Variable density in heat diffusion equation

Suppose I have a heat diffusion equation:

rho(x)*c(x)*dT/dt=d/dx(k(x)*dT/dx)

Where rho(x) is the density, c(x) is the heat capacity and k(x) is thermal diffusivity, all being position dependent. Would I need to specify these as vectors in pdepe instead of simple numbers?

[Torsten]

Just like the initial conditions: as simple numbers dependent on x.

Answer 1

[Matthew Hunt]

So I do the interp1 trick you demonstrated before?

Answer 2

[Matthew Hunt]

So I used the following code:

function [CC, FF, SS]=battery_GE(x,t,u,DuDx)
global rho; global c_th; global a_1;global k; global epsilon; global c_1;
global D; global a_2; global sigma; global b; global c_2; global x_vector;
D_var=interp1(x_vector,D,x);
k_var=interp1(x_vector,k,x);
sigma_var=interp1(x_vector,sigma,x);
epsilon_var=interp1(x_vector,epsilon,x);
rho_var=interp1(x_vector,rho,x);
c_var=interp1(x_vector,c_th,x);
CC=[rho_var.*c_var; 0; 1];
FF=[a_1*k_var; epsilon; c_1*D_var].*DuDx;
SS=[a_2*sigma_var.*DuDx(2); -b*u(3);c_2*(DuDx(3)-(b*u(3)./epsilon_var)-DuDx(1))];

And I got the following error messages:

Error using vertcat Dimensions of arrays being concatenated are not consistent.

Error in battery_model>battery_GE (line 83)
FF=[a_1*k_var; epsilon; c_1*D_var].*DuDx;

Error in pdepe (line 246)
[c,f,s] = feval(pde,xi(1),t(1),U,Ux,varargin{:});

Error in battery_model (line 50)
sol = pdepe(m,@battery_GE,@battery_ic,@battery_bc,x,t);

What went wrong?

[Matthew Hunt]

I see the error.

[Torsten]

FF=[a_1*k_var; epsilon_var; c_1*D_var].*DuDx;

instead of

FF=[a_1*k_var; epsilon; c_1*D_var].*DuDx;

Best wishes

Torsten.