Advances in Special Functions of Fractional Calculus: Special Functions in Fractional Calculus and Their Applications in Engineering

In the present article, we have 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. The modified adaptive control method is very affective and more convenient in comparison to the existing method for the synchronization of the fractional order chaotic systems. The chaotic attractors and synchronization of the systems are found for fractional order time derivatives described in Caputo sense. Numerical simulation results which are carried out using Adams-Boshforth-Moulton method show that the method is reliable and effective for synchronization and anti-synchronization of autonomous chaotic systems. 

In this paper, by using the generalized Taylor's formula we improved the generalized differential transform method, which is a useful tool for getting the approximate analytic solutions of fractional differential equations. With this improvement, solutions of a class of linear nonhomogeneous ordinary fractional differential equations, which could not be solved with generalized differential transform method before, will be achieved and the solutions obtained will contain more integers and fractional exponents

This chapter aims to introduce the incomplete K2-Function. Incomplete hypergeometric function, incomplete confluent hypergeometric function, and incomplete Mittag-Leffler function can be deduced as special cases of our findings. Some fractional integral formulae illustrate various avenues of their applications.

Hypergeometric functions are extensions and generalizations of the geometric series, and the process of generalization of hypergeometric series started in the 19th century itself. Thus, the subject of hypergeometrics has a rich history and led to renewed interest. Many mathematicians have presented the hypergeometric function in different ways and explained its properties. Recently, Srivastava et al. [9] represented hypergeometric functions in different forms with the help of incomplete pochhammer symbols. This paper is an attempt to present some new results for the incomplete hypergeometric function. 

This paper adopts the transcendental Bernstein series (TBS), a set of basis functions based on the Bernstein polynomials (BP), for approximating analytical functions. The TBS is more accurate than the BP method, particularly in approximating functions including one or more transcendental terms. The numerical results reveal also the applicability and higher computational efficiency of the new approach.

The object of this chapter is to find sufficient conditions under which z1F2(a, b, c; z) belongs to 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 this note we 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. By this discovery, the reason why the domains of Stoltz path and of convergence look similar is made clear. Also as a natural intrinsic property of Stieltjes integral, the use of partial summation in existing proofs is elucidated. Secondly, we shall reveal that a basic part of the proof of Paley-Wiener theorem is a version of the Laplace transform.

The present paper introduces a hybrid family of truncated exponential-Gould-Hopper-based Genocchi polynomials by means of generating function and series definition. Some significant properties of these polynomials are established. In addition, graphs of truncated exponential-Gould-Hopper-based Genocchi polynomials are drawn using Matlab. Thereafter, the distribution of zeros of these polynomials is shown.

In this paper, we derived a finite difference approximation equation from the discretization of the one-dimensional linear time-fractional diffusion equations with Caputo's time-fractional derivative. A linear system is generated by implementing Caputo's finite difference approximation equation on the specified solution domain. Then, the linear system is solved using the proposed half-sweep preconditioned Gauss-Seidel iterative method. The effectiveness of the method is studied, and the efficiency is analyzed compared to the existing preconditioned Gauss-Seidel, also known as the full-sweep preconditioned Gauss-Seidel and the classic Gauss-Seidel iterative method. A few examples of the mathematical problem are delivered to compare the performance of the proposed and existing methods. The finding of this paper showed that the proposed method is more efficient and effective than the full-sweep preconditioned Gauss-Seidel and Gauss-Seidel methods.

This chapter discusses Krasnoselskii-type fixed point results for monotone operators. It is well known that the monotone operators are not continuous on the whole domain, so we will find the solutions of discontinuous operator equations and inclusions. The presented fixed point results may be considered as variants of the Krasnoselskii fixed point theorem in a more general setting. The results of Darbo, Schauder and Bohnentblust-Karlin are also generalized. We prove these results for the case of single-valued and set-valued monotone operators. We use our main result for single-valued operators to obtain the existence of solutions of anti-periodic ABC fractional BVP. The fixed point result for set-valued monotone operators is used to discuss the existence of solutions of a given fractional integral inclusion in ordered Banach spaces. 

In this chapter, we are interested in a certain class of integral equations, namely the quadratic integral equation. In this case, the unknown function is treated by some operators, then a pointwise multiplication of such operators is applied. The study of such a kind of problem was begun in the early 60’s due to the mathematical modeling of radiative transfer. The main objective was to present a special method or technique and results concerning various existence for a certain quadratic integral equation. 

In this chapter, we established two existence theorems for the non-linear Volterra-Stieltjes integral inclusion. The continuous dependence of the solutions on the delay functions, gi (i = 1,2) and on the set of selections, will be proved. The nonlinear Chandrasekhar set-valued functional integral equation and a non-linear Chandrasekhar quadratic functional integral equation, also the set-valued fractional orders integral equation, are studied as an application. An initial value problem of fractional-orders set-valued integro-differential equation will be considered.

This article aims to establish Saigo fractional derivatives of extended hypergeometric functions. Some special cases of these integrals are also derived. 

This paper aims to establish some new formulas and results related to the Erdélyi-Kober fractional integral operator applied to the extended hypergeometric functions. The results are expressed as the Hadamard product of the extended and confluent hypergeometric functions. Some special cases of our main results are also derived.

This paper investigates the kinetic model with four different fractional derivatives. We obtain the solutions of the models by Sumudu transform and demonstrate our results with some figures. We prove the accuracy of the Sumudu transform by some theoretical results and applications.