mbiol.m

function mbiol
%MBIOL    Reaction-diffusion system from mathematical biology.
%   Solves the PDE and tests the energy decay condition.

m = 0;
xmesh = linspace(0,1,15);
tspan = linspace(0,0.2,10);
sol = pdepe(m,@mbpde,@mbic,@mbbc,xmesh,tspan);
u1 = sol(:,:,1);
u2 = sol(:,:,2);

subplot(221)
surf(xmesh,tspan,u1)
xlabel('x','FontSize',12)
ylabel('t','FontSize',12)
title('u_1','FontSize',14,'FontWeight','normal')

subplot(222)
surf(xmesh,tspan,u2)
xlabel('x','FontSize',12)
ylabel('t','FontSize',12)
title('u_2','FontSize',14,'FontWeight','normal')

% Estimate energy integral.
dx = xmesh(2) - xmesh(1);  % Constant spacing.
energy = 0.5*sum( (diff(u1,1,2)).^2 + (diff(u2,1,2)).^2, 2)/dx;
subplot(212)
plot(tspan',energy,'LineWidth',1)
xlabel('t','FontSize',12)
title('Energy','FontSize',14,'FontWeight','normal')

% ----------------------- Local functions -----------------------
function [c,f,s] = mbpde(x,t,u,DuDx)
c = [1; 1];
f = DuDx/2;
s = [1/(1+u(2)^2); 1/(1+u(1)^2)];

function u0 = mbic(x);
u0 = [1+0.5*cos(2*pi*x); 1-0.5*cos(2*pi*x)];

function [pa,qa,pb,qb] = mbbc(xa,ua,xb,ub,t)
pa = [0; 0];
qa = [1; 1];
pb = [0; 0];
qb = [1; 1];