Reproducing kernel Hilbert space method for solving fractal fractional differential equations

Based on reproducing kernel theory, an analytical approach is considered to construct numerical solutions for some basic fractional ordinary differential equations (FODEs, for short) under fractal fractional derivative with the exponential decay kernel. For the first time, the implemented approach, namely reproducing kernel Hilbert space method (RKHSM), is proposed in terms of analytic and numerical fractal fractional solutions. Through the convergence analysis, we illustrate the high competency of the RKHSM. Our results are compared with the exact solutions, and they show us how the fractal-fractional derivative when the kernel is exponential decay affects the obtained outcomes. And, they also confirm the superior performance of the RKHSM.