How to solve method of lines on one-dimensional heat equation using Euler's method?

I have tried but I couldn't solve it fully. Here i attach my code and the question.

% Parameters
k = 0.5;
% Evaluate BC
N = 3;
u(0)= 0;
u(1:N)= U;
u(N+1)=0;
h = 1/(N+1) ; %step size

%Initial condition
  for i=1:N
    u(i)=sin(pi*i);
  end

% Define du/dt
dudt = zeros (N+1,1);
for i=1:N
    dudt(i)=k/h^2*(u(i-1)-2*u(i)+u(i+1));
end

[Torsten]

u(0)= 0;

Array indices start with 1, not with 0. Thus u(0) will throw an error.

u(1:N)= U;

You did not yet define U.

u(i)=sin(pi*i);

i must be replaced by x(i) if 0=x(1)<x(2)<...<x(N+2)=1 is your grid in x-direction. And using N+2 grid points, your loop limits 1 and N are wrong.

dudt = zeros (N+1,1);

You use N+2 grid points.

for i=1:N

dudt(i)=k/h^2*(u(i-1)-2*u(i)+u(i+1));

end

Loop limits are wrong.

You must advance u in time by dudt, so something like u(n+1,i) = u(n,i) + dt * dudt(i) is missing where the "n" refers to the solution at time dt*n.

Answer 1

[Torsten]

% Parameters
k = 0.5;
% Evaluate BC
dx = 0.01;
x = 0:dx:1;
dt = 0.0001;
t = 0:dt:0.5; 
h = 1/(length(x)-1) ; %step size
u = zeros(length(t),length(x));
%Initial condition
for i=1:length(x)
    u(1,i)=sin(pi*x(i));    
end 
for j = 1:length(t)-1
    for i=2:length(x)-1
        u(j+1,i) = u(j,i) + dt* k/dx^2*(u(j,i-1)-2*u(j,i)+u(j,i+1));
    end    
end
figure(1)
plot(x,[u(1,:);u(500,:);u(1000,:);u(2000,:);u(5000,:)])

figure(2)
surf(x,t,u,'Edgecolor','none')

[Nannthini]

what is length(x) and length(t) refers to ?

[Torsten]

https://de.mathworks.com/help/matlab/ref/length.html

[Nannthini]

u(j+1,i) = u(j,i) + dt* k/dx^2*(u(j,i-1)-2*u(j,i)+u(j,i+1)); at this line, you fixed the dt to 0.1. But I want the code to be able to increase the dt with a step size of 0.1.

[Torsten]

This will cause divergence because of the CFL condition. Try it:

% Parameters
k = 0.5;
% Evaluate BC
dx = 0.01;
x = 0:dx:1;
nx = length(x);
h = 1/(nx-1) ; %step size
dt = 0.1;
nt = 100;

u = zeros(nt,nx);
%Initial condition
for i=1:nx
    u(1,i)=sin(pi*x(i));    
end 
t(1) = 0.0;
for j = 1:nt-1
    for i=2:nx-1
        u(j+1,i) = u(j,i) + dt* k/dx^2*(u(j,i-1)-2*u(j,i)+u(j,i+1));
    end    
    t(j+1) = t(j) + dt;
    dt = dt + 0.1;
end
figure(1)
plot(x,[u(1,:);u(5,:);u(10,:);u(20,:);u(50,:)])

figure(2)
surf(x,t,u,'Edgecolor','none')

[Ankitha]

@Torsten could you please have a look at my question as well https://de.mathworks.com/matlabcentral/answers/1888402-how-to-solve-heat-equation-using-euler-cauchy-method?s_tid=srchtitle ? iit would be really great. Thank you

[Nannthini]

okey Thank you so much it helps me a lot.