Given the increasing number of proposals and definitions of operators in the scope of
fractional calculus, it is important to introduce a systematic classification. Nonetheless, many …

Artificial neural network (ANN) is the backbone of machine learning, specifically deep
learning. The interpolating and learning ability of an ANN makes it an ideal tool for …

This book provides efficient and reliable numerical methods for solving fractional calculus
problems. It focuses on numerical techniques for fractional integrals, derivatives, and …

Fractional calculus, which has two main features—singularity and nonlocality from its origin—
means integration and differentiation of any positive real order or even complex order. It has …

Partial differential equations (PDEs) are used with huge success to model phenomena
across all scientific and engineering disciplines. However, across an equally wide swath …

This paper discusses the concepts underlying the formulation of operators capable of being
interpreted as fractional derivatives or fractional integrals. Two criteria for required by a …

Up to now we have introduced a fractional derivative definition for special simple function
classes. In the following section we will present common generalizations for arbitrary …

A class of second order approximations, called the weighted and shifted Grünwald
difference (WSGD) operators, are proposed for Riemann-Liouville fractional derivatives, with …

We examine a numerical method to approximate to a fractional diffusion equation with the
Riesz fractional derivative in a finite domain, which has second order accuracy in time and …

In this paper, a novel compact operator is derived for the approximation of the Riesz
derivative with order α∈(1,2. The compact operator is proved with fourth-order accuracy …