How do I create matrix A using initial and boundary conditions?

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SPerera 2024-8-29

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评论： SPerera 2024-8-30

Using below data, how do I create matrix A?

x = 0:0.2:2;
t = 0:30;

Initial condition:

Boundary conditions:

and

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SPerera 2024-8-29

编辑：SPerera 2024-8-29

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@Torsten @Star Strider Actually I'm trying to solve 1-D heat equation using BTCS method in following way. So my problem is, how do I define the initial and boundary conditions?

L = 1-0; % total length 0<x<1
T = 1-0; %total time 0<t<1
alpha = 1; % diffusivity constant
dx = 0.01; % spatial step size
dt = 0.00001; % time step size
lambda = (alpha*dt)/(dx)^2;
N = round(L/dx +1); % space nodes 
M = round(T/dt +1); % time nodes
x = 0:dx:1; % space domain
t = 0:dt:1; % time domain
U = zeros(M,N); 
U(:,1) = 0; % left boundary condition u(0,t)=0
U(:,N) = 0; % right boundary condition u(L,t)=0
U(1,2:N-1) = sin(pi*x(2:N-1)); % initial condition u(x,0)=sin(pi*x),applying without boundary nodes
A = zeros(N-2,N-2); % coeffient matrix
A(1,1) = 1+2*lambda;
A(1,2) = -lambda;
A(N-2,N-3) = -lambda;
A(N-2,N-2) = 1+2*lambda;
for i= 2:N-3
    A(i,i-1) = -lambda;
    A(i,i) = 1+2*lambda;
    A(i,i+1) = -lambda;
end
B = zeros(N-2,1); % known values
for n= 1:M-1 % no zero position of a vector so starting with 1 instead of 0
    B(1) = U(n,2)+lambda*U(n+1,1);
    B(N-2) = U(n,N-1)+lambda*U(n+1,N);
    for i = 2:N-3
        B(i) = U(n,i+1);
    end
    w = A\B; % unknown values
    U(n+1,2:N-1) = w;
end
U;

I used above method, is it correct? Or should I use this way?

U = zeros(N,M); 
U(1,:) = 0; % left boundary condition u(0,t)=0
U(N,:) = 0; % right boundary condition u(L,t)=0
  

And What do you think about creating A & B?

Torsten 2024-8-30

编辑：Torsten 2024-8-30

The easiest way is to make A of size (NxN) and B of size (Nx1) where you define

u_1^(i+1) = u_1^(i) (since u(0,t) remains constant at 0 for all t)

(u_j^(i+1) - u_j^(i))/dt = alpha * (u_(j-1)^(i+1)-2*u_j^(i+1)+u_(j+1)^(i+1)) (2 <= j <= N-1)

u_N^(i+1) = u_N^(i) (since u(1,t) remains constant at 0 for all t)

as N linear equations in the unknowns u_1^(i+1),u_2^(i+1),...,u_N^(i+1).

SPerera 2024-8-30

@Torsten and @Rahul Thank you all

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