In this paper, the homotopy analysis method is applied to obtain the solution of nonlinear fractional partial differential equations. The method has been successively provided for finding approximate analytical solutions of the fractional nonlinear Klein-Gordon equation. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ħ. The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.

This paper is a short description of our recent results on an important class of the so-called Mittag-Leffler functions, which became important as solutions of fractional order differential and integral equations, control systems and refined mathematical models of various physical, chemical, economical, management and bioengineering phenomena. We have studied the Mittag-Leffler functions as their typical representatives, including many interesting special cases that have already proven their usefulness in fractional calculus and its applications. We obtained a number of useful relationships between the Mittag-Leffler functions and the Wright functions. The Wright function plays an important role in the solution of a linear partial differential equation. The Wright function, which we denote by W ( z ; α , β ) , is so named in honor of Wright who introduced and investigated this function in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions.

The past decade has been characterized by the growing volumes of data due to the widespread use of the Internet of Things (IoT) applications, which introduced many challenges for efficient data storage and management. Thus, the efficient indexing and searching of large data collections is a very topical and urgent issue. Such solutions can provide users with valuable information about IoT data. However, efficient retrieval and management of such information in terms of index size and search time require optimization of indexing schemes which is rather difficult to implement. The purpose of this paper is to examine and review existing indexing techniques for large-scale data. A taxonomy of indexing techniques is proposed to enable researchers to understand and select the techniques that will serve as a basis for designing a new indexing scheme. The real-world applications of the existing indexing techniques in different areas, such as health, business, scientific experiments, and social networks, are presented. Open problems and research challenges, e.g., privacy and large-scale data mining, are also discussed.

A new solution technique for analytical solutions of fractional partial differential equations (FPDEs) is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as well as a more general fractional differential equation.