The solution of 1D diffusion equation on a half line (semi infinite) can be found with the help of Fourier Cosine Transform. Equation 3 is the attached figure is the solution of 1D diffusion equation (eq:1). I want to write a code for this equation in MATLAB/Python but I don't understand what value should I give for the dumy variable 'tau' . This variable is not defined at the start and only appears during solution process.
tau is the integration variable...that means that for computing h at a time t, you have to compute an integral function. Once a value t is fixed, the integral is definite and you have to integrate in dtau (the dummy variable can be denoted as you want, for example t')
@Filipo Maria Denaro, Can you please tell me what will be the code for D = 100, x = 40, t=30, and f(t) = 250. I know that with these values, the solution by Carslaw and Jaeger, gives the answer of 151.39.
Ather, the value f(t) = 250 is constant for all t or is f(t=30)=250? Note that from expression of solution h(x,t), if f is a positive constant, you have h(x,t) < 0. Gianluca
Ather, at following link you can find the MATLAB code solving Carslaw-Jaeger 1D diffusion equation. Gianluca
https://fr.mathworks.com/matlabcentral/fileexchange/47781-1-d-heat-diffusion-in-a-rod?requestedDomain=www.mathworks.com
Ather, f(tau) is a function that you need to integrate from 0 to 30, not a single value. Therefore only if f is a constant in time you can fix its value.This is a homework for a student...
Well, I have written following code
p = 100; % number of steps
ft1 = sin(linspace(0,45,p));
D = 100; x = 40;
t = 1:100;
ft2 = ft1*0.5;
ftau = -(ft2-ft1)/10 ;
out_of_integral = sqrt(D/pi);
fun = @(tau,ftau,t) ftau./(t-tau).*exp(-x^2./(4*D*(t-tau)));
h = zeros(1,p);
for i = 1:p
q = integral(@(tau)fun(tau,ftau(i),t(i)),0,t(i));
h(i) = q*out_of_integral;
end
plot(ft1)
hold on;
plot(ft2,'k')
plot(ftau,'c')
plot(h,'r')
legend('Input','','f( \tau)','Output (observed)')
xlabel('time(sec) at x = 40m')
ylabel('temperature')
But the result is little problematic i:e amplitude of observed signal at x=40 keeps on increasing with time. The input data (f(tau)) is not totally realistic but I am unable to understand this increase in observed amplitude. The result can be seen in attached figure.
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