Caputo derivative

Abstract. Let \(0 < \alpha < 1\), and define \(e_n(x):=x^{n\alpha}/\Gamma(n\alpha+1)\) for \(n\ge 0\). We prove that the algebraic direct sum \(\mathcal{G}_{\alpha}^{\mathrm{alg}}:=\bigoplus_{n=0}^{\infty}\mathbb{C}e_n\) is the distinguished \(\alpha\)-graded monomial space on which the order-\(\alpha\) Riemann–Liouville integral \(J_\alpha:={}_0 I_x^\alpha\) and the order-\(\alpha\) Caputo derivative \(C_\alpha:={}_0^{\mathrm{C}}D_x^\alpha\) act as a unilateral shift pair, namely \(J_\alpha e_n=e_{n+1}\) ...

Abstract. We prove that the ordinary differentiation operator on the finite-dimensional polynomial space \( P_n := \{p(x)\in \mathbb{C}[x] : \deg p \le n\} \) cannot serve as an internal model for classical fractional calculus. Here, by an internal model we mean a family of linear endomorphisms acting on the same …