Time-space fractional for the Stefan model

In this paper, we solve a particular time-space fractional Stefan problem including fractional order derivatives in time and space variables in the Fourier heat conduction equation. For this, we consider fractional time derivative of order α ∈ (0, 1] and fractional # space derivative of order 2β with β ∈ 2 1 , 1 , both in the Caputo sense. Including time and space fractional derivatives, the melt front advances as s ∼ t ξ , where ξ = ξ (α, β), and we can recover sub diffusion, classical diffusion and super-diffusion behaviors. The result for the proposed problem depends on the choice of order of fractional derivatives α and β provided that the choice satisfies the relation 2β/α = 1+β