Two Stefan's problems for the diffusion fractional equation are solved, where the fractional derivative of order α ∈ (0, 1) is taken in the Caputo sense. The first one has a constant condition on x = 0 and the second presents a flux condition T x (0, t) = q t α/2 . An equivalence between these problems is proved and the convergence to the classical solutions is analyzed when α 1 recovering the heat equation with its respective Stefan's condition.MSC 2010 : Primary 26A33; Secondary 33E12, 35R11, 35R35, 35R37, 80A22

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In this study, a new form of a quadratic spline is obtained, where the coefficients are determined explicitly by variational methods. Convergence is studied and parity conservation is demonstrated. Finally, the method is applied to solve integral equations.

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