Explicit Euler Method to Solve System of ODEs in MATLAB

In this tutorial, I am going to show a simple way to solve system of first order ordinary differential equations (ODE) by using explicit Euler method. Let' say, we have following three first order ODEs.

Here, we have 3 ODEs, 3 dependent variables (x, y, z), and 1 independent variable, t. The initial conditions when time is zero are, x(0) = y(0) = z(0) =1. Our goal is to solve these differential equations with explicit Euler approach and plot the solutions afterwards.

Step 1: Define the Equations

The first step is to define all the differential equations in MATLAB. I did this by using MATLAB function handle, which is shown below.

Step 2: Choose a Numerical Approach 

The next step is to select a numerical method to solve the differential equations. In this example, we will use explicit Euler method. I have created a function to implement the algorithm. The following image shows the application of the explicit Euler method.

Step 3: Call the Function

This is the final step where we need to call the function. In the previous step, we have created a function, which we are going to call in this step to solve the equations.

 

So, that's it. We are done with the process, and we can now visualize the solution. I have also attached the MATLAB codes for this problem at the end.

MATLAB Program:

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] = explicit_euler(f,[0,5],[1;1;1],0.001); % Calling Euler function

plot(x,y);

title('When Time Step is 0.001');

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

xlabel('t')

ylabel('Solutions')

function [x, y] = explicit_euler( f, xRange, y_initial, h )

% This function uses Euler’s explicit method to solve the ODE

% dv/dt=f(t,v); x refers to independent and y refers to dependent variables

% f defines the differential equation of the problem

% xRange = [x1, x2] where the solution is sought on

% y_initial = column vector of initial values for y at x1

% numSteps = number of equally-sized steps to take from x1 to x2

% x = row vector of values of x

% y = matrix whose k-th column is the approximate solution at x(k)

x(1) = xRange(1);

numSteps = ( xRange(2) - xRange(1) ) /h ;

y(:,1) = y_initial(:);

for k = 1 : numSteps

x(k + 1) = x(k) + h; 

y(:,k+1) = y(:,k) + h * f( x(k), y(:,k) );

end

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