How do i fix my Runge Kutta (4th order) method to solve a 2nd order ODE?

Hi,

I have a code for a Runge Kutta 2th order analysis of a 2nd order ODE but it has not give me the right answer.

% d2a/dt2=(16*a^(1/2))/35 - 18/(35*a) - (6*a^3)/35
%initial condition: da/dt(x=-∞)=0 و a(x=0)=0.705

Here is my code below:

clear all
clc
h=0.2;
t=-2:h:0;
Nt=numel(t);
A=zeros(2,Nt);
A(:,1)=[0.705,0];

f = @(t, A) [A(2);(16*A(1).^(1./2))./35 - 18./(35.*A(1)) - (6.*A(1).^3)./35];

for i=1:Nt-1
        k1 = h.*f(t(i), A(:,i)); 
        k2 = h.*f(t(i) + 0.5.*h, A(:,i) + 0.5.*k1); 
        A(:,i+1) = A(:,i) + (1.0./6.0).*(k1 + 2.*k2  );
end
figure(1)
plot(t, A, '-o');
xlabel('x/hu')
ylabel('h/hu')
grid on

Answer 1

[Mathieu NOE]

hello

attached a zip for fixed time steps ODE solvers - compare with your own implementation

[Mj]

thanks. but i am the beginner in matlab. i dont know how to use them in my matlab code.

[Mathieu NOE]

hey it's time to learn by practising !

look at the matlab demos / ode

http://web.mit.edu/voigtlab/BP205/Notes/BP205_Matlab_slides.pdf

and attachement

[Mj]

thanks. but i dont want to use ode45 function. i want to solve this by rung kutta method.

[Mathieu NOE]

https://fr.mathworks.com/matlabcentral/answers/36222-runge-kutta

https://www.matlabcoding.com/2019/01/runge-kutta-method-order-4-for-solving.html