On solving the Martínez-Kaabar fractal-fractional ordinary differential equations via the fractal-fractional differential transformation method

Abstract In this research work, we extend the Martinez-Kaabar fractal-fractional calculus (abbreviated simply as MK calculus) that has been proposed in 2024, which offers a novel extension of fractional calculus by involving both fractional-order memory effects ( $$\alpha $$ ) and fractal time deformation ( $$\gamma $$ ) within one unified operator to model systems that exhibit complex dynamics, to the context of differential transformation (DT) method with applications to the solutions of some ordinary differential equations. Since the DT method stemmed from the Taylor series expansion, this theory is developed for sufficiently MK differentiable functions. Thus, first the Taylor formula is introduced in the sense of the MK derivative, and the two most important forms of the residue are established: Lagrange and integral forms. A sufficient condition for an infinitely MK differentiable function to expand into a convergent infinite series, the so-called fractal-fractional Taylor series, is proposed. Next, the concept of fractal-fractional DT is defined, and its essential properties are investigated. The approach in this work significantly expands the analytical toolkit for solving linear and nonlinear differential equations with particular applications to engineering sciences, such as the Riccati and Lane-Emden equations. Furthermore, the fractal-fractional DT method has also been successfully applied to find numerical solutions for linear and nonlinear MK systems of ordinary differential equations. In comparison with previously employed methods, the MK approach is compatible with transformation techniques, enables series expansions for MK-differentiable functions, and provides flexible basis adaptation for numerical computation. These features make it especially compelling for modeling multi-scale physical systems with inherent fractal properties and memory, by offering an innovative direction towards both theoretical development and applied problem-solving in scientific computation.