I need to change the boundary condition in the code below

% Heat diffusion in one dimensional wire within the explicit FTCS method

clear;

% Parameters to define the heat equation and the range in space and time

D = 1*10^-4; % Heat Conductivity parameter in m2/s

K = 1*10^-4; % Heat Conductivity parameter in s^-1

L = 5.; % Total length in m

T = 43200.; % Final Time

% Parameters needed to solve the equation within the explicit method

Nt = 7200; % Number of time steps

Dt = T/Nt; % Time step

Nx = 51; % Number of space steps in m

Dx = L/(Nx-1); % Space step

b = 1-(2*D*Dt/Dx^2)- K*Dt; % beta parameter in the finite- difference implementation

% Remember that the FTCS method is stable for b=<1

disp(b);

% The initial condition: the initial temperature of the pipe

for i = 1:Nx+1

x(i) = (i-1)*Dx; % we also define vector x, due to the space discretization

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

end

% Boundary conditions: the temperature of the pipe at the boundaries at any time

for k =1:Nt+1

u(1,k) = 1.;

u(Nx+1,k) = 1.;

t(k) = (k-1)*Dt; % we also define vector t, due to the time discretization

end

% Implementation of the explicit FTCS method

for k =1:Nt % Time Loop

for i=2:Nx; % Space Loop

u(i,k+1) =u(i,k) + D*Dt/(Dx)^2*(u(i+1,k)-2*u(i,k)+u(i-1,k)); % Implementation of the FTCS Method

end

end

%Graphical representation of the temperature at different selected times

figure(1)

plot(x,u(:,1),'-b',x,u(:,round(Nt/100)),'--g',x,u(:,round(Nt/10)),':b',x,u(:,Nt),'-.r')

Title('Temperature of the pipe within the explicit FTCS method')

xlabel('X')

ylabel('T')

figure(2)

mesh(t,x,u)

title('temperature of the pipe within the explicit FTCS method')

xlabel('t')

ylabel('x')

Boundary and Initial Conditions: 𝐶|𝑥=0 = 1 mg/L, 𝑑𝐶 𝑑𝑥 |𝑥=𝐿 = 0, 𝐶|𝑡=0 = 1 mg/L

[Torsten]

But I already gave you the solution: C = 1 mg/l for all x and all times t.

No need to simulate anything.

[Naveen Krish]

So the above one is right? I don't need to change anything right?

[Torsten]

If you want to solve

dC/dt = D * d^2C/dx^2
C(0,t) = 1
dc/dx(L,t) = 0
C(x,0) = 1

then the solution is C(x,t) = 1.

No need to simulate anything.

Answer 1

[Kayemba Luwaga]

if true
  % code
% Heat diffusion in one dimensional wire within the explicit FTCS method
clear;
% Parameters to define the heat equation and the range in space and time
D = 1*10^-4;               % Heat Conductivity parameter in m2/s
K = 1*10^-4;               % Heat Conductivity parameter in s^-1
L = 5.;                    % Total length in m
T = 43200.;                   % Final Time
% Parameters needed to solve the equation within the explicit method
Nt = 7200;              % Number of time steps
Dt = T/Nt;              % Time step
Nx = 51;               % Number of space steps in m
Dx = L/(Nx-1);              % Space step
b = 1-(2*D*Dt/Dx^2)- K*Dt; % beta parameter in the finite- difference implementation
% Remember that the FTCS method is stable for b=<1
disp(b);
% The initial condition: the initial temperature of the pipe
for i = 1:Nx+1
      x(i) = (i-1)*Dx; % we also define vector x, due to the space discretization
      u(i,1) = sin (pi*x(i));
end
% Boundary conditions: the temperature of the pipe at the boundaries at any time
for k =1:Nt+1
    u(1,k) = 1.;  %% Edit: Lower BC at Temp = 1
    u(Nx+1,k) = 100.;   %% Edit: Upper BC at Temp = 100
    t(k) = (k-1)*Dt; % we also define vector t, due to the time discretization
end
% Implementation of the explicit FTCS method
for k =1:Nt             % Time Loop
  for i=2:Nx;         % Space Loop
      u(i,k+1) =u(i,k) + D*Dt/(Dx)^2*(u(i+1,k)-2*u(i,k)+u(i-1,k)); % Implementation of the FTCS Method
  end
end
%Graphical representation of the temperature at different selected times
figure(1)
plot(x,u(:,1),'-b',x,u(:,round(Nt/100)),'--g',x,u(:,round(Nt/10)),':b',x,u(:,Nt),'-.r')
title('Temperature of the pipe within the explicit FTCS method')
xlabel('X')
ylabel('T')
figure(2)
mesh(t,x,u)
% plot(t,u)
title('temperature of the pipe within the explicit FTCS method')
xlabel('t')
ylabel('x')

end

[Kayemba Luwaga]

Check lines 23 and 24 for editing the temperature boundary conditions/values, Hope it's what you want Cheers!