Project 2.5 Improved Euler Implementation

Transcription

1 Project 2.5 Improved Euler Implemetatio Figure i the text lists TI-85 ad BASIC programs implemetig the improved Euler method to approximate the solutio of the iitial value problem dy dx = x+ y, y( 0) = 1 (1) cosidered i Example 2 of Sectio 2.5 i the text. The commets provided i the fial colum should reder these programs itelligible eve if you have little familiarity with the BASIC ad TI programmig laguages. To apply the improved Euler method to a differetial equatio dy/dx = f (x, y), oe eed oly chage the iitial lie of the program, i which the fuctio f is defied. To icrease the umber of steps (ad thereby decrease the step size) oe eed oly chage the value of N specified i the secod lie of the program. We illustrate below the implemetatio of the improved Euler method i systems like Maple, Mathematica, ad MATLAB. To begi this project, you should implemet the improved Euler method o your calculator or i a programmig laguage of your choice. Test your program by applicatio first to the iitial value problem i (1), ad the to some of the problems for Sectio 2.5 i the text. The carry out oe or more of the followig ivestigatios. Famous Numbers, Agai The problems below describe the umbers e , l , ad π as specific values of certai iitial value problem solutios. I each case, apply the improved Euler method with = 10, 20, 40, subitervals (doublig each time). How may subitervals are eeded to obtai twice i successio the correct value of the target umber rouded off to 5 decimal places? 1. The umber e = y() 1 where y( x) is the solutio of the iitial value problem y = y, y( 0) = The umber l 2 = y( 2) where y( x) is the solutio of the iitial value problem y = 1/ x, y( 1) = The umber π = y() 1 where y( x) is the solutio of the iitial value problem 2 y = 4/( 1+ x ), y( 0) = 0. Project

2 Logistic Populatio Ivestigatio Apply your improved Euler program to the iitial value problem dy dt 1 = y08 y5, y( 0) = 1 3 of Example 3 i Sectio 2.5 i the text. I particular, verify (as claimed) that the approximate solutio with step size h = 1 levels off at y rather tha at the limitig value y = 8 of the exact solutio. Perhaps a table of values for 0 x 100 will make this apparet. For you ow logistic populatio to ivestigate, you might cosider the iitial value problem dy dt = 1 ym 0 y 5, y ( 0) = 1 where m ad are (for istace) the largest ad smallest digits i your studet ID umber. Does the improved Euler approximatio with step size h = 1 level off at the "correct" limitig value of the exact solutio? If ot, fid a smaller value of h such that it does. With Periodic Harvestig ad Restockig The differetial equatio dy dt 0 5 si 2π t = ky M y h P models a logistic populatio that is periodically harvested ad restocked with period P ad maximal harvestig/restockig rate h. A umerical approximatio program was used to plot the typical solutio curves for the case k = M = h = P = 1 that are show i Fig i the text. This figure suggests though it does ot suffice to prove the existece of a threshold iitial populatio such that Startig with a iitial populatio above this threshold, the populatio oscillates (perhaps with period P?) about the (uharvested) stable limitig populatio yt () M, while The populatio dies out if it starts with a iitial populatio below this threshold. 46 Chapter 2

3 Use a appropriate plottig utility to ivestigate your ow logistic populatio with periodic harvestig ad restockig (selectig typical values of the parameters k, M, h, ad P). Do the observatios idicated above appear to hold for your populatio? Usig Maple To apply the improved Euler method to the iitial value problem i (1), we first defie the right-had fuctio f (x, y) = x + y i the differetial equatio. f := (x,y) -> x + y; f : = ( x, y) x+ y To approximate the solutio with iitial value y(x 0 ) = y 0 o the iterval [x 0, x f ], we eter first the iitial values x0 := 0: y0 := 1: xf := 1: ad the the desired umber of steps ad the resultig step size h. := 10: h := evalf((xf - x0)/); After we iitialize the values of x ad y, x := x0: y := y0: h := the improved Euler method itself is implemeted by the followig for loop, which carries out the iteratio k1 = f( x, y), k2 = f( x + h, y + hk1), k = 1 21k1+ k26, y = y + hk, x = x + h times i successio to take steps across the iterval from x = x 0 to x = x f. for i from 1 to do k1 := f(x,y): k2 := f(x+h,y+h*k1): k := (k1 + k2)/2: y := y + h*k: x := x + h: prit(x,y); od: # the left-had slope # the right-had slope # the average slope # Euler step to update y # update x # display curret values Project

4 , , , , , , , , , , Note that x is updated after y i order that the computatio k 1 = f (x, y) ca use the lefthad values (with either yet updated). The output cosists of x- ad y-colums of resultig xi- ad yi-values. I particular, we see that the improved Euler method with = 10 steps gives y(1) for the iitial value problem i (1). The exact solutio is y(x) = 2e x x 1, so the actual value at x = 1 is y(1) = 2e Thus our improved Euler approximatio uderestimates the actual value by about 0.24% (as compared with the 7.25% error observed i the Euler approximatio of the Sectio 2.4 project). If oly the fial edpoit result is wated explicitly, the the prit commad ca be removed from the loop ad executed immediately followig it (just as we did with the Euler loop i the Sectio 2.4 project). For a differet iitial value problem, we eed oly eter the appropriate ew fuctio f (x, y) ad the desired iitial ad fial values i the first two commads above, the re-execute the subsequet oes. Usig Mathematica To apply the improved Euler method to the iitial value problem i (1), we first defie the right-had fuctio f (x, y) = x + y i the differetial equatio. f[x_,y_] := x + y To approximate the solutio with iitial value y(x 0 ) = y 0 o the iterval [x 0, x f ], we eter first the iitial values x0 = 0; y0 = 1; xf = 1; ad the the desired umber of steps ad the resultig step size h. = 10; h = (xf - x0)/ // N 48 Chapter 2

