Question

8.2 (2). Consider the linear equation Y'(x) = XY (2) + (1 - 1) cos(x) - (1+) sin(x), quady (0) = 1 The true solution is Y (1) = sin(1) + cos(r). Solve this problem using Euler's method with several values of and h, for 0 <<<10. Comment on the results. (a) X = -1; h=0.5, 0.25, 0.125 (b) X = 1; h = 0.5, 0.25, 0.125 (c) = -5; h=0.5, 0.25, 0.125, 0.0625 (d) = 5; h= 0.0625

Answer 1

1 2 - 3 - 4 - 5- 6 - clear x(1)=0; y(1)=1;lambda=5; h=0.0625; m=10/h; E for i=1:m x(i+1)=x (i) +h; y(i+1)=y(i)+h* (lambda*y(i) + (1-lambda) *cos (x(i)) - (1+lambda) *sin(x(i))); end for i=1:m+1 e(i)=abs (y(i)-cos (x(i))-sin(x(i))); end 7 8 9- 10 - 11 12 - error=max (e)

(a) x = -1, h = 0.5 Y'(x) = -1 Y(x) + (1-(-12) cosx - (1 + (-1)) sinn xo=o. or, y'(x) = -Y(«) + 2 cos x • Y (o)=1 Y=1, 08, F(x,y) = -Y(x) + 2 cosx. Euler method: Ynt - Ynt hf (xn, Yn) For n=0, y = Yo the f (xo, To) = yoth(-Yo +20osto) = 1 +0.5x (-1+2 coso) = 1.5 Now y = 9 (0.5) = 15 x = noth=0+ 0.5=0.5 D=0, Yz = y, thf(x,y) = y oth (-9, +220sx, ) =15+0.5(-1-5+2 Cos (0-5)) = 1.6276 For true solution : YCO) = 1 (using yasinx + cosx) Ylo:s) = sin(0-5) + Cos (0-5) =13570 Y (1:0) = since tcos (I) = 13818 in for h=0.5, a = -1 xozo, yo = y(o)=1 y (o)=1 My = 0.5, y = ylos) =15 y (0.5) = 1.3570 12 = 1:0, G2 = 9 (1.0) = 1.6276 y(1-0) = 1-3818 True solution

% Euler Method clear all clc; format short for k=1:3 lambda=-1 h=1/2^k f=@(X,Y)lambda*Y+(1-lambda)*cos(x)- (1+lambda)*sin(x); x = 0:h:10; Y = zeros(1,length(x)); Y(1)=1; % index has been taken as i instead of o for n=1:(length(x)-1) Y(n+1) = Y(n)+h*f(x(n),Y(n)); end j=1; for X=0:h:10 true_sol=cos(x)+sin(x); True_Sol(i)=true_sol; j=j+1; end x_Y_TrueSol=[x' Y' True_Sol'] % Solution of ODE at t=0 to 3 end OUTPUT lambda = -1 h = 0.5000 X_Y_TrueSol = 0 1.0000 1.0000 0.5000 1.5000 1.3570 1.0000 1.6276 1.3818 1.5000 1.3541 1.0682 2.0000 0.7478 0.4932 2.5000 -0.0423 -0.2027 3.0000 -0.8223 -0.8489 3.5000 -1.4011 -1.2872 4.0000 -1.6370 -1.4104 4.5000 -1.4722 -1.1883 5.0000 -0.9469 -0.6753 5.5000 -0.1898 0.0031 6.0000 0.6138 0.6808 6.5000 1.2671 1.1917 7.0000 1.6101 1.4109

