Analysis of Cauchy problem with fractal-fractional differential operators

Abstract Cauchy problems with fractal-fractional differential operators a power law, exponential decay, and the generalized Mittag-Leffler kernels are considered in this work. We start deriving some important inequalities, then by using linear growth Lipchitz conditions, we derive conditions under which these equations admit unique solutions. A numerical scheme was suggested for each case to solution equation. Some examples of were presented, their exact solutions obtained compared used scheme. nonlinear solved, presented graphically different values fractional orders fractal dimensions.