## Answers - One Dimensional Diffusion Model

```      % DIFF1.M    1-Dimensional Diffusion Solution
%
% Tom Huber, March 1997
%
ncells = 25;       % Number of Cells in the Array
NTimes = 20;       % Number of Time steps to perform

FracLeft  = 1/3;   % Fraction diffusing to the left
FracSame  = 1/3;   % Fraction which remains in same cell
FracRight = 1/3;   % Fraction diffusing to the right

Values = zeros(ncells,1);  % Initialize the Array to have zero concentration
VMax = 100;                % Maximum Concentration
Values(ncells/2) = VMax;   % Initially set the concentration at the midpoint

for i=1:NTimes  % Perform a total of NTimes time steps
NewValues = Values*FracSame;  % New values initially a fraction of original
for j=2:ncells-1  % For each cell except leftmost or rightmost
NewValues(j-1) = NewValues(j-1)+Values(j)*FracLeft;  % Diffuse Left
NewValues(j+1) = NewValues(j+1)+Values(j)*FracRight; % Diffuse Right
end % for j=1...

Values = NewValues;     % The updated values become the current values
plot(Values)            % Plot the values
axis([1 ncells 0 VMax]) % Set the minima and maxima on axes
xlabel('Position')      % Label the axes and the title
ylabel('Value')
title(['After Time Step: ' num2str(i)])
drawnow                % Make Matlab display the graph
% (Normally it only displays at the end of a program)
end  % for i=1...
```

1. Run the M-file diff1 as written (this performs undriven diffusion, with equal probabilities to the left, right and same cells)
2. Modify the M-file to better model the flow in the river by starting the initial concentration a quarter of the way along the array and changing the fraction to the left to 1/10, the fraction staying in the same location to 3/10, and the fraction to the right to 6/10. Run the program several times with different values for the relative fractions, NMax, NCells, etc.
3. ```      % DIFF1.M    1-Dimensional Diffusion Solution
%
% Tom Huber, March 1997
%
ncells = 25;       % Number of Cells in the Array
NTimes = 20;       % Number of Time steps to perform

FracLeft  = 1/10;   % Fraction diffusing to the left
FracSame  = 3/10;   % Fraction which remains in same cell
FracRight = 6/10;   % Fraction diffusing to the right

Values = zeros(ncells,1);  % Initialize the Array to have zero concentration
VMax = 100;                % Maximum Concentration
Values(ncells/2) = VMax;   % Initially set the concentration at the midpoint

for i=1:NTimes  % Perform a total of NTimes time steps
NewValues = Values*FracSame;  % New values initially a fraction of original
for j=2:ncells-1  % For each cell except leftmost or rightmost
NewValues(j-1) = NewValues(j-1)+Values(j)*FracLeft;  % Diffuse Left
NewValues(j+1) = NewValues(j+1)+Values(j)*FracRight; % Diffuse Right
end % for j=1...

Values = NewValues;     % The updated values become the current values
plot(Values)            % Plot the values
axis([1 ncells 0 VMax]) % Set the minima and maxima on axes
xlabel('Position')      % Label the axes and the title
ylabel('Value')
title(['After Time Step: ' num2str(i)])
drawnow                % Make Matlab display the graph
% (Normally it only displays at the end of a program)
end  % for i=1...
```
4. Start the initial concentration in Cell number 1 - what happens? The model as written has a problem with the leftmost and rightmost points, namely we cannot add onto the points to the left or to the right of the endpoints [which would be NewValues(0) or NewValues(ncells+1) respectively]. We got around this problem by making our loop for j=2:ncells-1, however this causes the program to miss diffusion to the right from the leftmost cell (and to the left from the rightmost cell). Correct this problem by making our loop run from for j=1:ncells and include if statements whereby we have transport to the left if j>1 and transport to the right if j<ncells.
```      % DIFF1.M    1-Dimensional Diffusion Solution
%
% Tom Huber, March 1997
%
ncells = 25;       % Number of Cells in the Array
NTimes = 20;       % Number of Time steps to perform

FracLeft  = 1/10;   % Fraction diffusing to the left
FracSame  = 3/10;   % Fraction which remains in same cell
FracRight = 6/10;   % Fraction diffusing to the right

Values = zeros(ncells,1);  % Initialize the Array to have zero concentration
VMax = 100;                % Maximum Concentration
Values(ncells/4) = VMax;   % Initially set the concentration at the midpoint

for i=1:NTimes  % Perform a total of NTimes time steps
NewValues = Values*FracSame;  % New values initially a fraction of original
for j=1:ncells  % For each cell except leftmost or rightmost
if j>1
NewValues(j-1) = NewValues(j-1)+Values(j)*FracLeft;  % Diffuse Left
end
if j

```

Electronic Copy: http://physics.gac.edu/~huber/envision/tutor2/diff1an.htm
Created: 4-APR-1997 by Tom Huber, Physics Department, Gustavus Adolphus College.