1D diffusion equation in spherical coordinates with moving boundary condition

Hi,

I was wondering how I could solve the 1D diffusion equation in spherical coordinates with a moving boundary condition using MATLAB. Is it possible to do this with pdepe? The differential equation is

Thank you in advance.

Answer 1

[Yoki G]

% define mesh

x = linspace(0,1,25);

t = linspace(0,1,25);

m = 2; % you have m=1 for cylindrical coordinates and m=2 for spherical

cb = pdepe(m,@heatcyl,@heatic,@heatbc,x,t); % run solver

%plot across a secton for different times

for i = 1:length(t)

col=[i/length(t) 0 1-i/length(t)];

plot(x,cb(i,:),'color',col,'linewidth',2); hold on; grid on; box on;

end

xlabel('r')

ylabel('c')

% define functions

function [c,f,s] = heatcyl(x,t,u,dudx) % diffusion equation equation

Dsb = 1; % diffusion coefficient

c = 1;

f = Dsb*dudx;

s = 0;

end

function u0 = heatic(x) % initial condition

cb0 =1; % initial concentration

u0 = cb0+0*x;

end

function [pl,ql,pr,qr] = heatbc(xl,ul,xr,ur,t) %BCs

Dsb = 1; % diffusion coefficient

cl1 = @(t) 1-t; % example of a boundary function

pl = 0;

ql = -Dsb;

pr = -cl1(t);

qr = -Dsb; % for your neumann condition at r=1

end

Here is an example. If you look up pdepe documentation they have an example for cylindrical coordinates with two Dirichlet condition. You need to move the diffusion coefficient inside the derivative (assuming constant) and solve the resulting equation.