Modified Adaptive Synchronization and Anti-Synchronization Method for Fractional Order Chaotic Systems with Uncertain Parameters
Page: 1-38 (38)
Author: S. K. Agrawal, Lalit Batra, V. Mishra* and D. Datta
DOI: 10.2174/9789815079333123010003
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Abstract
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.
Improved Generalized Differential Transform Method for a Class of Linear Nonhomogeneous Ordinary Fractional Differential Equations
Page: 39-51 (13)
Author: İ. Onur KIYMAZ* and Ayşegül ÇETİNKAYA
DOI: 10.2174/9789815079333123010004
PDF Price: $15
Abstract
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
Incomplete K2-function
Page: 52-61 (10)
Author: Dharmendra Kumar Singh and Vijay Laxmi Verma*
DOI: 10.2174/9789815079333123010005
PDF Price: $15
Abstract
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.
Some Results on Incomplete Hypergeometric Functions
Page: 62-72 (11)
Author: Dharmendra Kumar Singh* and Geeta Yadav
DOI: 10.2174/9789815079333123010006
PDF Price: $15
Abstract
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.
Transcendental Bernstein Series: Interpolation and Approximation
Page: 73-93 (21)
Author: Z. Avazzadeh, H. Hassani*, J.A. Tenreiro Machado, P. Agarwal and E. Naraghirad
DOI: 10.2174/9789815079333123010007
PDF Price: $15
Abstract
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.
Some Sufficient Conditions for Uniform Convexity of Normalized 1F2 Function
Page: 94-111 (18)
Author: Deepak Bansal* and Shilpi Jain
DOI: 10.2174/9789815079333123010008
PDF Price: $15
Abstract
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.
From Abel Continuity Theorem to Paley-Wiener Theorem
Page: 112-120 (9)
Author: S. Yu, P. Agarwal and S. Kanemitsu*
DOI: 10.2174/9789815079333123010009
PDF Price: $15
Abstract
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.
A New Class of Truncated Exponential-Gould-Hopper-based Genocchi Polynomials
Page: 121-135 (15)
Author: Ghazala Yasmin and Hibah Islahi*
DOI: 10.2174/9789815079333123010010
PDF Price: $15
Abstract
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.
Computational Preconditioned Gauss-Seidel via Half-Sweep Approximation to Caputo's Time-Fractional Differential Equations
Page: 136-156 (21)
Author: Andang Sunarto*, Jumat Sulaiman and Jackel Vui Lung Chew
DOI: 10.2174/9789815079333123010011
PDF Price: $15
Abstract
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.
Krasnoselskii-type Theorems for Monotone Operators in Ordered Banach Algebra with Applications in Fractional Differential Equations and Inclusion
Page: 157-184 (28)
Author: Nayyar Mehmood* and Niaz Ahmad
DOI: 10.2174/9789815079333123010012
PDF Price: $15
Abstract
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.
General Fractional Order Quadratic Functional Integral Equations: Existence, Properties of Solutions, and Some of their Applications
Page: 185-218 (34)
Author: Ahmed M.A. El-Sayed* and Hind H.G. Hashem
DOI: 10.2174/9789815079333123010013
PDF Price: $15
Abstract
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.
Non-linear Set-Valued Delay Functional Integral Equations of Volterra-Stieltjes Type: Existence of Solutions, Continuous Dependence and Applications
Page: 219-243 (25)
Author: A. M. A. El-Sayed, Sh. M Al-Issa* and Y. M. Y. Omar
DOI: 10.2174/9789815079333123010014
PDF Price: $15
Abstract
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.
Certain Saigo Fractional Derivatives of Extended Hypergeometric Functions
Page: 244-258 (15)
Author: S. Jain*, R. Goyal, P. Agarwal and S. Momani
DOI: 10.2174/9789815079333123010015
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Abstract
This article aims to establish Saigo fractional derivatives of extended
hypergeometric functions. Some special cases of these integrals are also derived.
Some Erdélyi-Kober Fractional Integrals of the Extended Hypergeometric Functions
Page: 259-269 (11)
Author: S. Jain*, R. Goyal, P. Agarwal, Clemente Cesarano and Juan L.G. Guirao
DOI: 10.2174/9789815079333123010016
PDF Price: $15
Abstract
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.
On Solutions of the Kinetic Model by Sumudu Transform
Page: 270-283 (14)
Author: Esra Karatas Akgül*, Fethi Bin Muhammed Belgacem and Ali Akgül
DOI: 10.2174/9789815079333123010017
PDF Price: $15
Abstract
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.
Subject Index
Page: 284-288 (5)
Author: Praveen Agarwal and Shilpi Jain
DOI: 10.2174/9789815079333123010018
Introduction
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 to their remarkable properties. This volume expands our understanding of special functions by highlighting recent trends in numerical analysis. are demonstrated by 15 chapters. Many chapters highlight the importance of fundamental results and techniques of the theory of complex analysis for partial differential equations Contributions 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. This volume is a timely update for mathematicians and researchers interested in advanced numerical methods and computational techniques used to solve complex problems List of Chapters 1. Modified Adaptive Synchronization and Anti Synchronization method for Fractional order chaotic systems with uncertain parameters 2. Improved generalized differential transform method for a class of linear non homogeneous ordinary fractional differential equation 3. Incomplete K2-Function 4. Some Results On Incomplete Hypergeometric Functions 5. Transcendental Bernstein Series: Interpolation and Approximation 6. Some Sufficient Conditions For Uniform Convexity Of Normalized 1F2 Function 7. From Abel continuity theorem to Paley-Wiener theorem 8. A New Class of Truncated Exponential-Gould-Hopper basedGenocchi Polynomials 9. Computational preconditioned Gauss-Seidel via half-sweep approximation to Caputo's time fractional differential equations 10. Krasnoselskii-type Theorems for Monotone Operators in Ordered Banach Algebra with Applications in Fractional Differential Equations and Inclusion 11. General fractional order quadratic functional integral equations: Existence, properties of solutions and some of its Applications 12.Nonlinear set-valued delay functional integral equations of Volterra-Stieltjes type: Existence of solutions, continuous dependence and applications 13.Certain Saigo Fractional Derivatives Of Extended Hypergeometric Functions 14. Some Erdelyi-kober Fractional Integrals Of The Extended Hypergeometric Functions 15. On solutions of Kinetic Model by Sumudu transform