Runge-Kutta 3

So I'm doing a project for school, where i have to solve the x''-x'-x=sin(t) equation with Runge-Kutta.

intervalmin=0;

intervalmax=1;

numnodes=5;

f=@(t,x) [x(2); x(2)+x(1)+sin(t)];

inival=0;

h=0.1;

t=zeros(1,numnodes);

t(1)=intervalmin;

x=zeros(1,numnodes);

X=zeros(1,numnodes);

x(1)=inival;

X(1)=inival;

for i = 2:numnodes

k1 = f(t(i-1),x(i-1));

k2 = f(t(i-1)+h/2,x(i-1)+(h/2)*k1);

k3 = f(t(i-1)+h,x(i-1)-h*k1+2*h*k2);

x(i)=(x(i-1)+h/6)*(k1+4*k2+k3);

X(i)=(X(i-1)+h/6)*((f(t(i-1),X(i-1)))+4*(f(t(i-1)+h/2,X(i-1)+(h/2)*k1))+(f(t(i-1)+h,X(i-1)-h*k1+2*h*k2)));

t(i) = t(i-1)+h;

end

figure

plot(t,x,'.-b',t,X,'-.r');

The code is this so far, but it has multiple errors like:

Index exceeds the number of array elements (1).

Error in Untitled5>@(t,x)[t(2);t(2)+t(1)+sin(x)]

Error in Untitled5 (line 14) k1=f(t(i-1),x(i-1));

I'm pretty new to matlab, and just can't seem to find a way to make it work.

Answer 1

[darova]

Here is the mistake

SOlution

[Dávid Bak]

Thank you, but after these corrections, I get another error: Unable to perform assignment because the left and right sides have a different number of elements.

Error in masodikprojekt_elsocsoport (line 18)

x(i)=(x(i-1)+h/6)*(k1+4*k2+k3);

[darova]

Should be changed everywhere

[Dávid Bak]

intervalmin=0;

intervalmax=1;

h=0.1;

g=0.01;

numnodes=ceil(((intervalmax-intervalmin)/h)+1);

numnodes2=ceil(((intervalmax-intervalmin)/g)+1);

inival=0;

%0.1

t=zeros(1,numnodes);

x=zeros(2,numnodes);

t(1)=intervalmin;

x(:,1)=inival;

f=@(t,x) [x(2); x(2)+x(1)+sin(t)];

for i = 2:numnodes

t(i) = t(i-1)+h;

k1 = f(t(i-1), x(:,i-1));

k2 = f(t(i-1)+h/2, x(:,i-1) + (h/2)*k1);

k3 = f(t(i-1)+h/2, x(:,i-1) + (h/2)*k2);

x(:,i)=x(:,i-1)+(h/6)*(k1+4*k2+k3);

end

%0.01

o = zeros(1,numnodes2);

z = zeros(2,numnodes2);

o(1) = intervalmin;

z(:,1) = inival;

m=@(o,z) [z(2); z(2)+z(1)+sin(o)];

for i = 2:numnodes

o(i) = o(i-1)+g;

k1 = m(o(i-1), z(:,i-1));

k2 = m(o(i-1)+h/2, z(:,i-1) + (g/2)*k1);

k3 = m(o(i-1)+h/2, z(:,i-1) + (g/2)*k2);

z(:,i)=z(:,i-1)+(g/6)*(k1+4*k2+k3);

end

[TT,XX] = ode45(f,[intmin intmax],x(:,1));

figure

plot(t,x(2,:),'.-',o,z(2,:),'.-',TT,XX(:,2),'.-')

hold on

syms x(t)

dz = diff(x);

ode = diff(x,t,2) == (x(2)+x(1)+sin(t));

cond1 = x(0) == 2;

cond2 = dz(0) == 0;

dsolve(ode,cond1,cond2)

tt = linspace(0,1);

yy = 4 - 2./(tan(tt/2) + 1) - tt;

legend('h=0.1','h=0.01','ode45')

ax=gca;

ax.XAxisLocation='origin';

ax.YAxisLocation='origin';

grid on;

Whit this code, can you tell me if the red curve is supposed to be that small? I can't really work out the math part of this, so that's why i'm wondering.

[darova]

Please use code button

[darova]

Mistake

[Dávid Bak]

Oh, now everything's alright, thank you very much, you helped a lot!