A. J. Lotka and V. Volterra. Consider the problem of predicting the population of two species, one of which is 4

5 a predator, whose population at time t is x 2 (t), feeding on the other, which is the prey, whose population is x 1 (t). We will assume that the prey always has an adequate food supply and that its birth rate at any time is proportional to the number of prey alive at that time; that is, birth rate (prey) is k 1 x 1 (t). The death rate of the prey depends on both the number of prey and predators alive at that time. For simplicity, we assume death rate (prey) = k 2 x 1 (t)x 2 (t). The birth rate of the predator, on the other hand, depends on its food supply, x 1 (t), as well as on the number of predators available for reproductive purposes. For this reason, we assume that the birth (predator) is k 3 x 1 (t)x 2 (t). The death rate of the predator will be taken as simply proportional to the number of predators alive at the time; that is, death rate (predator) = k 4 x 2 (t). Since x 1(t) and x 2(t) represent the change in the prey and predator populations, respectively, with respect to time, the problem is expressed by the system of nonlinear differential equations x 1(t) = k 1 x 1 (t) k 2 x 1 (t)x 2 (t) and x 2(t) = k 3 x 1 (t)x 2 (t) k 4 x 2 (t). Solve this system for 0 t 4, assuming that the initial population of the prey is 1000 and of the predators is 500 and that the constants are k 1 = 3, k 2 = 0.002, k 3 = , and k 4 = 0.5. Sketch a graph of the solutions to this problem, plotting both populations with time, and describe the physical phenomena represented. Is there a stable solution to this population model? If so, for what values x 1 and x 2 is the solution stable? Solution. Running the Runge-Kutta for systems algorithm, we get that at time t = 4, the number of prey is approximately and the number of predators is approximately There are two stable solutions to the population model occurring when x 1 and x 2 are both 0. One occurs when both x 1 and x 2 are 0. The other occurs when both k 1 k 2 x 2 (t) = 0 and k 3 x 1 (t) k 4 = 0 implying that x 2 (t) = k 1 k 2 and x 1 (t) = k 4 k 3. With the current values of k 1, k 2, k 3, and k 4, this gives x 1 (t) = and x 2 (t) = which cannot happen since we cannot have one third of one prey. Burden & Faires Stability 4. Consider the differential equation a. Show that y = f(t, y), a t b, y(a) = α. y (t i ) = 3y(t i) + 4y(t i+1 ) y(t i+2 ) 2h for some ξ, where t i < ξ i < t i+2. b. Part (a) suggests the difference method use this method to solve + h2 3 y (ξ 1 ), w i+2 = 4w i+1 3w i 2hf(t i, w i ), for i = 0, 1,..., N 2. y = 1 y, 0 t 1, y(0) = 0, with h = 0.1. Use the starting values w 0 = 0 and w 1 = y(t 1 ) = 1 e 0.1. c. Repeat part (b) with h = 0.01 and w 1 = 1 e d. Analyze this method for consistency, stability, and convergence. Solution. a. Applying the three point method for approximating the derivative at t i given the values at t i, t i+1 = t i + h, and t i+2 = t i + 2h, we get the following equation: for some ξ i, where t i < ξ i < t i+2. y (t i ) = 3y(t i) + 4y(t i+1 ) y(t i+2 ) 2h + h2 3 y (ξ i ), 5

6 b. Applying this method with h = 0.1 we obtain the following results. t i w i c. Applying this method with h = 0.01 we obtain the following results

7

8 d. From part (a), we get the following: τ i+1 (h) = y(t i+1) 4y(t i ) + 3y(t i 1 ) h + 2f(t i 1, y(t i 1 )) = 2h2 3 y (ξ i 1 ). Thus this method is consistent. The method has characteristic equation given by: P (λ) = λ 2 4λ + 3. This equation has roots 1 and 3, but since 3 > 1, the method is unstable. Thus the method is not convergent. 5. Given the multistep method w i+1 = 3 2 w i + 3w i w i 2 + 3hf(t i, w i ), for i = 2,..., N 1, with starting values w 0, w 1, w 2 : a. Find the local truncation error. b. Comment on consistency, stability, and convergence. Solution. a. Using the polynomial approximation agreeing with y(t) at t i+1, t i, t i 1, and t i 2, we have: y(t) = P (t) + y(4) (ξ(t)) (t t i+1 )(t t i )(t t i 1 )(t t i 2 ). 24 Plugging P (t) into τ i+1 (h) = y(t i+1) y(t i) 3y(t i 1 ) y(t i 2) h we get 0 and plugging in the error term we get: 1 4 h3 y (4) (ξ(t i )). 3f(t i, y(t i )), b. Part (a) shows that the method is consistent. The characteristic equation is given by P (λ) = λ λ2 3λ = 0. Factoring, we get ( (λ 1) λ λ 1 ) =

