On the Martínez–Kaabar Fractal–Fractional Reduced Pukhov Differential Transformation and Its Applications

This paper addresses the extension of the Martinez–Kaabar fractal–fractional calculus (simply expressed as MK calculus) to the context of reduced differential transformation, with applications to the solution of some partial differential equations. Since this differential transformation is derived from the Taylor series expansion of real-valued functions of several variables, it is necessary to develop this theory in the context of such functions. Firstly, classical elements of the analysis of functions of several real variables are introduced, such as the concept of partial derivative and Clairaut’s theorem, in terms of the MK partial $\alpha ,\gamma$-derivative. Next, we establish the fractal–fractional (FrFr) Taylor formula with Lagrange residue and discuss a sufficient condition for a function of class $C_{\alpha ,\gamma }^{\infty }$ on an open and bounded set $D\subset R^2$ to be expanded into a convergent infinite series, the so-called FrFr Taylor series. The theoretical study is completed by defining the FrFr reduced differential transformation and establishing its fundamental properties, which will allow the construction of the FrFr reduced Pukhov differential transformation method (FrFrRPDTM). Based on the previous results, this new technique is applied to solve interesting non-integer order linear and non-linear partial differential equations that incorporate the fractal effect. Finally, the results show that the FrFrRPDTM represents a simple instrument that provides a direct, efficient, and effective solution to problems involving this class of partial differential equations.