Convert higher order to first order system

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kingsley
kingsley on 17 May 2017
Answered: VBBV on 20 Nov 2021
I'm trying to solve the higher order ode by using RK4 method. Here is the code I have so far.
function [y] = rk4_high_ode(a,t0,n,h,f)
%f = @(t,y) 2*y(2)-y(3)+2*y(4);
% Do I need to define y(2) derivative first?
F=@(t,y)[y(2:end);f]; % Convert the higher order to the 1st order system
t(1)=t0;
for i=1:n
% update time
t(i+1)=t(i)+h;
k1=F(t(i) ,y(i) );
k2=F(t(i)+0.5*h,y(i)+0.5*h*k1);
k3=F(t(i)+0.5*h,y(i)+0.5*h*k2);
k2=F(t(i)+h ,y(i)+h*k1 );
y(i+1)=y(i)+h/6*(k1+2*k2+2*k3+k4);
end
end
And this is the test program:
clear
% test y=sin(t)
% y^(4) = 2*y'-y"+2*y^(3)
t0 = 0.1; n = 100; h = 1e-2;
a = [sin(t0) cos(t0) -sin(t0) -cos(t0)]';
f = @(t,y) 2*y(2)-y(3)+2*y(4);
y = rk4_high_ode(a,t0,n,h,f);
ye = sin(t0+n*h);
error2 = abs(y-ye)
There is an error " Undefined function or variable 'y'". Does that mean I need to define y(2) first? or something else.

Answers (2)

Torsten
Torsten on 17 May 2017
You will have to define
f=@(t,y) [y(2) y(3) y(4) 2*y(2)-y(3)+2*y(4)];
Additionally note that "rk4_high_ode" must be modified because at the moment, it is only capable of solving a single ODE.
Best wishes
Torsten.

VBBV
VBBV on 20 Nov 2021
clear
y = @(t) sin(t)
y = function_handle with value:
@(t)sin(t)
y(2)
ans = 0.9093
% y^(4) = 2*y'-y"+2*y^(3)
t0 = 0.1; n = 100; h = 1e-2;
a = [sin(t0) cos(t0) -sin(t0) -cos(t0)]';
f = 2*y(2)-y(3)+2*y(4)
f = 0.1639
[y t] = rk4_high_ode(a,t0,n,h,f)
F = function_handle with value:
@(t,y)f
y = 1×100
0 0.0016 0.0033 0.0049 0.0066 0.0082 0.0098 0.0115 0.0131 0.0147 0.0164 0.0180 0.0197 0.0213 0.0229 0.0246 0.0262 0.0279 0.0295 0.0311 0.0328 0.0344 0.0361 0.0377 0.0393 0.0410 0.0426 0.0442 0.0459 0.0475
t = 1×100
0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100
ye = sin(t0+n*h);
error2 = abs(y-ye)
error2 = 1×100
0.8912 0.8896 0.8879 0.8863 0.8847 0.8830 0.8814 0.8797 0.8781 0.8765 0.8748 0.8732 0.8715 0.8699 0.8683 0.8666 0.8650 0.8633 0.8617 0.8601 0.8584 0.8568 0.8552 0.8535 0.8519 0.8502 0.8486 0.8470 0.8453 0.8437
plot(y,'linewidth',2)
hold on; plot(error2,'ro')
function [y t] = rk4_high_ode(a,t0,n,h,f)
%f = @(t,y) 2*y(2)-y(3)+2*y(4);
% Do I need to define y(2) derivative first?
F=@(t,y) f % Convert the higher order to the 1st order system
t = zeros(1,n);
y = zeros(1,n);
t(1)=t0;
for i=2:n
% update time
t(i-1)=t(i)+h;
k1=F(t(i-1),y(i-1));
k2=F(t(i-1)+0.5*h,y(i-1)+0.5*h*k1);
k3=F(t(i-1)+0.5*h,y(i-1)+0.5*h*k2);
k4=F(t(i-1)+h,y(i-1)+h*k1);
y(i)=y(i-1)+h/6*(k1+2*k2+2*k3+k4);
end
end

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