Numerical solution of Caputo fractional differential equations with infinity memory effect at initial condition

•A new method for computing numerical solutions of fractional differential equations in the sense of Riemann-Liouville definition.•Use of Caputo definition in the case of non zero initial conditions.•The infinite memory effect of fractional calculus is adequately dealt with. The simulation of linear and nonlinear fractional-order systems on digital computers is investigated. The Grünwald-Letnikov definition of the fractional-order derivative is analyzed in the light of the initial conditions and as a consequence a new modified scheme for the discretization and simulation of fractional order systems is proposed. For this new scheme, it will be shown a new result where Riemann-Liouville derivative with the lower limit at infinity is related with a Caputo derivative with the lower limit at a finite real value allowing the infinite memory effect of fractional calculus to be adequately dealt with. To illustrate the use of the proposed method, the numerical solution of a linear fractional-order system is compared to the available analytical solution and, in the case of nonlinear fractional systems, the solution is compared to one provided by using the Adams method proposed by Diethelm [1, 2, 3].