7.5000 1.5590 1.2846 8.0000 1.1261 0.8439 8.5000 0.4176 0.1965 9.0000 -0.3932 -0.4990 9.5000 1.1077 -1.0723 10.0000 -1.5510 -1.3831 lambda = -1 h = 0.2500 x_Y_TrueSol = 0 1.0000 1.0000 0.2500 1.2500 1.2163 0.5000 1.4220 1.3570 0.7500 1.5053 1.4133 1.0000 1.4948 1.3818 1.2500 1.3912 1.2643 1.5000 1.2011 1.0682 1.7500 0.9362 0.8057 2.0000 0.6130 0.4932 2.2500 0.2517 0.1499 2.5000 -0.1253 -0.2027 2.7500 -0.4946 -0.5426 3.0000 -0.8331 -0.8489 3.2500 -1.1198 -1.1023 3.5000 -1.3369 -1.2872 3.7500 -1.4709 -1.3921 4.0000 -1.5135 -1.4104 4.2500 -1.4619 -1.3411 4.5000 -1.3195 -1.1883 4.7500 -1.0950 -0.9617 5.0000 -0.8025 -0.6753 5.2500 -0.4600 -0.3468 5.5000 -0.0890 0.0031 5.7500 0.2876 0.3529 6.0000 0.6463 0.6808 6.2500 0.9648 0.9663 6.5000 1.2233 1.1917 6.7500 1.4058 1.3431 7.0000 1.5008 1.4109 7.2500 1.5026 1.3910 7.5000 1.4109 1.2846 7.7500 1.2315 1.0984 8.0000 0.9755 0.8439 8.2500 0.6589 0.5369 8.5000 0.3013 0.1965

8.7500 -0.0750 -0.1561 9.0000 -0.4467 -0.4990 9.2500 -0.7906 -0.8109 9.5000 -1.0853 -1.0723 9.7500 -1.3126 -1.2671 10.0000 -1.4582 -1.3831 b) % Euler Method clear all clc; format short for k=1:3 lambda=1 h=1/24k f=@(X,Y)lambda*Y+(1-lambda)*cos(x)- (1+lambda)*sin(x); x = 0:h:10; Y = zeros(1,length(x)); Y(1)=1; % index has been taken as i instead of o for n=1:(length(x)-1) Y(n+1) = Y(n)+h*f(x(n), y(n)); end j=1; for X=0:h:10 true_sol=cos(x)+sin(x); True_Sol(i)=true_sol; j=j+1; end X_Y_TrueSol=[x' Y' True_Sol'] % Solution of ODE at t=0 to 3 end OUTPUT lambda = 1 h = 0.5000 x_Y_TrueSol = 0 1.0000 1.0000 0.5000 1.5000 1.3570 1.0000 1.7706 1.3818 1.5000 1.8144 1.0682 2.0000 1.7241 0.4932 2.5000 1.6768 -0.2027 3.0000 1.9168 -0.8489 3.5000 2.7341 -1.2872 4.0000 4.4519 -1.4104

4.5000 7.4346 -1.1883 5.0000 12.1294 -0.6753 5.5000 19.1531 0.0031 6.0000 29.4352 0.6808 6.5000 44.4322 1.1917 7.0000 66.4332 1.4109 7.5000 98.9928 1.2846 8.0000 147.5512 0.8439 8.5000 220.3374 0.1965 9.0000 329.7076 -0.4990 9.5000 494.1492 -1.0723 10.0000 741.2990 -1.3831 lambda = 1 h = 0.2500 X_Y_TrueSol = 0 1.0000 1.0000 0.2500 1.2500 1.2163 0.5000 1.4388 1.3570 0.7500 1.5588 1.4133 1.0000 1.6077 1.3818 1.2500 1.5888 1.2643 1.5000 1.5116 1.0682 1.7500 1.3907 0.8057 2.0000 1.2464 0.4932 2.2500 1.1033 0.1499 2.5000 0.9901 -0.2027 2.7500 0.9384 -0.5426 3.0000 0.9822 -0.8489 3.2500 1.1572 -1.1023 3.5000 1.5006 -1.2872 3.7500 2.0511 -1.3921 4.0000 2.8497 -1.4104 4.2500 3.9405 -1.3411 4.5000 5.3731 -1.1883 4.7500 7.2052 -0.9617 5.0000 9.5061 -0.6753 5.2500 12.3621 -0.3468 5.5000 15.8821 0.0031 5.7500 20.2054 0.3529 6.0000 25.5109 0.6808 6.2500 32.0283 0.9663 6.5000 40.0520 1.1917 6.7500 49.9574 1.3431 7.0000 62.2217 1.4109

