Finite explicit method for heat differential equation

I'm get struggles with solving this problem: Using finite difference explicit and implicit finite difference method solve problem with initial condition: u(0,x)=sin(x) and boundary conditions: ,

So, I tried but get struggles and really need advises. Even I'm not sure how to describe this differential equation or choose number of time steps/space steps in Matlab. Here is what I have tried:

L = 1.; % Length of the wire
T =1; % Final time
% Parameters needed to solve the equation within the explicit method
maxk = 50; % Number of time steps
dt = T/maxk;
n = 10; % Number of space steps
dx = L/n;
a = 1; 
b = 2.*a*dt/(dx*dx); 
% Initial temperature of the wire: a sinus.
for i = 1:n+1
x(i) =(i-1)*dx;
u(i,1) =sin(x(i));
end
% Temperature at the boundary
for t=1:maxk+1
u(1,t) = exp(t);
u(n+1,t) = sin(1)*exp(t);
time(t) = (t-1)*dt;
end
% Implementation of the explicit method
for t=1:maxk % Time Loop
for i=2:n; % Space Loop
u(i,t+1) =u(i,t) + b*(u(i-1,t)+u(i+1,t)-2.*u(i,t));
end
end

% Graphical representation of the temperature at different selected times
figure(1)
plot(x,u(:,1),'-',x,u(:,10),'-',x,u(:,45),'-',x,u(:,30),'-',x,u(:,60),'-')
title('Temperature within the explicit method')
xlabel('X')
ylabel('T')
figure(2)
mesh(x,time,u')
title('Temperature within the explicit method')
xlabel('X')
ylabel('Temperature')

[Furqan Ahmad]

Dear Respected sir, i appreciate your work .Dear sir, I'm student of BS(Hons)Mathematics in GC University,Lahore.Sir,I'm last year student and my research topic is "Numerical solution of 1D Wave equation using by Finite Difference approximation" (Explicit and implicit Schemes).Sir,I need your help .Sir can you make me a big favour? Sir ,i want you to make 3 codes for me. 1:Matlab code for Analytic soltuion of 1D Wave equation 2: Matlab codes for Explicit and Implicit methods. Sir, i can send you the details of my topic. Please be kind with me I will be gratefull to you

Answer 1

[Thiago Henrique Gomes Lobato]

You were actually pretty close, the main problem was that in the boundary condition you was exponentiating your index, not your actually time. Besides this you missed the x and t terms in the equation and the time step needed to be smaller for stability. This code produces reasonable results

L = 1.; % Length of the wire
T =1; % Final time
% Parameters needed to solve the equation within the explicit method
maxk = 210; % Number of time steps
dt = T/maxk;
n = 10; % Number of space steps
dx = L/n;
a = 1;
b = a*dt/(dx*dx); % b should be less than 0.5
% Initial temperature of the wire: a sinus.
for i = 1:n+1
    x(i) =(i-1)*dx;
    u(i,1) =sin(x(i));
end
% Temperature at the boundary
for t=1:maxk+1
    time(t) = (t-1)*dt;
    u(1,t) = exp(time(t));
    u(n+1,t) = sin(1)*exp(time(t));
    
end
% Implementation of the explicit method
for t=1:maxk % Time Loop
    for i=2:n; % Space Loop
        u(i,t+1) =u(i,t) + b*(u(i-1,t)+u(i+1,t)-2.*u(i,t)) + dt*(x(i)-time(t));
    end
end
% Graphical representation of the temperature at different selected times
figure(1)
plot(x,u(:,1),'-',x,u(:,10),'-',x,u(:,45),'-',x,u(:,30),'-',x,u(:,60),'-')
title('Temperature within the explicit method')
xlabel('X')
ylabel('T')
figure(2)
mesh(x,time,u')
title('Temperature within the explicit method')
xlabel('X')
ylabel('Temperature')

[Ewona CZ]

Thank you, it's really helpful, but why you using dt* here before (x(i)-time(t)?

for i=2:n; 
        u(i,t+1) =u(i,t) + b*(u(i-1,t)+u(i+1,t)-2.*u(i,t)) + dt*(x(i)-time(t));
        end

[Thiago Henrique Gomes Lobato]

if you discretize the equation you end up with a dt term multiplying the right side. You had this term in your b, but the (x-t) term didn't, therefore to reproduce the equation you wrote it this dt is needed

Answer 2

[Shahidul]

L = 1.; % Length of the wire

T =1; % Final time

% Parameters needed to solve the equation within the explicit method

maxk = 210; % Number of time steps

dt = T/maxk;

n = 10; % Number of space steps

dx = L/n;

a = 1;

b = a*dt/(dx*dx); % b should be less than 0.5

% Initial temperature of the wire: a sinus.

for i = 1:n+1

x(i) =(i-1)*dx;

u(i,1) =sin(x(i));

end

% Temperature at the boundary

for t=1:maxk+1

time(t) = (t-1)*dt;

u(1,t) = exp(time(t));

u(n+1,t) = sin(1)*exp(time(t));

end

% Implementation of the explicit method

for t=1:maxk % Time Loop

for i=2:n; % Space Loop

u(i,t+1) =u(i,t) + b*(u(i-1,t)+u(i+1,t)-2.*u(i,t)) + dt*(x(i)-time(t));

end

end

% Graphical representation of the temperature at different selected times

figure(1)

plot(x,u(:,1),'-',x,u(:,10),'-',x,u(:,45),'-',x,u(:,30),'-',x,u(:,60),'-')

title('Temperature within the explicit method')

xlabel('X')

ylabel('T')

figure(2)

mesh(x,time,u')

title('Temperature within the explicit method')

xlabel('X')

ylabel('Temperature')