A novel method for fractal-fractional differential equations

Abstract

We consider the reproducing kernel Hilbert space method to construct numerical solutions for some basic fractional ordinary differential equations (FODEs) under fractal fractional derivative with the generalized Mittag–Leffler (M-L) kernel. Deriving the analytic and numerical solutions of this new class of differential equations are modern trends. To apply this method, we use reproducing kernel theory and two important Hilbert spaces. We provide three problems to illustrate our main results including the profiles of different representative approximate solutions. The computational results are compared with the exact solutions. The results obtained clearly show the effect of the fractal fractional derivative with the M-L kernel in the obtained outcomes. Meanwhile, the compatibility between the approximate and exact solutions confirms the applicability and superior performance of the method.