Abstract
We drive efficient and reliable finite difference methods for fractional differential equations (FDEs) based on recently defined conformable fractional derivative. We first derive fractional Euler and fractional Taylor methods based on the fractional Taylor expansion. This fractional Taylor series are the generalized fractional Taylor series that are independent of initial point. We show that the proposed methods are more efficient and faster by applying these methods on first order FDEs and second order oscillatory FDEs. Our second approach is based on inverting FDEs to a weakly singular integral equation that is approximated by product integration rule. This new definition has no special functions and thus the proposed numerical methods will be more accurate and easier to implement than existing methods for FDEs. We prove the stability and convergence of the proposed methods. Numerical examples are given to support the theoretical results.
Highlights
Fractional differential equations (FDEs) become more attractive and have been developed in theory and applications in science and mathematics
Some applications of fractional differential equations (FDEs) can be founded in chemistry, mechanics, physics, control theory and so on
Much simpler and compatible definition of fractional derivative obeying chain rule and semi-group properties based on the basic limit processing so called the conformable fractional derivative has been given in [4]
Summary
Fractional differential equations (FDEs) become more attractive and have been developed in theory and applications in science and mathematics. Almost all the definitions of the fractional derivative have been defined globally and in non local sense so that they involve fractional integral equations with weakly singular kernels and some special functions such as Gamma and Mittag-Leffler functions. All these definitions does not obey some standard rules and important properties of ordinary derivative such as chain rule or semi-group property. The main difference in between classical derivative and fractional derivative is the non local properties of the fractional calculus This leads to intense computational methods and high order numerical methods that are very limited in literature. The weakly singular kernel of the Volterra type integral equations makes it difficult to have an efficient and high order numerical method. Throughout, the notations C and c, with or without a subscript, denote generic constants, which may differ at different occurrences, but are always independent of the mesh size
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