% Exercise 6.
% Solution of the diffusion transport equation 
%
%         - mu*u_xx + b*u_x = K,  u(0)=0, u(L)=1. 
%
% by a linear finite elements, constant stepsize.  

L = 1; 
mu = 0.1;
b = 1; 
K = 1/100;

% Number of elements, and the stepsize. 
M = 10; 
h = L/M;

phi = 0;  % Numerical diffusion.

N = M-1; 
Ad = gallery('tridiag',N,-1,2,-1)/h;
At = gallery('tridiag',N,-1,0,1)/2;

A = mu*(1+phi)*Ad+b*At;
F = h*K*ones(N,1); 
F(end) = F(end) + mu*(1+phi)/h - b/2;   % Boundary conditions. 

u = A\F; 
u = [0;u;1]; 
x = linspace(0,L,M+1); 

% Exact solution
tt = linspace(0,L,1001);
ue = (1-K*L/b)*exp(-b*(L-tt)/mu).*(1-exp(-b*tt/mu))/(1-exp(-b*L/mu))+K*tt/b;
plot(tt,ue,x,u,'-or')
legend('Exact','Numerical',2)