Forward Euler method for Higher order differential equation

Helloo!

I am trying to solve a third order differential equation using forward euler method. The differential equation is : y''' = 2 y'' + 6y ; y(0) =1, y'(0) = 0, y''(0)=1, interval is [0, 5], step size= 0.001. here the derivative is with respect to time (t). I dont know how to proceed further and do the iteration. I tried to change the above differential equation into a first order differential equation but then I dont know how to proceed with the code.

y(1) = y ;

y(2) = y' = y(1)'

y(3) = y'' = y(2)'

y(3)' = 2 y(3) + 6 y(1)

Below is a code which I used to solve first order differential equation using euler method. Please help.

clear
close all
clc
%%
t0 = 0;              %initial value
y0 = 1;          %initial condition(given)
tEnd = 5;            %end of time step
h = 0.001;          %step size
N = (tEnd-t0)/h;     %number of interval
T = t0:h:tEnd;
Y = zeros(1, N+1);
Y(1) = y0;
for i = 1:N
    fi =               %function
    Y(i+ 1) = Y(i) + h * fi;
end

%%
%ploting===================================================================
plot(T,Y,'.','LineWidth',2);
title('Approximate solution Euler Explicit Method')

[Cr0ke]

@Hrishikesh Das Did you found the problem and did you make it?

Answer 1

[James Tursa]

You need to think of Y as your three element column vector as you have defined. So at each step you are dealing with a column vector. E.g.,

Y = zeros(3, N+1); % <-- a series of 3-element column vectors
Y(:,1) = [1;0;1]; % <-- initial value is a 3-element column vector of [ y ; y' ; y'' ]
for i = 1:N
    fi =               %function, which needs to calculate a 3-element derivative column vector
    Y(:,i+1) = Y(:,i) + h * fi;  % <-- each step works with columns of Y

[Hrishikesh Das]

In case of a third order derivative how to write the function? As I have converted it into a first order using other vriables. plz help.

[James Tursa]

Just take your definitions:

y(2) = y' = y(1)'

y(3) = y'' = y(2)'

y(3)' = 2 y(3) + 6 y(1)

and write a 3-element fi vector based on that. The outline is

fi = [ y(1)' ; y(2)' ; y(3)' ]; % <-- you fill this in based on the above three lines

Now you fill in the above three elements based on your definitions in terms of y(1), y(2), and y(3), and that becomes your code. E.g., that first spot for y(1)' using your definitions is simply y(2), so y(2) would be the first element of fi. But remember we are dealing with columns of Y in your actual code, so for you actual code y(2) at time point i is Y(2,i). So you would have

fi = [ Y(2,i) ; y(2)' ; y(3)' ]; % <-- you fill this in based on the above three lines

Do this for the other two elements.

Answer 2

[Ameer Hamza]

You can download the zip file given in this answer: https://www.mathworks.com/matlabcentral/answers/98293-is-there-a-fixed-step-ordinary-differential-equation-ode-solver-in-matlab-8-0-r2012b#answer_107643 and study the code of ode1.m. It is the implementation of the Euler method provided by Mathworks in very early releases of MATLAB. It is no longer included in MATLAB by default, but it is still useful to understand the implementation of the Euler method for higher-order ODEs.

[Hrishikesh Das]

thank you for your answer. I just downloaded the files.

[Ameer Hamza]

Glad to be of help. That code will help you to correct your implementation.

[Hrishikesh Das]

Are you talking about this file? I took a quick look but I am not getting this properly.

[Ameer Hamza]

You can see James' answer. You can follow that hint to implement multiple ODEs using Euler.