Code covered by the BSD License

# Matlab code for my Graduate Thesis

by

### Troy (view profile)

Numerically solves the diffusion equations as it pertains to medical imaging.

crankNicolson(varargin)
```function w = crankNicolson(varargin)
% crankNicolson: uses Crank-Nicolson algorithm to approximate the solution
%   to the parabolic PDE:
%       u_{t}(x,t) - alpha^2 u_{xx}(x,t) = 0, 0<x<L, 0<t<T
%   subject to the boundary conditions
%       u(0,t) = u(L,t) = 0, 0<t<T
%   and the initial conditions
%       u(x,0) = f(x), 0<=x<=L
%
% arguments:
%   L (scalar) - upper bound of spatial (x) variable
%       (Default L = 1)
%   T (scalar) - upper bound of time (t) variable
%       (Default T = 1)
%   alpha (scalar) - square root of coefficient of u_{xx} term
%       (Default alpha = 1)
%   m (scalar) - number of discrete spatial intervals
%       (Default m = 100)
%   n (scalar) - number of discrete time intervals
%       (Default n = 100)
%
%   w (m+1 x n) - approximation to u(x,t) at discrete space/time positions
%

% author: Troy J. Winkstern
% email: tjw8191@rit.edu
% date: 30 Jan 2011

% parse input arguments
[L,T,alpha,m,n] = parseInputs(varargin{:});

% initialize h, k, lambda, and w
h = L./m;
k = T./n;
lambda = (alpha.^2).*k./(h.^2);
w = zeros(m+1,n);

% initialize first column of w
w(2:m,1) = sin(pi.*h.*(1:(m-1)).');

% construct matrices A and B based on lambda
Bdiags = repmat([lambda./2 1-lambda lambda./2],m-1,1);
B = spdiags(Bdiags,[-1 0 1],m-1,m-1);

% loop through columns of w, solving linear system:
% A w(:,i+1) = B w(:,i)
for i=1:(n-1)
w(2:m,i+1) = A\(B*w(2:m,i));
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% subfunction parseInputs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [L,T,alpha,m,n] = parseInputs(varargin)

% check number of input arguments
nargs = length(varargin);
error(nargchk(0,5,nargs));

% get/set n
if nargs<5
n = [];
else
n = varargin{5};
end
if isempty(n)
n = 100;
end

% check n has valid value
if (numel(n)>1) || (n<1) || ~isequal(round(n),n)
error('n must be a positive integer.');
end

% get/set m
if nargs<4
m = [];
else
m = varargin{4};
end
if isempty(m)
m = 100;
end

% check m has valid value
if (numel(m)>1) || (m<1) || ~isequal(round(m),m)
error('m must be a positive integer.');
end

% get/set alpha
if nargs<3
alpha = [];
else
alpha = varargin{3};
end
if isempty(alpha)
alpha = 1;
end

% check alpha has valid value
if numel(alpha)>1
error('alpha must be a real scalar.');
end

% get/set T
if nargs<2
T = [];
else
T = varargin{2};
end
if isempty(T)
T = 1;
end

% check T has valid value
if (numel(T)>1) || (T<=0)
error('T must be a positive real scalar.');
end

% get/set L
if nargs<1
L = [];
else
L = varargin{1};
end
if isempty(L)
L = 1;
end

% check L has valid value
if (numel(L)>1) || (L<=0)
error('L must be a positive real scalar.');
end

% warning if L not integer
if ~isequal(round(L),L)
warning('L should be an integer so that the inital condition takes on the value of 0 for t = 0 and t = T.');
end
```