A novel numerical method for solving autonomous initial value problems of fractional order differential equations

Due to the recent growth of fractional calculus in many practical scientific problems, a new numerical method for solving autonomous initial value problems of fractional order differential equations is presented. This method is based on using a circular sampling technique with the Fourier transform to obtain the coefficients to construct an approximate solution to the problem in a form of Taylor series expansion. Laplace transform of fractional order integrals and Laplace transform of fractional order derivatives are used in the method. Since the process of this method is mainly performed by the substitution and the symbolic integration, there is no need to change in the body of the structure for any different problems in this type. The accuracy of the method can be adjusted in several ways. Moreover, the error due to the past time dependence in the present time calculation, which occurs in numerical methods for solving fractional order differential equations, does not appear in this solution method. In the end, the results illustrate that the approximate solutions can be within satisfactory criteria if the relevant parameters are properly chosen.