Generalized Caputo-Type Fractional-Order Differential Equations with the Mittag–Leffler Function and Their Exact Solutions via the Laplace Integral Transform Operator

In this study, within the framework of recent advances in fractional-order differential and integral operators, a fractional differential equation defined by means of a generalized Caputo-type fractional derivative is investigated. The proposed approach extends the classical Caputo fractional derivative by introducing a generalized kernel structure, thereby allowing a more flexible and comprehensive description of memory and hereditary effects. From this perspective, this study contributes to the theoretical development of fractional-order operators and their analytical properties.

The considered fractional differential equation involves the Mittag–Leffler function, which plays a fundamental role in the analysis of fractional-order dynamical systems. This special function naturally arises in the solution structure of fractional differential equations and is essential for characterizing the qualitative and quantitative behavior of fractional-order models. To derive an explicit analytical solution, the Laplace transform method is employed as a powerful and systematic tool for handling generalized Caputo-type operators. By applying the Laplace transform, the fractional differential equation is transformed into an algebraic equation in the Laplace domain, significantly simplifying the analytical treatment of the problem.

Subsequently, the inverse Laplace transform is applied to obtain the exact solution in closed form, expressed in terms of the Mittag–Leffler function. This result clearly demonstrates the intrinsic connection between generalized fractional derivatives and special functions commonly used in fractional calculus. Moreover, it is shown that, under suitable parameter selections, the proposed model reduces to the classical Caputo fractional differential equation, and several well-known results available in the literature are recovered as special cases.

The methodology and results presented in this work provide a unified and effective framework for the analysis of a broader class of fractional differential equations involving generalized kernels. The proposed model and solution technique offer valuable insights into the behavior of fractional-order systems and have potential applications in mathematical physics and various applied sciences where complex memory-dependent processes are encountered.