Certain problems in the theory of fractional-order linear differential operators (Q1363889)

A transformation operator is constructed which replaces the solution of the equation \[ D_{0x}^{(\sigma)}u- \{\lambda+q(x)\}u=0 \tag{1} \] with the solution of \[ D_{0x}^{(\sigma)}u- \lambda u=0.\tag{2} \] Here \(D_{0x}^{(\sigma)}\) is an operator of fractional (in the Riemann-Liouville sense) differentiation of order \(1<\sigma\leq 2\). A function which solves a problem \[ D_{0x}^{(\sigma-2)} u|_{x=0}= h_0, \qquad D_{0x}^{(\sigma-1)} u|_{x=0}= h_1 \] for equation (1) is constructed with the aid of a solution of equation (2). Properties of the fractional operator are discussed.