9 The two roots other than 1 are given by but 5 ± 33, > 1, which implies that the method is unstable and hence not convergent. 8. Consider the problem y = 0, 0 t 10, y(0) = 0, which has the solution y = 0. If the difference method of Exercise 4 is applied to the problem, then w i+1 = 4w i 3w i 1, for i = 1, 2,..., N 1, w 0 = 0, and w 1 = α 1. Suppose w 1 = α 1 = ɛ, where ɛ is a small rounding error. Compute w i exactly for i = 2, 3, 4, 5, 6 to find how the error ɛ is propagated. Solution. Applying the method, we have w 0 = 0 w 1 = ε w 2 = 4w 1 3w 0 = 4ε w 3 = 13ε w 4 = 40ε w 5 = 121ε w 6 = 364ε Burden & Faires Stiff Differential Equations 1. Solve the following stiff initial-value problem using Euler s method, and compare the result with the actual solution. c. Solution. y = 20y + 20 sin t + cos t, 0 t 2, y(0) = 1, with h = 0.25; actual solution y(t) = sin t + e 20t. c. Applying the Euler s method we get the following: t i w i y(t i )

10 2. Repeat Exercise 1 using the Runge-Kutta fourth-order method. Solution. c. Applying the Runge-Kutta method to the previous differential equation, we get the following: t i w i y(t i ) Solve the following stiff initial-value problem using the Runge-Kutta fourth-order method with (a) h = 0.1 and (b) h = u 1 = 32u u t + 2 3, 0 t 0.5, u 1(0) = 1 3 ; u 2 = 66u 1 133u t 1 3, 0 t 0.5, u 2(0) = 1 3. Compare the results to the actual solution, u 1 (t) = 2 3 t e t 1 3 e 100t and u 2 (t) = 1 3 t 1 3 e t e 100t. Solution. a. When h = 0.1, the Runge-Kutta fourth-order method gives: t i w 1, i u 1 (t i ) w 2, i u 2 (t i ) b. When h = 0.025, the Runge-Kutta fourth-order method gives: 10

11 t i w 1, i u 1 (t i ) w 2, i u 2 (t i ) Show that the fourth-order Runge-Kutta method, k 1 = hf(t i, w i ), k 2 = hf(t i + h/2, w i + k 1 /2), k 3 = hf(t i + h/2, w i + k 2 /2), k 4 = hf(t i + h, w i + k 3 ), w i+1 = w i (k 1 + 2k 2 + 2k 3 + k 4 ), when applied to the differential equation y = λy, can be written in the form w i+1 = (1 + hλ + 12 (hλ) (hλ) ) (hλ)4 w i. 11

12 Solution. We calculate: k 1 = hλw ( i k 2 = hλ w i + k ) 1 2 = (hλ + 12 ) (hλ)2 w i ( k 3 = hλ w i + 1 ) 2 k 2 = (hλ + 12 (hλ) ) (hλ)3 w i k 4 = hλ(w i + k 3 ) = (hλ + (hλ) (hλ) ) (hλ)4 w i w i+1 = w i (k 1 + 2k 2 + 2k 3 + k 4 ) = w i + 1 ( hλ + 2hλ + (hλ) 2 + 2hλ + (hλ) (hλ)3 = + hλ + (hλ) (hλ)3 + 1 ) 4 (hλ)4 w i (1 + hλ + 12 (hλ) (hλ) ) (hλ)4 w i 8. The Backward Euler one-step method is defined by w i+1 = w i + hf(t i+1, w i+1 ), for i = 0,..., N 1. a. Show that Q(hλ) = 1/(1 hλ) for the Backward Euler method. b. Apply the Backward Euler method to the differential equation given in Exercise 1. Solution. a. Applying the Backward Euler method to the differential equation y = λy, we get: w i+1 = w i + hf(t i+1, w i+1 ) = w i + hλw i+1 (1 hλ)w i+1 = w i w i+1 = w i 1 hλ = Q(hλ)w i Q(hλ) = 1 1 hλ b. Applying the Backward Euler method to the differential equation from Exercise 1, we get: Thus the results are: w i+1 = w i + hf(t i+1, w i+1 ) = w i + h( 20w i sin(t i+1 ) + cos(t i+1 )) (1 + 20h)w i+1 = w i + 20h sin(t i+1 ) + h cos(t i+1 ) w i+1 = w i + 20h sin(t i+1 ) + h cos(t i+1 ) h 12

13 t i w i y(t i )

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