If I have a sigmoid equation depend on time and I want to input to Runge Kutta Orde 4Th equation. So, the graph show the sigmoid from Runge_kutta . How to call the function?

This is my sigmoid equation depend on time ( j is looping for time)

%input for time
t(1)=0;
dt=0.1; %time interval
t=0:dt:100; %time span
T=zeros(length(t)+1,1); %empty array for t
M1=zeros(length(t)+1,1); %empty array for M1
for j = 1:length(t)-1 
    T(j+1)=T(j)+dt;
    M1(j+1)= M1(j)+1./(1+exp(-T(j))); %this is sigmoid equation
end
figure 
plot (T,M1,'r','Linewidth',3)
xlabel('time')
ylabel('M1')

That equation I want to input into Runge kutte orde 4 and show the sigmoid graph from RK 4

Thank you

Answer 1

[Sam Chak]

Hi @cindyawati cindyawati

I'm unsure if I understand the issue correctly. Do you expect a plot like this?

tspan  = linspace(0, 100, 10001);
M0     = 0;     % initial value
[t, M] = ode45(@odefcn, tspan, M0);

% Solution Plot
plot(t, M, 'linewidth', 1.5), 
grid on, xlabel('t'), ylabel('M(t)')

% System
function Mdot = odefcn(t, M)
    Mdot = 1/(1 + exp(- t));
end

[Sam Chak]

The integration method in your code looks like the forward Euler method and it produce similar result like the RK4.

t(1) = 0;
dt   = 1e-2;                      % time interval
t    = 0:dt:100;                  % time span
T    = zeros(length(t) + 1, 1);   % empty array for t
M1   = zeros(length(t) + 1, 1);   % empty array for M1
for j = 1:length(t)-1 
    T(j+1)  = T(j)  + dt;
    M1(j+1) = M1(j) + dt./(1 + exp(- T(j)));
end

figure 
plot(t, M1(1:end-1), '--'), grid on
xlabel('time')
ylabel('M1')

[cindyawati cindyawati]

hi @Sam Chak Thank you for your response. I want plot sigmoid graph but using runge kutta orde 4, is it possible?

Answer 2

[Sam Chak]

Hi @cindyawati cindyawati

The RK4 algorithm by Cleve Moler, is available in File Exchange.

https://www.mathworks.com/matlabcentral/fileexchange/54611-solving-odes-in-matlab?s_tid=srchtitle

t    = linspace(0, 100, 1001);
M0   = 0;     % initial value
Mout = ode4(@odefcn, 0, 0.1, 100, M0);

% Solution Plot
plot(t, Mout, 'linewidth', 1.5), 
grid on, xlabel('t'), ylabel('M(t)')

% System
function Mdot = odefcn(t, M)
    Mdot = 1/(1 + exp(- t));
end

% Classical Runge-Kutta 4th-order ODE solver
function yout = ode4(F, t0, h, tfinal, y0)
      y = y0;
      yout = y;
      for t = t0 : h : tfinal-h
         s1 = F(t,y);
         s2 = F(t+h/2, y+h*s1/2);
         s3 = F(t+h/2, y+h*s2/2);
         s4 = F(t+h, y+h*s3);
         y = y + h*(s1 + 2*s2 + 2*s3 + s4)/6;
         yout = [yout; y];
      end
end