Approximation of transient 1D conduction in a finite domain using parametric fractional derivatives

A solution to the problem of transient one-dimensional heat conduction in a finite domain is developed through the use of parametric fractional derivatives. The heat diffusion equation is rewritten as anomalous diffusion, and both analytical and numerical solutions for the evolution of the dimensionless temperature profile are obtained. For large slab thicknesses, the results using fractional order derivatives match the semi-infinite domain solution for Fourier numbers, F_oε[0,1/16]. For thinner slabs, the fractional order solution matches the results obtained using the integral transform method and Green's function solution for finite domains. A correlation is obtained to display the variation of the fractional order p as a function of dimensionless time (F_o) and slab thickness (l) at the boundary l=0.