How to solve fick's 2nd law of diffusion equation?

Hello everybody,

I am trying to solve a PDE which has the form:

dC/dt = (1+c)^2/{(1+c)^2+1}d2C/dx2

Can Matlab solve such a equation like that? If it can, how can I set it up?

Thanks for any help.

Answer 1

[Precise Simulation]

Modeling and simulation of convection and diffusion is certainly possible to solve in Matlab with the FEA Toolbox, as shown in the model example below:

    % Set up 1D domain from 0..1 with 20 elements.
    fea.sdim = { 'x' };
    fea.grid = linegrid( 20, 0, 1);

    % Add covection and diffusion physics mode.
    fea = addphys( fea, @convectiondiffusion, {'C'} );

    % Define diffusion coefficient.
    fea.phys.cd.eqn.coef{2,end} = {'d_coef'};
    fea.expr = { 'c',      {'1.23'} ;
                 'd_coef', {'(1+c)^2/((1+c)^2+1)'} };

    % Use c = -1 on right boundary, and insulation
    % flux boundary conditions on the left.
    fea.phys.cd.bdr.sel = [ 1 3 ];
    fea.phys.cd.bdr.coef{1,end}{1} = -1;

    % Check, parse, and solve problem 
    % with initial condition 'C=2*x'.
    fea = parsephys( fea );
    fea = parseprob( fea );

    [fea.sol.u,tlist] = ...
        solvetime( fea, 'dt', 0.1, 'tmax', 1, 'init', {'2*x'} );
    % Alternatively, solvestat can be used for stationary problems.

    % Postprocessing.
    isol = length(tlist);
    postplot( fea, 'surfexpr', 'C', 'solnum', isol )
    title( ['Solution at time, t = ',num2str(tlist(isol))] )
    ylabel( 'Concentration, C' )

[Nonlinear]

Thank you very much for your help.