function pdex1
%PDEX1  Example 1 for PDEPE
%   This is a simple example with an analytical solution that illustrates 
%   the straightforward formulation, computation, and plotting of the solution
%   of a single PDE.
%
%   In the form expected by PDEPE, the single PDE is
%
%   [pi^2] .*  D_ [u] =  D_ [   Du/Dx   ] +      [0]
%              Dt        Dx
%    ----         ---       -------------    -------------
%     c            u        f(x,t,u,Du/Dx)   s(x,t,u,Du/Dx)
%
%   The equation is to hold on an interval 0 <= x <= 1 for times t >= 0.
%   There is an analytical solution u(x,t) = exp(-t)*sin(pi*x). Boundary
%   conditions are specified so that it will be the solution of the initial-
%   boundary value problem.  Obviously the initial values are sin(pi*x). 
%   Two kinds of boundary conditions are chosen so as to show how they
%   appear in the form expected by the solver.  The left bc is u(0,t) = 0:
%
%     [u]    +     [0] .* [       Du/Dx          ] = [0]
% 
%     ---          ---    ------------------------   ---
%   p(0,t,u)      q(0,t)        f(0,t,u,Du/Dx)        0
%
%   The right bc is taken to be pi*exp(-t) + Du/Dx(1,t) = 0:
%
%   [ pi*exp(-t) ] + [1] .* [   Du/Dx    ] = [0]
%
%   --------------   ---    --------------   ---
%      p(1,t,u)     q(1,t)   f(1,t,u,Du/Dx)   0
%
%   The problem is coded in subfunctions PDEX1PDE, PDEX1IC, and PDEX1BC.
%
%   See also PDEPE, @.

%   Lawrence F. Shampine and Jacek Kierzenka
%   Copyright 1984-2000 The MathWorks, Inc. 
%   $Revision: 1.4 $  $Date: 2000/08/30 20:52:06 $

m = 0;
x = linspace(0,1,20);
t = linspace(0,2,5);

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);
% Extract the first solution component as u.  This is not necessary
% for a single equation, but makes a point about the form of the output.
u = sol(:,:,1);

% A surface plot is often a good way to study a solution.
figure;
surf(x,t,u);    
title('Numerical solution computed with 20 mesh points.');
xlabel('Distance x');
ylabel('Time t');

figure;
surf(x,t,exp(-t)'*sin(pi*x));
title('True solution plotted with 20 mesh points.');
xlabel('Distance x');
ylabel('Time t');

% A solution profile can also be illuminating.
figure;
plot(x,u(end,:),'o',x,exp(-t(end))*sin(pi*x));
title('Solutions at t = 2.');
legend('Numerical, 20 mesh points','Analytical',0);
xlabel('Distance x');
ylabel('u(x,2)');

% --------------------------------------------------------------------------

function [c,f,s] = pdex1pde(x,t,u,DuDx)
c = pi^2;
f = DuDx;
s = 0;

% --------------------------------------------------------------------------

function u0 = pdex1ic(x)
u0 = sin(pi*x);
    
% --------------------------------------------------------------------------

function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
pl = ul;
ql = 0;
pr = pi * exp(-t);
qr = 1;
  

