A hybrid algorithm for Caputo fractional differential equations

•We revisited the Riemann–Liouville, Grnwald–Letnikov and Caputo definitios of fractional calculus.•We reviewed fractional differential equations and fractional initial value problems for Caputo definition.•A hybrid method to solve fractional differential equations in Caputo’s sense is proposed.•The hybrid method is compared with the method based on Adams-Bashforth-Moulon for Caputo derivatives.•The hybrid method is faster than based on Adams-Bashforth-Moulon with no longer computing time.