Question

using matlab solve numerically dy/dt = sin t, y(0)=0 for 0<=t<=4π

the exact solution is y(t) = 1 - cos t. Compare the exact and numerical solution.

Answer 1

MATLAB CODE

% NUMERICAL METHOD : Runge-Kutta 4th order method
clear all
format short
clc;

f=@(t,y)sin(t); % y'=f(t,y)
y(1)=0; % index has been taken as i instead of 0

h=0.1;
a=0;b=4*pi; % Range of t a<=t<=b
t=a:h:b;
N=(b-a)/h;

for n=1:(length(t)-1)
k1 = h*f(t(n),y(n));
k2 = h*f(t(n)+0.5*h,y(n)+0.5*k1);
k3 = h*f((t(n)+0.5*h),(y(n)+0.5*k2));
k4 = h*f((t(n)+h),(y(n)+k3));
  
y(n+1) = y(n) + (1/6)*(k1+2*k2+2*k3+k4) ;
end
T=t;
y'; % displays all y values
%%% Exact Solution
k=1;
for t = 0:0.1:4*pi;
y_e=1-cos(t);
yext(k)=y_e;
k=k+1;
end
yext';
Comparison_t_y_ext=[T' y' yext']

Comparison_t_y_ext =

0 0 0
0.1000 0.0050 0.0050
0.2000 0.0199 0.0199
0.3000 0.0447 0.0447
0.4000 0.0789 0.0789
0.5000 0.1224 0.1224
0.6000 0.1747 0.1747
0.7000 0.2352 0.2352
0.8000 0.3033 0.3033
0.9000 0.3784 0.3784
1.0000 0.4597 0.4597
1.1000 0.5464 0.5464
1.2000 0.6376 0.6376
1.3000 0.7325 0.7325
1.4000 0.8300 0.8300
1.5000 0.9293 0.9293
1.6000 1.0292 1.0292
1.7000 1.1288 1.1288
1.8000 1.2272 1.2272
1.9000 1.3233 1.3233
2.0000 1.4161 1.4161
2.1000 1.5048 1.5048
2.2000 1.5885 1.5885
2.3000 1.6663 1.6663
2.4000 1.7374 1.7374
2.5000 1.8011 1.8011
2.6000 1.8569 1.8569
2.7000 1.9041 1.9041
2.8000 1.9422 1.9422
2.9000 1.9710 1.9710
3.0000 1.9900 1.9900
3.1000 1.9991 1.9991
3.2000 1.9983 1.9983
3.3000 1.9875 1.9875
3.4000 1.9668 1.9668
3.5000 1.9365 1.9365
3.6000 1.8968 1.8968
3.7000 1.8481 1.8481
3.8000 1.7910 1.7910
3.9000 1.7259 1.7259
4.0000 1.6536 1.6536
4.1000 1.5748 1.5748
4.2000 1.4903 1.4903
4.3000 1.4008 1.4008
4.4000 1.3073 1.3073
4.5000 1.2108 1.2108
4.6000 1.1122 1.1122
4.7000 1.0124 1.0124
4.8000 0.9125 0.9125
4.9000 0.8135 0.8135
5.0000 0.7163 0.7163
5.1000 0.6220 0.6220
5.2000 0.5315 0.5315
5.3000 0.4456 0.4456
5.4000 0.3653 0.3653
5.5000 0.2913 0.2913
5.6000 0.2244 0.2244
5.7000 0.1653 0.1653
5.8000 0.1145 0.1145
5.9000 0.0725 0.0725
6.0000 0.0398 0.0398
6.1000 0.0167 0.0167
6.2000 0.0035 0.0035
6.3000 0.0001 0.0001
6.4000 0.0068 0.0068
6.5000 0.0234 0.0234
6.6000 0.0498 0.0498
6.7000 0.0856 0.0856
6.8000 0.1306 0.1306
6.9000 0.1843 0.1843
7.0000 0.2461 0.2461
7.1000 0.3155 0.3155
7.2000 0.3916 0.3916
7.3000 0.4739 0.4739
7.4000 0.5615 0.5615
7.5000 0.6534 0.6534
7.6000 0.7487 0.7487
7.7000 0.8466 0.8466
7.8000 0.9460 0.9460
7.9000 1.0460 1.0460
8.0000 1.1455 1.1455
8.1000 1.2435 1.2435
8.2000 1.3392 1.3392
8.3000 1.4314 1.4314
8.4000 1.5193 1.5193
8.5000 1.6020 1.6020
8.6000 1.6787 1.6787
8.7000 1.7486 1.7486
8.8000 1.8111 1.8111
8.9000 1.8654 1.8654
9.0000 1.9111 1.9111
9.1000 1.9477 1.9477
9.2000 1.9748 1.9748
9.3000 1.9922 1.9922
9.4000 1.9997 1.9997
9.5000 1.9972 1.9972
9.6000 1.9847 1.9847
9.7000 1.9624 1.9624
9.8000 1.9304 1.9304
9.9000 1.8892 1.8892
10.0000 1.8391 1.8391
10.1000 1.7806 1.7806
10.2000 1.7143 1.7143
10.3000 1.6408 1.6408
10.4000 1.5610 1.5610
10.5000 1.4755 1.4755
10.6000 1.3853 1.3853
10.7000 1.2913 1.2913
10.8000 1.1943 1.1943
10.9000 1.0954 1.0954
11.0000 0.9956 0.9956
11.1000 0.8958 0.8958
11.2000 0.7970 0.7970
11.3000 0.7003 0.7003
11.4000 0.6065 0.6065
11.5000 0.5167 0.5167
11.6000 0.4317 0.4317
11.7000 0.3524 0.3524
11.8000 0.2796 0.2796
11.9000 0.2139 0.2139
12.0000 0.1561 0.1561
12.1000 0.1068 0.1068
12.2000 0.0664 0.0664
12.3000 0.0353 0.0353
12.4000 0.0138 0.0138
12.5000 0.0022 0.0022