Finite differencing scheme on 2-D plate

%% Initialisation
dt=0.01;
lt=5/dt+1;
dx=2*pi/100;
dy=2*pi/100;
a=0.1;
B=0.1;
L=4*pi;
H=2*pi;
n=round(L/dx) +1;
m=round(H/dy) +1;
%% setting up x vector
x=zeros(n,1);
for i=1:n
    x(i)=(i-1)*dx;
end
%% setting up y vector
y=zeros(m,1);
for j=1:m
    y(j)=(j-1)*dy;
end
%% Initial Condition
for i=1:n
    for j=1:m
        T(i,j,1)=20*cos((i-1)*dx)*sin((j-1)*dy);
    end
end
%%  Realisations - seeting up t vector
t=zeros(lt,1);
for k=1:lt
T(k)=(k-1)*dt;
end
%% Time Loop for velocity distribution
v=zeros(n,m,lt);
for i=1:n
for k=2:lt
    V(i,k)=5*(1-(x(i)-2*pi)^2/(4*pi^2))*cos(pi*t(k))*sin(x(i));
end

end
%% Discretisation
as=((-B*(dt/dy^2)) + V*(dt/2*dy));
aw=(-a*dt/dx^2);
ap=(1+2*a*(dt/dx^2)+2*B*(dt/dy^2));
an=((-B*(dt/dy^2))-V*(dt/2*dy));
ae=(-a*(dt/dx^2));

for k=2:lt
%% T1
for i=2:n-1
    for j=2:m-1
        pointer=(j-1)*n+i;
        A(pointer,pointer-n)=as(i,k);
        A(pointer,pointer-1)=aw;
        A(pointer, pointer)=ap;
        A(pointer,pointer+1)=ae;
        A(pointer,pointer+n)=an(i,k);
    end
end

%% T2
for i=1
    for j=2:m-1
        pointer=(j-1)*n+i;
        A(pointer,pointer)=ap;
        A(pointer,pointer+1)=(aw+ae);
        A(pointer,pointer+n)=an(i,k);
        A(pointer,pointer-n)=as(i,k);
    end
end

%% T3
for i=n
    for j=2:m-1
         pointer=(j-1)*n+i;
        A(pointer, pointer)=ap;
        A(pointer, pointer-1)=(aw+ae);
        A(pointer, pointer-n)=as(i,k);
        A(pointer, pointer+n)=an(i,k);
    end
end
%% T4
for i=2:n-1
    for j=1
        pointer=(j-1)*n+i;
        A(pointer,pointer)=ap;
        A(pointer,pointer+(m-2)*n)=as(i,k);
        A(pointer,pointer-1)=aw;
        A(pointer,pointer+1)=ae;
        A(pointer,pointer+n)=an(i,k);
    end
end

%% T5
for i=2:n-1
    for j=m
        pointer=(j-1)*n+i;
         A(pointer,pointer-m)=as(i,k);
         A(pointer,pointer-1)=aw;
         A(pointer,pointer)=ap;
         A(pointer,pointer+1)=ae;
         A(pointer,pointer-(m+2)*n)=an(i,k);
    end
end

%% T6
for i=1
    for j=1
        pointer=(j-1)*n+i;
        A(pointer,pointer+1)=(aw+ae);
        A(pointer,pointer+(m-2)*n)=as(i,k);
        A(pointer,pointer)=ap;
        A(pointer,pointer+n)=an(i,k);
    end
end

%% T7
for i=n
    for j=1
        pointer=(j-1)*n+i;
          A(pointer,pointer+(m-2)*n)=as(i,k);
          A(pointer,pointer)=ap;
          A(pointer,pointer-1)=(aw+ae);
          A(pointer,pointer+n)=an(i,k);
    end
end

%% T8
for i=1
    for j=m
        pointer=(j-1)*n+i;
          A(pointer,pointer-n)=as(i,k);
          A(pointer,pointer)=ap;
          A(pointer,pointer+1)=(aw+ae);
          A(pointer,pointer+n)=an(i,k);
    end
end
   %% T9
for i=n
    for j=m
        pointer=(j-1)*n+i;
          A(pointer,pointer-n)=as(i,k);
          A(pointer,pointer)=ap;
          A(pointer,pointer-1)=(aw+ae);
          A(pointer,pointer+n)=an(i,k);
    end
end
        pointer=(j-1)*n+i;
        b(pointer)=f(i,j,k-1);
        Save A\b
end

Plot(A\b)


    (1-((i-1)*dx)^2/4*pi^2)*cos(pi*k)*sin((i-1)*dx);

 Contour(x,y)