Fourth Order Runge Kutta Method by MATLAB to Solve System of Differential Equations

The fourth-order Runge-Kutta method (RK4) is a widely used numerical approach to solve the system of differential equations. In this module, we will solve a system of three ordinary differential equations by implementing the RK4 algorithm in MATLAB.

x' = -5x + 5y

y' = 14x - 2y - zx

z' = -3z + xy

The initial conditions are, x(0) = y(0) = z(0) = 1

The following MATLAB program implements RK4 scheme to solve the above system of three differential equations.

close all;

clc;

format long;

% Defining the three differential equations of the problem

f=@(t,y) [-5*y(1)+5*y(2), 14*y(1)-2*y(2)-y(1)*y(3), -3*y(3)+y(1)*y(2)];

[x,y]=runge_kutta_4(f,[0,5],[1,1,1],0.1); % Calling RK4 function defined below

plot(x,y);

title('When Time Step is 0.1');

legend('x(t)', 'y(t)', 'z(t)', 'Location', 'NorthEast')

xlabel('t')

ylabel('Solutions')

figure;

hold on;

for h=[0.1, 10^-3, 10^-6] % Implementing 3 different time steps

[x,y]=runge_kutta_4(f,[0,5],[1,1,1],h);

plot(x,y(:,1));

end

title('Plot of x(t) for 3 Different Time Steps');

legend('h=0.1', 'h=10^{-3}', 'h=10^{-6}');

xlabel('t')

ylabel('x(t)')

% Fourth order Runge-Kutta method

function [x,y]=runge_kutta_4(f,tspan,y0,h)

x = tspan(1):h:tspan(2); % Calculates upto y(3)

y = zeros(length(x),3);

y(1,:) = y0; % Initial Conditions

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

k_1 = f(x(i),y(i,:));

k_2 = f(x(i)+0.5*h,y(i,:)+0.5*h*k_1);

k_3 = f((x(i)+0.5*h),(y(i,:)+0.5*h*k_2));

k_4 = f((x(i)+h),(y(i,:)+k_3*h));

y(i+1,:) = y(i,:) + (1/6)*(k_1+2*k_2+2*k_3+k_4)*h;

end

end

Program Output: