MATLAB: PDE propagating from point source

boundarydiffusionpdepointpropagation

I want to solve numerically a nonlinear diffusion equation from an instantaneous point source. Thus, I have initial conditions, but not boundary conditions. How should I go about writing a code to solve circular propagation from a point?

Thanks!!

Best Answer

I assume you want the point source appear at r=0.

Choose the interval of integration as [0:R] where R is big enough to ensure that C=C(t=0) throughout the period of integration.

As initial condition, choose an approximation to the delta function.

As boundary conditions, choose dC/dr = 0 at both ends.

Best wishes

Torsten.

Related Solutions

MATLAB: PDEPE function

Yes it is possible in pdefun :)

Look at example:

function diffusion
m = 0;
x = linspace(0,1,20);
t = linspace(0,2,5);
sol = pdepe(m,@eqtn,@ic,@bc,x,t);
u = sol(:,:,1);
figure;
surf(x,t,u);    
xlabel('Distance x');
ylabel('Time t');
% --------------------------------------------------------------------------



function [c,f,s] = eqtn(x,t,u,DuDx)
c = 1;
f = Dfsn(u)*DuDx;
s = 0;
% --------------------------------------------------------------------------
function u0 = ic(x)
u0 = 1:length(x);
% --------------------------------------------------------------------------
function [pl,ql,pr,qr] = bc(xl,ul,xr,ur,t)
pl = ul;
ql = 2;
pr = ur;
qr = 2;
% --------------------------------------------------------------------------
function d = Dfsn(u)
d = sqrt(u+1);

Where Dfsn is your non linear diffusion coefficient function.

MATLAB: Bvp with constant vector in function definition

function main
rinter = ...;
ninter = ...;
c= ...;
g = @(ya,yb) [ya(1); yb(1)-c];
solinit=bvpinit(linspace(0,r_max,10),[0 0]);
sol = bvp4c(@(r,U)f(r,U,rinter,ninter),g,solinit);
function dUdr = f(r,U,rinter,ninter)
nr = interp1(rinter,ninter,r);
dUdr = zeros(2,1);
dUdr(1) = U(2);
dUdr(2) = -4*pi*r*nr;