1. (Exercise Lecture 09) This exercise revisits the pizza baking problem from Lecture 9 (formal notes). The temperature of the pizza, y(t), is assumed to satisfy (3: : 0.0572(400 y) subject to y(0) : 32. (t in minutes) a. Find the exact solution and compute y{60}. Lecture 9 $8.2 Accuracy of Numerical Methods Example : Consider a pizza biking in an oven set to 400 degrees. Suppose that the temperature of the pizza satisfies 17= 0.0572 (400-y) where y is the temperature of the pizza and tis in minutes. Use Euler's method to predict the temperature of the pizza after 20 minutes assuming y(0) = 32. Solution: Let's try 4t= 20. to = to + At = D+ 20=20 400 yI = yo+ flto, yo) At = 32 + 0.0572(400-32) (20 ) = 453. This is impossible because the existence-uniqueness theorem guarantees that the colution satisfying y/01 = 32 cannot cross the equilibrium solution yIt)= 400. 400- equilibrium solution Eder's 300+ solution with ylol = 32 Method Approximation 200 100- 32 The actual solution must be concave down , while Euler's method uses straight lines over each interval. Naturally, Euler's method will lead to an overestimate of yit) for all t. This is one of the shortcomings of Euler's method . 0 Error Estimate for Euler's Method @ The Taylor series provides an expression for y(t At) as an infinite series. yet At ) = yet/ + y'(t ) ( txtAt +x) + zy"(tx/ (tx+At -tx)2+ (higher order terms) = yltx) + y'ltx) At + = y" (t, ) At" + (terms with At3). Notice that y'/th) = fitkyltkl) and y" /tel = [f (tryAN) = ( tyE ) + 86 (tyk) fitmyF ) Hence , y( t +At ) = ylth) + f(tmyItx1) At + = of (tx, YHA)) + SERYE) fltryIxl)|4+ + (terms with At3 ) 2 Euler's method estimates yet+At) using the tangent line. Y(Ex tAt) = YETI = yetk) + flex, YIK) ) .At Notice that the first two terms match up with the Taylor series expression . The error of a single step is thus level = (4(tx+At ) - YRM/ = 1 2 of (ta, YEW) + BE (argyll f(tx,y/x))| At" + ( terms with At3) ~ CAL 3 In order to use Euler's method over an interval actsb one must implement N= ba steps. The total error is thus At Etotal = Elerl s N.CAT = C (b-a) AtThis analysis has a practical application for Euler's Method . In order to reduce the error of approximation by a factor of '2 ( or 10 ) one must reduce At by a factor of 2 ( or 10 ) . 8 8.3 Improved Euler's Method The improved Euler's method attempts to correct Euler's method by using ( txt At , ym + f ( tx / 41 ) ) two slope estimates for each step . 1 Given IVP: dy = fit,y) subject to yltol= yo. (tx , 4k ) @ Start at (tx, YK) and compute the slope. M, = f(t ,YK ) (2 Compute new point using Euler's method: (txtAt, YK+ flex, yr)At) Compute the slope. Mz = f (txtAt, YK + f (try=) At) (3 Use the average slope to predict YK+1. YKH = yo + At f ( tkyk) + f ( txtAt , YK+ f ( tryplat )| = yk + 4t ( mit m2 ) The improved Euler's Method has a tolal emor satisfying Etotal ~ C(b-a ) At, so reducing At by a factor of 2 ( or 10) will reduce the error by a factor of 4 ( or 100 ) . Example: Use the improved Euler's method to predict the pizza temperature after 20 minutes. Solution: Let's use At= 20 again, Notice that ti= to+At = 0+20= 20. The first slope estimate is My = f ( to, yo) = f (0/ 32) = 0. 0572(400-32) - 21.04 leading to the point ( 20 , 32 + 21. 04 ( 20 ) ) = ( 20 , 453 ) . The second slope is given by M2 = f (to+At, yo+ M,At) = f (20, 453)- -3,03 Hence , y1= yo + 4 (mitm2) = 32+ 20 (21,04 -3,03) = 212.0 degrees. Example: Find the exact pizza temperature after 20 minutes. Solution: "or =5-10572 at -Inly-4001 = - 0.0572+ + C ->y(+) = 400 + A-010572t The initial condition implies that 32= 400 + A or A= - 368. Hence, y(t)= 400-368e572t Finally, y(20) = 400 - 368 -0,0572(20) = 283. Thus At =20 is probably still too big. (2)Below are comparisons of Euler's method, improved Euler's Method, and the exact solution. @ At=10 Step Size: 10.00 400 350 300 250 200 150 100 Exact 50 Euler e- Improved 40 50 10 20 30 2 At = 5 Step Size: 5.00 400 350 300 250 200 150 100 Exact 50 Euler -e- Improved 20 40 50 60 10 30 3