7.2500 77.4487 1.3910 7.5000 96.3993 1.2846 7.7500 120.0301 1.0984 8.0000 149.5404 0.8439 8.2500 186.4308 0.5369 8.5000 232.5772 0.1965 8.7500 290.3222 -0.1561 9.0000 362.5904 -0.4990 9.2500 453.0319 -0.8109 9.5000 566.2030 -1.0723 9.7500 707.7913 -1.2671 10.0000 884.8989 -1.3831 lambda = 1 h = 0.1250 X_Y_TrueSol = 0 1.0000 1.0000 0.1250 1.1250 1.1169 0.2500 1.2345 1.2163 0.3750 1.3269 1.2968 0.5000 1.4012 1.3570 0.6250 1.4565 1.3961 0.7500 1.4923 1.4133 0.8750 1.5084 1.4085 1.0000 1.5051 1.3818 1.1250 1.4829 1.3334 1.2500 1.4426 1.2643 1.3750 1.3857 1.1754 1.5000 1.3137 1.0682 1.6250 1.2286 0.9444 1.7500 1.1325 0.8057 1.8750 1.0281 0.6546 2.0000 0.9180 0.4932 2.1250 0.8055 0.3241 2.2500 0.6936 0.1499 2.3750 0.5858 -0.0266 2.5000 0.4856 -0.2027 2.6250 0.3966 -0.3756 2.7500 0.3227 -0.5426 2.8750 0.2677 -0.7012 3.0000 0.2353 -0.8489 3.1250 0.2294 -0.9833 3.2500 0.2539 -1.1023 3.3750 0.3127 -1.2042 3.5000 0.4096 -1.2872 3.6250 0.5485 -1.3502

3.7500 0.7333 -1.3921 3.8750 0.9678 -1.4123 4.0000 1.2562 -1.4104 4.1250 1.6024 -1.3866 4.2500 2.0108 -1.3411 4.3750 2.4859 -1.2746 4.5000 3.0325 -1.1883 4.6250 3.6560 -1.0835 4.7500 4.3620 -0.9617 4.8750 5.1571 -0.8249 5.0000 6.0484 -0.6753 5.1250 7.0442 -0.5151 5.2500 8.1538 -0.3468 5.3750 9.3877 -0.1732 5.5000 10.7583 0.0031 5.6250 12.2795 0.1794 5.7500 13.9673 0.3529 5.8750 15.8403 0.5209 6.0000 17.9196 0.6808 6.1250 20.2294 0.8300 6.2500 22.7974 0.9663 6.3750 25.6554 1.0875 6.5000 28.8394 11917 6.6250 32.3905 1.2773 6.7500 36.3556 1.3431 6.8750 40.7875 1.3878 7.0000 45.7465 1.4109 7.1250 51.3005 1.4120 7.2500 57.5266 1.3910 7.3750 64.5117 1.3483 7.5000 72.3538 1.2846 7.6250 81.1635 1.2009 77500 91.0654 1.0984 7.8750 102.2000 0.9788 8.0000 114.7250 0.8439 8.1250 128.8183 0.6958 8.2500 144.6797 0.5369 8.3750 162.5340 0.3695 8.5000 182.6340 0.1965 88.6250 205.2636 0.0203 88.7500 230.7422 -0.1561 8.8750 259.4288 -0.3301 9.0000 291.7268 -0.4990 9.1250 328.0896 -0.6601 9.2500 369.0270 -0.8109 9.3750 415.1119 -0.9490 9.5000 466.9885 -1.0723 9.6250 525.3808 -1.1789 9.7500 591.1032 -1.2671

9.8750 665.0709 -1.3355 10.0000 748.3136 -1.3831