Editors: Mehmet Yavuz, Necati Özdemir

Series Title: Current Developments in Mathematical Sciences

Fractional Calculus: New Applications in Understanding Nonlinear Phenomena

Volume 3

eBook: US $79 Special Offer (PDF + Printed Copy): US $126
Printed Copy: US $87
Library License: US $316
ISSN: 2589-2711 (Print)
ISSN: 2589-272X (Online)
ISBN: 978-981-5051-94-0 (Print)
ISBN: 978-981-5051-93-3 (Online)
Year of Publication: 2022
DOI: 10.2174/97898150519331220101


In the last two decades, many new fractional operators have appeared, often defined using integrals with special functions in the kernel as well as their extended or multivariable forms. Modern operators in fractional calculus have different properties which are comparable to those of classical operators. These have been intensively studied for modelling and analysing real-world phenomena. There is now a growing body of research on new methods to understand natural occurrences and tackle different problems.

This book presents ten reviews of recent fractional operators split over three sections:

  1. - Chaotic Systems and Control (covers the Caputo fractional derivative, and a chaotic fractional-order financial system)
  2. - Heat Conduction (covers the Duhamel theorem for time-dependent source terms, and the Cattaneo–Hristov model for oscillatory heat transfer)
  3. - Computational Methods and Their Illustrative Applications (covers mathematical analysis for understanding 5 real-word phenomena: HTLV-1 infection of CD4+ T-cells, traveling waves, rumor-spreading, biochemical reactions, and the computational fluid dynamics of a non-powered floating object navigating in an approach channel)

This volume is a resource for researchers in physics, biology, behavioral sciences, and mathematics who are interested in new applications of fractional calculus in the study of nonlinear phenomena.

Audience :

Academics interested in mathematical analysis; researchers in natural sciences and social science.


The investigation of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between operators of integer order. This field has covered the classical fractional operators such as Riemann–Liouville, Weyl, Caputo, Grunwald–Letnikov, and so on. Also, especially in the last two decades, many new fractional operators have appeared, often defined using integrals with special functions in the kernel as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, because of their different properties and behaviours, which are comparable to those of the classical operators.

This book contains ten chapters in three sections. The first section, Chaotic Systems and Control, contains three chapters. In Chapter 1, Sene proposed a numerical procedure and its applications to a fractional-order chaotic system represented with the Caputo fractional derivative. In Chapter 2, Okundalaye et al. gave a new multistage optimal homotopy asymptotic method for solutions to a couple of fractional optimal control problems. In Chapter 3, Farman et al. studied a complex chaotic fractional-order financial system in price exponent with control and modelling.

The second part of the book, Heat Conduction, contains two chapters. In Chapter 4, Hristov proposed an attempt to demonstrate that the Duhamel theorem applicable for time-dependent boundary conditions (or time-dependent source terms) of heat conduction in a finite domain and the use of the Fourier method of separation of variable (superposition version) naturally leads to appearance of the Caputo–Fabrizio operators in the solution. In Chapter 5, Avcı and İskender Eroğlu considered the oscillatory heat transfer due to the Cattaneo–Hristov model on the real line modelled by a fractional-order derivative with a non-singular kernel.

The third section of the book, Computational Methods and Their Illustrative Applications, contains five chapters related to different types of real-life problems. In Chapter 6, Ghoreishi et al. applied the optimal homotopy analysis method for a nonlinear fractional-order model to HTLV-1 infection of CD4+ T- cells. In Chapter 7, Durur et al. investigated the behavior analysis and asymptotic stability of the traveling wave solution of the Kaup-Kupershmidt equation with the conformable operator. In Chapter 8, Baishya et al. took into account the Caputo fractional order derivative in the mathematical analysis of a rumor- spreading model and presented interesting numerical results. In Chapter 9, Veeresha et al. studied a unified approach for the fractional system of equations arising in the biochemical reaction without a singular kernel. In Chapter 10, Bora et al. investigated the hydro-morphodynamic effects induced by a non-powered floating object navigating in an approach channel using the CFD (Computational Fluid Dynamics) process.

We are very much thankful to all the contributors to this book for their valuable and productive works. The foreword for this book has been written by Prof. Dumitru Baleanu and Prof. Jordan Hristov. We would like to express our sincere gratitude for their guidance and support.

We are extremely grateful to Ms. Humaira Hashmi and Ms. Fariya Zulfiqar, the Editorial Manager Publications of Bentham Science Publishers who helped us in the publication process. We are also extending our thanks to Bentham Science Publishers for publishing this book.

We wish that this book will be especially useful to scientists doing research in the field of fractional calculus and to researchers at graduate level in this field.

Mehmet Yavuz
Department of Mathematics and Computer Sciences
Necmettin Erbakan University
Konya, 42090


Necati Özdemir Department of Mathematics
Balıkesir University
Çağiş Campus, 10145