Solve x′′ = −x − α(x2 − 1)x′, 0 ≤ t ≤ 30, with the initial conditions x(t = 0) = 0.5 and x′(t = 0) = 0, using RK4 method in your computer where α = 1. Plot the solution.

Urgent. Please solve it with explanation as soon as possible.

[James Tursa]

What have you done so far? What specific problems are you having with your code?

[KAUSHIK JAS]

I am a beginner of matlab.Not able to write this code. If you can do then answer it.

[James Tursa]

There are many people on this forum that can write the code for this, but we don't do that for homework problems. You must show some effort first, and then we can help you solve your coding problems. To code an RK4 scheme from scratch, I would first suggest you look at the equations here and try to code them up:

https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods

The RK4 equations are spelled out. Just remember that your "y" and "k"s will be 2-element vectors since you have a 2nd order ODE to solve.

Answer 1

[KAUSHIK JAS]

I solve it correctly by my own. It is in below.

%RK2 of two variables

h=0.5;

t=0:h:30;

x=zeros(1,length(t));

z=zeros(1,length(t));

alpha=1.0;

x(1)=0.5;

z(1)=0;

f=@(p,q,r) (r);

g=@(p,q,r) (-q-alpha*(q^2-1)*r);

for i=1:(length(t)-1)

k11=h*f(t(i),x(i),z(i));

k12=h*g(t(i),x(i),z(i));

k21=h*f(t(i)+0.5*h,x(i)+0.5*k11,z(i)+0.5*k12);

k22=h*g(t(i)+0.5*h,x(i)+0.5*k11,z(i)+0.5*k12);

k31=h*f(t(i)+0.5*h,x(i)+0.5*k21,z(i)+0.5*k22);

k32=h*g(t(i)+0.5*h,x(i)+0.5*k21,z(i)+0.5*k22);

k41=h*f(t(i)+h,x(i)+k31,z(i)+k32);

k42=h*g(t(i)+h,x(i)+k31,z(i)+k32);

x(i+1)=x(i)+(1/6)*(k11+2*k21+2*k31+k41);

z(i+1)=z(i)+(1/6)*(k12+2*k22+2*k32+k42);

end

subplot(2,1,1);

plot(t,x);

subplot(2,1,2);

plot(t,z);