On the combined Shooting-Pseudo-Arclength method for finding frequency response of nonlinear fractional-order differential equations

In this paper a numerical method for finding and drawing amplitude–frequency curves of nonlinear fractional differential equations is introduced. This method is based on the combined Shooting and Pseudo-Arclength methods. The Shooting method and the Finite Difference method (FDM) are employed to find periodic solution of nonlinear fractional differential equations and the Pseudo-Arclength method is used for continuation of periodic solutions. For investigating the accuracy and effectiveness of the proposed Shooting-Pseudo-Arclength (SPA) method, it is compared to other methods. It is shown that, the SPA method has the best agreement with numerical simulations obtained by finite difference in comparison with the Averaging method and the Harmonic Balance method. Also, some techniques are introduced to improve the accuracy and reduce the computational time. The proposed method can be used for fractional differential equations and partial fractional differential equations. It should be noted that, the proposed method computes both stable and unstable solutions.