Operator-Based Approach for the Construction of Solutions to ( C D (1/ n ) ) k -Type Fractional-Order Differential Equations

A novel methodology for solving Caputo D ( 1 / n ) C k -type fractional differential equations (FDEs), where the fractional differentiation order is k / n , is proposed. This approach uniquely utilizes fractional power series expansions to transform the original FDE into a higher-order FDE of type D ( 1 / n ) C k n . Significantly, this perfect FDE is then reduced to a k -th-order ordinary differential equation (ODE) of a special form, thereby allowing the problem to be addressed using established ODE techniques rather than direct fractional calculus methods. The effectiveness and applicability of this framework are demonstrated by its application to the fractional Riccati-type differential equation.