Hermite Wavelet Method for Nonlinear Fractional Differential Equations

Abstract

Nonlinear fractional differential equations (FDEs) constitute the basis for many dynamical systems in various areas of engineering and applied science. Obtaining the numerical solutions to those nonlinear FDEs has quickly gained importance for the purposes of accurate modelling and fast prototyping among many others in recent years. In this study, we use Hermite wavelets to solve nonlinear FDEs. To this end, utilizing Hermite wavelets and block-pulse functions (BPF) for function approximation, we first derive the operational matrices for the fractional integration. The novel contribution provided by this method involves combining the orthogonal Hermite wavelets with their corresponding operational matrices of integrations to obtain sparser conversion matrices. Sparser conversion matrices require less computational load, and also converge rapidly. Using the generated approximate matrices, the original nonlinear FDE is converted into an algebraic equation in vector-matrix form. The obtained algebraic equation is then solved using the collocation points. The proposed method is used to find a number of nonlinear FDE solutions. Numerical results for several resolutions and comparisons are provided to demonstrate the value of the method. The convergence analysis is also provided for the proposed method.