Question: Under what condition the Caputo fractional derivative. Dβu(x) = 1. Î(1 - β). â« x. 0 u (t)(x - t)âβdt, 0
Question: Under what condition the Caputo fractional derivative Z x 1 u0 (t)(x − t)−β dt, 0 < β < 1 Dβ u(x) = Γ(1 − β) 0 of a polynomial of degree n is a polynomial of degree at most n? Answer: The Caputo derivative of xn is Z x 1 Dβ (xn ) = (tn )0 (x − t)−β dt Γ(1 − β) 0 Z x 1 ntn−1 (x − t)−β dt = Γ(1 − β) 0 Z 1 n = (ux)n−1 (x − ux)−β xdu {t → ux} Γ(1 − β) 0 Z 1 nxn−β = un−1 (1 − u)−β du Γ(1 − β) 0 Z 1 nxn−β a b = B(1 − β, n) u (1 − u) du = B(b + 1, a + 1) Γ(1 − β) 0 xn−β Γ(1 − β)nΓ(n) Γ(a)Γ(b) = B(a, b) = Γ(1 − β) Γ(n − β + 1) Γ(a + b) Γ(n + 1) xn−β {nΓ(n) = Γ(n + 1)} = Γ(n − β + 1) So, the Caputo fractional derivative of a polynomial of degree n is a polynomial of degree at most n when 0 < β < 1.
