How to specify a 3 element column vector in Euler's Method for ODE

I am writing code that will approximate the solution to an ODE IVP. I want the initual condition to be a 3D column array supplied by the user rather than one number becuase I am simulating a 3D vector , u(t) that changes in time. I am unsure how to make this initial condition 3D vector.

% u'(t) = F(t, u(t))  where u(t) = t^3 + 12t^2- 20t +1
% u(0) = v % note v is a vector
% solve du/dt = t^3 + 12t^2- 20t  + 1 using euler method
% Euler's Method
% Initial conditions and setup
h=input('Enter the step size') % step size
t=0:h:4;%(starting time value 0):h step size
%(the ending value of t3 );  % the range of t
u=zeros(size(t)); % allocate the result y
%v=input('Enter the intial vector of 3 components using brackets')  ??????????
u(1,1,1)=v; % the initial u as 3D.  I GET ERROR AT THIS LINE
n=numel(u); % the number of u values
% The loop to solve the ODE
for i = 1:n-1
    dudt= *t(i).^3 +12*t(i).^2 -20*t(i)+1 ; %the expression for u' in the ODE
    u(i+1) = u(i)+dudt*h ;
    fprintf('="Y"\n\t   %0.01f',u(i));
end
%%fprintf('="U"\n\t   %0.01f',u);
plot(t,u);
xlabel('t')
ylabel('u(t)')
grid on;

Answer 1

[Torsten]

Here is a hint:

https://de.mathworks.com/matlabcentral/answers/1676979-how-do-i-use-euler-s-forward-difference-method-and-the-second-order-runge-kutta-method-to-work-out-a?s_tid=srchtitle

[Chris Horne]

Thanks.

[Chris Horne]

I added the input command and a vector of initial conditons and recieved this error:

% solve u'(t) = F(t,u(t)) where u(t)= -2t^3 + 12t^2- 20t + 9 using euler method

% Euler's Method for a 3D vector

% Initial conditions and setup

h=input('Enter the step size') % step size

t=0:h:4;%(starting time value 0):h step size

%(the ending value of t3 ); % the range of t

%u=zeros(size(t)); % allocate the result u ?????

v=input('Enter the intial vector of 3 components using brackets')

u(1,:)= v; % the initial u value

Unable to perform assignment because the size of the left side is 1-by-14 and the size of the right side is 1-by-3.

Error in Euler2 (line 9)

u(1,:)= v; % the initial u value

[Torsten]

How did you get u as 1x14 ?

If you used the hint, u should be initialized as 14x3 :

u = zeros(size(t,2),3)

[Chris Horne]

I got the vector dimensions to agree. Thanks. I am learning Matlab. Now in the loop I get another error that the index exceeds the number of outputs, u. I am not sure this method will solve more than one u(t) since u is a VECTOR function (3D).

h=input('Enter the step size') % step size

t=0:h:4;%(starting time value 0):h step size

%(the ending value of t3 ); % the range of t

u = zeros(size(t,2),3); % allocate the result u

v=input('Enter the intial vector of 3 components using brackets')

u(1,:)= v; % the initial u value

n=numel(u); % the number of u values

% The loop to solve the ODE

for i = 1:n-1

dudt= -2*t(i).^3 +12*t(i).^2 -20*t(i)+8.5 ; %the expression for u' in the ODE

u(i+1,:) = u(i,:)+dudt*h ;

Index exceeds the number of array elements (9)

Error in Euler2 (next to last line)

dudt= -2*t(i).^3 +12*t(i).^2 -20*t(i)+8.5 ; %the expression for u' in the ODE

[Torsten]

dudt is a scalar. It must be an 1x3 vector because you wanted to solve 3 ODEs simultaneously.

So

neqn = 3;
h = 0.01;
t = (0:h:1).';
nt = size(t,1);
f = @(t,u) [-2*t^3+12*t^2-20*t+8.5,-2*t^3+12*t^2-20*t+8.5,-2*t^3+12*t^2-20*t+8.5];
u = zeros(nt,neqn);
u(1,:) = [1, 3, -4];
for i = 1:nt-1
  u(i+1,:) = u(i,:) + h*f(t(i),u(i,:));
end
plot(t,u(:,1))

[Chris Horne]

Thanks again.

[Chris Horne]

What if I want to plot each of the 3 outputs, u1(t), u2(t) , and u3(t) on the same Matlab figure. plot(t, ??)

[Chris Horne]

Like this to plot 3 vector functions which are solution to the vector ODE?

plot(t,u(:,1),'color','r')% plot the first column vector of function values

hold on;

plot(t,u(:,2),'color','g')% plot the second column vector of function values

hold on;

plot(t,u(:,3),'color','b') % plot the third column vector of function values

hold off

title('Euler Method with Vector Functions')

[Torsten]

Yes.

Or:

plot(t,y,'Color', 'rgb')

[Chris Horne]

TNX which is morse code language for Thanks

Answer 2

[Chris Horne]

Is the term 'forward Euler' the same as 'Euler' in terms of the algorithm? Here is my method for solving 3 equaitons as a vector:

% This code solves u'(t) = F(t,u(t)) where u(t)= t, cos(t), sin(t)

% using the FORWARD EULER METHOD

% Initial conditions and setup

neqn = 3; % set a number of equations variable

h=input('Enter the step size: ') % step size will effect solution size

t=(0:h:4).';%(starting time value 0):h step size

nt = size(t,1); % size of time array

%(the ending value of t ); % the range of t

% define the function vector, F

F = @(t,u)[t,cos(t),sin(t)]; % define the function 'handle', F

% with hard coded vector functions of time

u = zeros(nt,neqn); % initialize the u vector with zeros

v=input('Enter the intial vector values of 3 components using brackets [u1(0),u2(0),u3(0)]: ')

u(1,:)= v; % the initial u value and the first column

%n=numel(u); % the number of u values

% The loop to solve the ODE (Forward Euler Algorithm)

for i = 1:nt-1

u(i+1,:) = u(i,:) + h*F(t(i),u(i,:)); % Euler's formula for a vector function F

end