We Have Numbers Of Free Samples

a) Line of Sight Data Analysis:

Matrix method for abel inverse.

h(1)=0;

y(1)=0;

for i=2:51

y(i)=y(i-1)+0.1;

end

for i=1:51

fm(i)=sqrt(2*3.14)*((y(i)*y(i))+1)*exp(-(y(i)*y(i))/2);

end

Tikonov regularization is used for accuracy issues. As in this case no accuracy measures have been taken Tikonov regularization can be used.

b) Comutation of abel inverse

%%inverse abel using fourier transform%%

for i=1:51

if(mod(i,2)==0)

A(i)=0;

else

A(i)=(2/(i*3.14))*((-1)^((i-1)/2));

end

% A(i)=randi(10);

end

fo(1)=1;

for i=2:51

fo(i)=1-((-1)^i)*cos((i*3.14*y(i))/51);

end

for i=1:51

fou(i)=A(i)*fo(i);

end

Comparisons between fourier and matrix method for abel inverse.

Discussion of code

%% Finite Difference Explicit Method (Iteration) 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

% Iteration using Taylor’s Finite Difference

for j = 1:nt-1 % t counter

for i = 1:nx-2 % x counter

c(i+1,j+1) = c(i+1,j)+((dt/(dx)^2)*D*(c(i+2,j)-2*c(i+1,j)+c(i,j))); % c(x+dx,t+dt)

% Neumann boundary (zero flux): cx(xi,t)=0 and cx(xf,t)=0 -> c(xi,t)=c(xi+dx,t) and c(xf,t)=c(xf-dx,t)

c(1,j+1) = c(2,j+1); % change c(xi) = c(xi+dx)

c(nx,j+1) = c(nx-1,j+1); % change c(xf) = c(xf-dx)

end

end

b) Monte Carlo simulation converted from the tiny_mc.c to matlab.

D) 1. code prepared to compute the problem

2. Code applied to finalsdata.dat to fit line.

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