Evaluating a 2nd order ODE using the Runge-Kutta method

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Emir Alp Karslioglu
Emir Alp Karslioglu 2020-12-31
评论： Emir Alp Karslioglu 2021-1-1
Greetings,
I've been working on a 2nd order ODE: y''(t) = -e^(3t)*y'(t) - y(t) + (5-2e^(-3t))*e^(-2t) +1
With initial conditions y(0) = 2 ; y'(0) = -2 where 0<t<1
I need to use a third order Runge Kutta method, which I can code for a 1st order ODE.
However given a 2nd order differential equation, I'm having difficulties implementing the ODE into my Runge Kutta code.
I've tried writing the 2nd order ODE in a linear system of 1st order ODEs but im still stuck.
By the way, i do not want to use ode45 or ode23 commands. Thank you for your feedback!
Here is what I've been working on so far:
function RK3 = func(a,b,n)
a = 0;
b = 1;
n = input('Enter the number of intervals:');
h = (b-a)/n;
y(1) = 2;
y(2) = -2;
for i=1:n
t = (i-1)*h;
k1 = f(t,y(:,i));
k2 = f(t+0.5*h,y(:,i)+0.5*k1*h);
k3 = f(t+h,y(:,i)-k1*h+(2*k2*h));
y(i+1)= y(:,i)+h/6*(k1+4*k2+k3); %approximated value of the ODE using RK3
end
for k=1:n+1
t=a+(k-1)*h;
exact(k)=exp(-2*t)+1; %exact value of the ODE
end
error = max(abs(y-exact));
max(exact)
max(y)
error
end
function fty = f(t,y) %matrix form of the 2nd order ODE
y = [0 1; -1 -exp(-3*t)];
u = [0; 5-2*exp(-3*t)*exp(-2*t)+1];
fty = y + u;
end

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采纳的回答

Alan Stevens
Alan Stevens 2020-12-31
More like this:
a = 0;
b = 1;
%n = input('Enter the number of intervals:');
n = 100;
h = (b-a)/n;
y = [2; -2];
t = 0:h:n*h;
for i=1:n
k1 = f(t(i),y(:,i));
k2 = f(t(i)+0.5*h,y(:,i)+0.5*k1*h);
k3 = f(t(i)+h,y(:,i)-k1*h+2*k2*h);
y(:,i+1)= y(:,i)+h/6*(k1+4*k2+k3); %approximated value of the ODE using RK3
end
exact = exp(-2*t)+1;
disp(max(abs(exact-y(1,:))))
plot(t,y(1,:),'r',t,exact,'b--'),grid
xlabel('t'),ylabel('y')
legend('RK3','Exact')
function fty = f(t,y) %matrix form of the 2nd order ODE
Y = [0 1; -1 -exp(-3*t)]*y;
u = [0; (5-2*exp(-3*t))*exp(-2*t)+1];
fty = Y + u;
end
  1 个评论
Emir Alp Karslioglu
Emir Alp Karslioglu 2021-1-1
This code works like a charm! Thank you so much for your assistance. I need to work more on matlix manipulation in matlab it seems. Much appreciated!

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