Error in computing numerical solution?

I am enable to compute numerical solution, as it generates weird Initial conditions (may be) compared to the inintial value of U_exact

t0  = 0;
tn  = 1;
x0  = 0;
xn  = 1;
N = 16;
dx = 1/N;
dt = 0.004;
x = (x0:dx:xn)';
t = (t0:dt:tn)';
beta =  1;
L = 1;
nx = (xn-x0)/dx;
nt = (tn-t0)/dt;

u  = @(x,t) (0.2*pi/3)*(((sin(pi*x))*exp(-(pi^2)*t)));

%% construction of diagonal matrix
v =((1:nx)*pi/L).^2;
A = full(diag(v,0));

%%
U=zeros(nx+1,nt+1);
U_exact=zeros(nx+1,nt+1);

for i=1:nx+1
fun = @(x) 2*x.*(1-x).*sin(i*(pi).*x);
q(i) = integral(fun,0,L,'ArrayValued',true);
end
U(:,1) = q';
U(1,:)=0;
U(nx+1,:)=0;

%% numerical solution construction
for k=1:nt
    U(2:nx+1,k+1) = (eye(N)+dt*A)\U(2:nx+1,k);
end

%% exact solution construction
for k=1:nt+1
    U_exact(1:nx+1,k) =u(x,t(k));
end

Numerical_sol = U;
Analytical_sol = U_exact;

%Plots
figure(1)
subplot(2,2,1)
mesh(t,x,U)
xlabel('Temporal dim')
ylabel('Spatial dim')
zlabel('Numerical sol')

subplot(2,2,2)
mesh(t,x,U_exact)
xlabel('Temporal dim')
ylabel('Spatial dim')
zlabel('Analytical sol')

subplot(2,2,3)
mesh(t,x,U_exact-U)
xlabel('Temporal dim')
ylabel('Spatial dim')
zlabel('Error in  sol')

Help me to resolve the error. Thanks

[Nagasai Bharat]

Hi Usman,

After looking into your code, I think there might be an incorrect numerical approximation of the analytical expression. Please do check on that.