RK2    clear all; A = [0 1 ; 0.2*(y(1))^2-1 0]; n=32/101; time_period = 0:n:32; initial = [1 0]; [t,y] = ode45(@myode45function, time_period, initial); qstar = zeros(2,length(t)); % Preallocate array qstar(:,1) = initial; % Initial condition gives solution at t=0. for i=1:(length(t)-1) k1 = A*qstar(:,i); % Approx for y gives approx for deriv q1 = qstar(:,i)+k1*(n/2); % Intermediate value k2 = A*q1; % Approx deriv at intermediate value. qstar(:,i+1) = qstar(:,i) + k2*n; % Approx solution at next value of q end plot(t,y(:,1),t,qstar(1,:)); % qstar = first row of qstar legend('Exact','Approximate');

0 5 10 15 20 25 30 35-8000

-6000

-4000

-2000

0

2000

4000ExactApproximate

0 5 10 15 20 25 30 35-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1ExactApproximate

0 5 10 15 20 25 30 35-1.5

-1

-0.5

0

0.5

1

1.5ExactApproximate

0 5 10 15 20 25 30 35-0.5

0

0.5

1ExactApproximate

0 5 10 15 20 25 30 35-1.5

-1

-0.5

0

0.5

1

1.5ExactApproximate

0 5 10 15 20 25 30 35-0.5

0

0.5

1ExactApproximate

Adams-Bashworth clear all clc % Solve y''(t)+py'(t)+w^2(t)=0, y'(0)=0, y(0)=1 q0 = [1; 0]; % Initial Condition w=1; p=0.5; n=32/101; % Time step t = 0:n:32; % t goes from 0 to 32 seconds. A = [0 1 ; -w^2 -p]; % A Matrix c1=0; c2=1; yexact = c1*sin(t)+c2*cos(t); %Exact solution qstar = zeros(2,length(t)); % Preallocate array %Runge-Kutta 2nd Order to approximate i=2 qstar(:,1) = q0; % Initial condition gives solution at t=0. k1 = A*qstar(:,1); % Approx for y gives approx for deriv q1 = qstar(:,1)+k1*(n/2); % Intermediate value k2 = A*q1; % Approx deriv at intermediate value. qstar(:,2) = qstar(:,1) + k2*n; % Approx solution at next value of q %Adam-Bashforth 2nd Order for i=2:(length(t)-1) qstar(:,i+1) = qstar(:,i) + (1.5*qstar(:,i)-0.5*qstar(:,i-1))*n; end plot(t,yexact,t,qstar(1,:)); % qstar = first row of qstar legend('Exact','Approximate');

 

0 5 10 15 20 25 30 35

#1012

0

2

4

6

8

10

12

14

16

18

20ExactApproximate