On Martínez-Kaabar fractal-fractional double Laplace transformation

An extended version of the Martinez-Kaabar fractal-fractional (MKF-F) calculus theory is investigated in this work, to the integral transformations and employed in solving some fractal-fractional (F-F) partial integro-differential equations. Specifically, a new concept of double Laplace transformation, named as the MKF-F double Laplace transformation, is formulated, which involves the MK double integral operator into the ordinary double Laplace transformation. Regarding this F-F transformation proposed, fundamental results are established, such as the sufficient condition for its existence, the transformation of elementary functions of two real variables, the relationship between the transformation of a function of two real variables and its successive MKF-F partial derivatives, and the important convolution theorem. Furthermore, the MKF-F Laplace transformation method is constructed and employed in solving various types of non-integer partial integro-differential equations exhibiting a fractal effect. Finally, several examples are provided to validate the results obtained.