Example of the Euler Method in 2D

for initial conditions , , and .

% Set up initial conditions and parameters

x0=2;

y0=0;

gamma=0.1;

% Set up time stepping

h = 1/100;

tfinal=20;

nsteps=tfinal/h;

t=linspace(0,tfinal,nsteps+1);

% Initialize x & y

x=zeros(size(t));

y = zeros(size(t));

x(1)=x0; y(1)=y0;

% Loop through and run the Euler method

for k=1:nsteps

x(k+1) = x(k) + h * y(k);

y(k+1) = y(k) + h * (-gamma*y(k)-sin(x(k)));

end

% plot the results

plot(t,x,t,y);

xlabel('t'); legend('x','y')

Plot the solution in the phase plane

plot(x,y)

xlabel('x');ylabel('y');