1D - Heat equation
Finite differences applied to the solution of a 1D heat equation PDE.
Consider the following 1D heat problem where the temperature function depends on space and time 
with initial condition 
and boundary conditions 
.
.Solve this problem for the parameter values
and 
and The finite differences equation are
if we define 
then it can written in terms of components as:
It is also equivalent to write:
kc = 0.02; %conductivity coefficient
xini = 0;
xfin = 1;
L = xfin-xini; %bar length
nDiv = 100; %number of spatial subdivisions
dx = L/nDiv; % spatial step
tini=0;
tfin = 1.5; %final time for the simulation
dt = 0.002; %time step
r = kc*dt/dx^2; %stability parameter (r<=0.5)
if (r > 0.5)
disp('unstable simulation');
return;
end
x = xini:dx:xfin;
u = 20+40*x';
unext = u;
for t = dt:dt:tfin
for i = 2:mx
unext(i) = r*u(i-1)+(1-2*r)*u(i)+r*u(i+1);
end
% boundary conditions
unext(1) = 20*exp(-t);
unext(end) = 60*exp(-2*t);
% plot solution
plot(x, unext);
axis([-0.1,1.1, 0, 61]);
xlabel('x value');
ylabel('temperature')
text(0.2,50,['time = ' num2str(t)]);
grid;
drawnow;
u = unext;
end