Preface

In recent years special functions have been developed and applied in a variety of fields, such as combinatory, astronomy, applied mathematics, physics, and engineering due mainly to their remarkable properties. The main purpose of this Special Issue is to be a forum of recently-developed theories and formulas of special functions with their possible applications to some other research areas. This Special Issue provides readers with an opportunity to develop an understanding of recent trends of special functions and the skills needed to apply advanced mathematical techniques to solve complex problems in the theory of partial differential equations. Subject matters are normally related to special functions involving the mathematical analysis and its numerous applications, as well as to more abstract methods in the theory of partial differential equations. The main objective of this book is to highlight the importance of fundamental results and techniques of the theory of complex analysis for PDEs, and emphasize articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions.

In chapter 1, the authors investigated the Adaptive synchronization and Anti synchronization between fractional-order 3D autonomous chaotic system and novel 3D autonomous chaotic system with quadratic exponential term using modified adaptive control method with unknown parameters.

In chapter 2, the authors improved the generalized differential transform method by using the generalized Taylor's formula.

In chapter 3, the authors introduced an incomplete K2-Function.Incomplete hypergeometric function, incomplete hypergeometric function, incomplete confluent hypergeometric function, incomplete Mittag-Leffler function can be deduced as special cases of our finding.

In chapter 4, the authors present some new results for the in-complete hypergeometric function.

In chapter 5, the authors adopt the transcendental Bernstein series (TBS), a set of basis functions based on the Bernstein polynomials (BP), for approximating analytical functions.

In chapter 6, the authors find sufficient conditions under which 1F2 (a; b, c; z) belongsto UCV (α, β) and Sp(α, β). Here, 1F2 (a; b, c; z) is a special case of generalized hypergeometric function for p = 1 and q = 2.

In chapter 7, the authors reveal that the missing link among a few crucial results in analysis, Abel continuity theorem, convergence theorem on (generalized) Dirichlet series, Paley-Wiener theorem is the Laplace transform with Stieltjes integration.

In chapter 8, the authors introduces a hybrid family of truncated exponential-Gould-Hopper based Genocchi polynomials by means of generating function and series definition. Some signicant properties of these polynomials are established.

In chapter 9, the authors derived a finite difference approximation equation from the discretization of the one-dimensional linear time-fractional diffusion equations with Caputo's time-fractional derivative.

In chapter 10, authors derived some important theorems like Krasnoselskii-type Theorems for Monotone Operators in Ordered Banach Algebra with Applicationsin Fractional Differential Equations and Inclusion.

In chapter 11, authors studied general fractional order quadratic functional integral equations: Existence, properties of solutions and some of its applications.

In chapter 12, the authors consider a nonlinear set-valued delay functional integralequations of Volterra-Stieltjes type.

In chapter 13, the authors establish Saigo fractional derivatives of extended hypergeometric functions. Some special cases of these integrals are also derived.

In chapter 14, the authors establish some new formulas and new results related to the Erdelyi-Kober fractional integral operator which was applied on the extended hypergeometric functions.

In 15 chapter, the authors investigated the kinetic model with four different fractional derivatives. They obtained the solutions of the models by Sumudu transform. They demonstrated results by some figures and prove the accuracy of the Sumudu transform by some theoretical results and applications.