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...
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