However, it is exceedingly challenging to develop analytical solutions to many types of FDEs due to the extra complexity caused by arbitrary orders of derivation and integration. These real-valued orders of derivatives and integrals enable FDEs to model physical and applied scientific phenomena more precisely. The real-valued orders of derivatives and integrals are used in fractional differential equations (FDEs). The convergence analysis is also provided for the proposed method. Numerical results for several resolutions and comparisons are provided to demonstrate the value of the method. The proposed method is used to find a number of nonlinear FDE solutions. The obtained algebraic equation is then solved using the collocation points. Using the generated approximate matrices, the original nonlinear FDE is converted into an algebraic equation in vector-matrix form. Sparser conversion matrices require less computational load, and also converge rapidly. 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. To this end, utilizing Hermite wavelets and block-pulse functions (BPF) for function approximation, we first derive the operational matrices for the fractional integration.
In this study, we use Hermite wavelets to solve nonlinear FDEs. 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. Nonlinear fractional differential equations (FDEs) constitute the basis for many dynamical systems in various areas of engineering and applied science.
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