How to use convolution to solve Fick's 2nd Law?

∂C/∂t=D∇^2 C

I am tying to solve Fick's 2nd law in one dimension with a convolution kernal and convn for fast simulations. But I am not getting the correct results. The final result should appear something like the attached image, with the oxygen diffusing down concentration gradient with time. I cannot find the bug in my code. I think it is either with my boundary conditions or how I am implementing the convn() function. Can someone find the error in this code?

%This program simulates Fickian diffusion of molecular oxygen from a single
%PFOB droplet though water in radial coordinates
%Addjusting time shift seems to adjust position, not rate of reduction.

clear all; clf; close all;
tic
% % diffusivity or diffusion coefficient [m^2/s].  Oxygen in water.
D = 2.5E-9; %m2/s
xi = 1E-6; xf = 1000E-6; dx = 1E-6;  %position, meters
ti = 0; tf = 1000; dt = 1; %time, seconds
cf = (2.828E6)/330; % ambient concentration value [ng/L]
ci = 2.828E6; % initial concentration value [ng/L]
X = xi:dx:xf;
rr=round(xf/dx);
R=1E-6;
dyn=round(tf/dt)
T=ci*ones(rr,dyn,'double');

for t=1:dyn
    for r=1:rr
        w=1/(4*D*dt*t)*(exp(-((X(r))*(X(r)))./(4*D*dt*t)));
        G2(r,t)=w;
    end
end

%initial condition
T(1,1)=ci;
T(:,1)=cf;

tall(G2);tall(T);
Tout=convn(G2,T(:,1),'same');
gather(G2);gather(T); gather(Tout);

for t=1:dyn
    disp(t)
    plot(X,Tout(:,t))
    title(num2str(t))
    %ylim([cf ci])
    pause
end

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