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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Discussion on MATLAB Function and Function Handle

In MATLAB, the functions and function handles (@) are used frequently, especially when writing codes in m-files. We can create many functions, but when we call a particular function, and if that function has another function inside the input parameters - the function handle (@) is the way to address and call the second function. For example, we have created a function to find the roots of a polynomial using the Newton-Raphson method and named it as 'newtonraphson'. The function input parameters are: polynomial, initial guess, error tolerance and maximum number of iterations. And the output is the roots of the polynomial.

Now, to define the polynomial, we can create another function and name it as 'polyfunction'. This function only has the polynomial, and we are interested to find its roots. Let's see how we can call the function 'newtonraphson' in MATLAB command prompt.

>> newtonraphson(@ polyfunction, 1.0, 1e-5, 100)

 

Here,

newtonraphson = a function for finding roots of a polynomical

@ = function handle to call another function

1.0 = Initial guess

1e-5 = Error tolerance

100 = Maximum number of iterations 

If you do not put the function handle (@) in front of the 'polyfunction', there will be an error you will see instantly in the command prompt. As you can see, a function handle is used to pass information to another function. There are other uses of the function handle in MATLAB, but the above is most frequently and widely used expression.

The aforementioned example shows the application of the function handle (@) to a known or defined function by the user (e.g. 'polyfunction'). Now, if there is an anonymous equation that we need to define in our coding, then we can also use the function handle to do the job. For example, in our program, we need to define an equation: 3x - 4y + 2z. The x, y, and z are the unknowns. To define it with the function handle, we can write as,

>> L = @ (x, y, z) (3*x - 4*y + 2*z)

Here,

L = is any parameter referring the equation

@ = function handle defining x, y, z with respect to the equation for evaluation

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A Short Introduction to MATLAB for Beginners

In this post, you will find series of basic tutorial videos on MATLAB. As you may know that MATLAB is a very powerful computational tool, which has been widely used in academics as well as industries. You may find these videos useful if you haven't yet used this tool.