Kernel-Specific Structures in Non-Singular Fractional Derivatives

Abstract

This paper develops a unified analytical framework for evaluating a broad class of recent non-singular fractional derivatives acting on generalized Mittag–Leffler functions. Using the four-parameter Mittag–Leffler function as a representative function, kernel-specific representations are derived for different non-singular fractional operators. The proposed formulation expresses each operator through an associated kernel-specific function, enabling direct structural comparison. Kernel-specific bounds, a general uniform convergence theorem, and corresponding Laplace-transform representations are established. Numerical illustrations highlight the influence of different kernels on the resulting fractional derivatives and support the theoretical results.