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s11 - Ma 449 Numerical Applied Mathematics Model Solutions...

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Ma 449: Numerical Applied Mathematics Model Solutions to Homework Assignment 11 Prof. Wickerhauser Exercise 4 of Section 9.1, p.463. (a) If y(t) = C exp(-2t) + t exp(-2t), then y'(t)= -2 C exp(-2t) + exp(-2t) -2t exp(-2t), while f(t,y)=exp(-2t)-2y= exp(-2t)-2C exp(-2t)-2t exp(-2t). These are evidently equal, so y'=f. (b) f(t,y)=exp(-2t)-2y is differentiable in y at all (t,y), and f_y(t,y)=-2, so by theorem 9.1, L=2 is the smallest Lipschitz constant for f on any rectangle R. Exercise 6 of Section 9.1, p.463. The slope field consists of vectors tangent to the circles x^2+y^2=r^2 for various radii r, and the solutions are quarter-circles of radius sqrt(C). Exercise 7 of Section 9.2, p.473. The Euler's method relation y_{k+1} = y_k + h f(t_k) for this IVP may be reapplied to get y_{k+1} = y_{k-1} + h f(t_{k-1}) + h f(t_k) y_{k+1} = y_{k-2} + h f(t_{k-2}) + h f(t_{k-1}) + h f(t_k) and so on. After k reapplications, the f terms may be grouped to get y_{k+1} = y_0 + h Sum_{j=0 to k} f(t_j) Substituting n for k+1, n-1 for k, and 0 for y_0=y(a), gives the desired formula: y_n = h Sum_{j=0 to n-1} f(t_j) Exercise 6 of Section 9.3, p.480. The Heun's method relation y_{k+1}=y_k +(h/2)[f(t_k)+f(t_{k+1))] for this IVP may be reapplied to get y_{k+1} = y_{k-1} + (h/2)[f(t_{k-1})+f(t_{k))] + (h/2)[f(t_k)+f(t_{k+1))] y_{k+1} = y_{k-2} + (h/2)[f(t_{k-2})+f(t_{k-1))] + (h/2)[f(t_{k-1})+f(t_{k))] + (h/2)[f(t_k)+f(t_{k+1))] and so on. After k reapplications, the f terms may be grouped to get
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