5 0.1 After we iitialize the values of x ad y, x = x0; y = y0; the improved Euler method itself is implemeted by the followig Do loop, which carries out the iteratio k1 = f( x, y), k2 = f( x + h, y + hk1), k = 1 21k1+ k26, y = y + hk, x = x + h times i successtio to take steps across the iterval from x = x 0 to x = x f. Do[ k1 = f[x,y]; (* left-had slope *) k2 = f[x+h, y+h*k1]; (* right-had slope *) k = (k1 + k2)/2; (* average slope *) y = y + h*k; (* improved Euler step *) x = x + h; (* update x *) Prit[x," ",y], (* display x ad y *) {i,1,} ] Note that x is updated after y i order that the computatio k 1 = f (x, y) use the lefthad values (with either yet updated). The output cosists of x- ad y-colums of resultig xi- ad yi-values. I particular, we see that the improved Euler method with = 10 steps gives y(1) for the iitial value problem i (1). The exact solutio is y(x) = 2e x x 1, so the actual value at x = 1 is y(1) = 2e Thus our improved Euler approximatio uderestimates the actual value by about 0.24% (as compared with the 7.25% error observed i the Euler approximatio of the Sectio 2.4 project). If oly the fial edpoit result is wated explicitly, the the prit commad ca be removed from the loop ad executed immediately followig it (just as we did with the Project

6 Euler loop i the Sectio 2.4 project). For a differet iitial value problem, we eed oly eter the appropriate ew fuctio f (x, y) ad the desired iitial ad fial values i the first two commads above, the re-execute the subsequet oes. Usig MATLAB To apply the improved Euler method to the iitial value problem i (1), we first defie the right-had fuctio f (x, y) i the differetial equatio. User-defied fuctios i MATLAB are defied i (ASCII) text files. To defie the fuctio f (x, y) = x + y we save the MATLAB fuctio defiitio fuctio yp = f(x,y) yp = x + y; % yp = y' i the text file f.m. To approximate the solutio with iitial value y(x 0 ) = y 0 o the iterval [x 0, x f ], we eter first the iitial values x0 = 0; y0 = 1; xf = 1; ad the the desired umber of steps ad the resultig step size h. = 10; h = (xf - x0)/ h = After we iitialize the values of x ad y, x = x0; y = y0; ad the colum vectors X ad Y of approximate values X = x; Y = y; the improved Euler method itself is implemeted by the followig for loop, which carries out the iteratio k1 = f( x, y), k2 = f( x + h, y + hk1), k = 1 21k1+ k26, y = y + hk, x = x + h times i successio to take steps across the iterval from x = x 0 to x = x f. 50 Chapter 2

7 for i = 1 : k1 = f(x,y); k2 = f(x+h,y+h*k1); k = (k1 + k2)/2; y = y + h*k; x = x + h; X = [X; x]; Y = [Y; y]; ed % for i = 1 to do % left-had slope % right-had slope % average slope % Euler step to update y % update x % adjoi ew x-value % adjoi ew y-value Note that x is updated after y i order that the computatio k = f (x, y) ca use the lefthad values (with either yet updated). As output the loop above produces the resultig colum vectors X ad Y of x- ad y-values that ca be displayed simultaeously usig the commad [X,Y] as = I particular, we see that y(1) for the iitial value problem i (1). If oly this fial edpoit result is wated explicitly, the we ca simply eter [X(+1), Y(+1)] as = The idex +1 (istead of ) is required because the iitial values x 0 ad y 0 are the iitial vector elemets X(1) ad Y(1), respectively. The exact solutio of the iitial value problem i (1) is y(x) = 2e x x 1, so the actual value at x = 1 is y(1) = 2e Thus our improved Euler approximatio uderestimates the actual value by about 0.24% (as compared with the 7.25% error observed i the Euler approximatio of the Sectio 2.4 project). Project

8 For a differet iitial value problem, we eed oly defie the appropriate fuctio f (x, y) i the file f.m, the eter the desired iitial ad fial values i the first commad above ad re-execute the subsequet oes. Automatig the Improved Euler Method The for loop above is a bit log for ready etry i MATLAB's commad mode. The followig fuctio impeuler was defied by simple editig of the fuctio euler1 of Project 2.4. fuctio [X,Y] = impeuler(x,xf,y,) h = (xf - x)/; X = x; Y = y; for i = 1 : k1 = f(x,y); k2 = f(x+h,y+h*k1); k = (k1 + k2)/2; y = y + h*k; x = x + h; X = [X;x]; Y = [Y;y]; ed % step size % iitial x % iitial y % begi loop % left-had slope % right-had slope % average slope % improved Euler step % ew x % update x-colum % update y-colum % ed loop With this fuctio saved i the text file impeuler.m, we eed oly assume also that the fuctio f (x, y) has bee defied ad saved i the file f.m. The fuctio impeuler applies the improved Euler method to take steps from x to x f startig with the iitial value y of the solutio. For istace, with f as previously defied, the commad [X,Y] = impeuler(0,1, 1, 10); is a oe-lier that geerates the table [X,Y] displayed above to approximate the solutio of the iitial value problem y = x + y, y(0) = 1 o the x-iterval [0, 1]. 52 Chapter 2