Integrating Heat Equation Coefficient with time-dependent forcing.

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Salman Salamn 2021-12-6

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https://ww2.mathworks.cn/matlabcentral/answers/1604200-integrating-heat-equation-coefficient-with-time-dependent-forcing

SmallSphereNitrogen.csv

I have solved the heat equation with time-dependent boundary conditions and I got the following equation for the coefficient, and the temperature of the system, where

is the temperature data from an experiement. The data is given attached below.

Now I tried to use finite difference method to find

. My code follows like this. My issue is that I am getting a constant value for u(r,t); I think it is because C(i) is actually constant at every step. I am not eniterly sure why this the case. So if anyone has an idea what I am doing wrong here, that would be much apperciated.

Note: The equations above are both correct and I have already checked them.

clear; clc; 
Time = xlsread('SmallSphereNitrogen.csv','A2:A1442');
T_r0 = xlsread('SmallSphereNitrogen.csv','B2:B1442'); % temperature at r = 0
T_R2 = xlsread('SmallSphereNitrogen.csv','C2:C1442'); % temperature at r = R/2
R = 2.5*2.52/100; 
K = 14; 
Tr = 3; 
C = zeros(200,1); 
P = zeros(200,1); 
for n = 1:25 % modes 
    for i = 1:200-1 % time - Regime 1 
        for j = 1:200-2 % distance -  Regime 1 
            P(i) = T_r0(i); 
            P2(i) = T_R2(i); 
            C(i) = 2 * R * (P(i) - Tr) * (-1)^n / pi / n + ( P2(i) - P(i) ) * exp(-K * (n*pi/R)^2 * i);
        end
        u(j,i) = C(i)/j * sin(n*pi/R * j) * exp(-K * (n*pi/R)^2 * i) + P2(i) - P(i); 
    end
end