# Finite Difference Implicit Method for Fick's 2nd Law

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PDE solution using the implicit method
Updated 26 Apr 2019

%% Finite Difference Implicit Method (Matrix Equations) with D = diffusivity: Fick's 2nd Law of Diffusion
% by Prof. Roche C. de Guzman
clear; clc; close('all');
%% Given
xi = 0; xf = 0.6; dx = 0.04; % x range and step size = dx [m]
xL = 0; xU = 0.1; % initial value x lower and upper limits [m]
ti = 0; tf = 0.05; dt = 4e-4; % t range and step size = dt [s]
ci = 2; % initial concentration value [ng/L]
cLU = 8; % initial concentration value within x lower and upper limits [ng/L]
D = 1.5; % diffusivity or diffusion coefficient [m^2/s]
%% Calculations
% Independent variables: x and t
X = xi:dx:xf; nx = numel(X); T = ti:dt:tf; nt = numel(T); % x and t vectors and their number of elements
[x,t] = meshgrid(X,T); x = x'; t = t'; % x and t matrices
% Dependent variable: c
c = ones(nx,nt)*ci; % temporary c(x,t) matrix with rows: c(x) and columns: c(t)
% Initial values and Dirichlet boundary
I = find((X>=xL)&(X<=xU)); % index of lower and upper limits
c(I,1) = cLU; % c at t = 0 for lower and upper limits
% Matrix of coefficients
muD = (dt/dx^2)*D;
M = diag([1 ones(1,nx-2)*(1+2*muD) -1])... % main diagonal
+ diag([-1 ones(1,nx-2)*-muD],1)... % 1st superdiagonal
+ diag([ones(1,nx-2)*-muD 1],-1); % 1st subdiagonal
% Vector of unknowns per time point
for j = 1:nt-1
K = [0; c(2:nx-1,j); 0]; % column vector of constants
U = M\K; % column vector of unknowns
c(:,j+1) = U; % concentration matrix
end

### Cite As

Roche de Guzman (2024). Finite Difference Implicit Method for Fick's 2nd Law (https://www.mathworks.com/matlabcentral/fileexchange/71358-finite-difference-implicit-method-for-fick-s-2nd-law), MATLAB Central File Exchange. Retrieved .

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