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Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena

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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (5 June 2022) | Viewed by 34574

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Dr. Hijaz Ahmad

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Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
Interests: numerical analysis; PDEs; fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear colleagues,

This Special Issue will cover new aspects of the recent theoretical developments in fractional calculus and its illustrative applications in applied mathematics, engineering, finance, and health sciences. It aims to point out new techniques that can be applied to the real-life problems which are modelled with fractional differential equations. Moreover, it aims to provide new analytical and numerical methods to solve the real-life problems of both integer and fractional order differential equations and to understand their complicated behaviors in nonlinear phenomena. The Special Issue also presents the cutting-edge developments in linear and nonlinear modelling, optimization and solution strategies that can be applied to engineering and biological systems.

This Issue will provide readers with new theories and methods in nonlinear dynamical systems of fractional order. It will also help in the development of new techniques for the solution of complex engineering, biological, financial, and life sciences problems. The Special Issue also aims to help readers gain new insights into novel modelling and optimization processes and understand the relation between theory and practice.

The topics of the Special Issue include, but are not limited to:

Fractional modelling in real-world phenomena;
New analytical and numerical methods for fractional differential equations;
Fractal and fractional differential equations;
Fractional derivatives with and without non-singular kernels;
Memory kernels: identification, construction, and definition of new fractional operators;
Deterministic and stochastic fractional differential equations;
Applications in bioengineering, biology, and health sciences;
Discrete fractional calculus;
Local fractional derivatives and applications;
Variable order fractional differential equations;
Fractional optimal control problems;
Fractional calculus in modelling and controller design;
Fractional variational principles;
Fractional order diffusion models;
Heat, mass, and momentum transfer (fluid dynamics) with relaxations;
Biomechanical and biomedical applications of fractional calculus;
Fractional functional differential systems;
Fractals and related topics;
Fractional impulsive systems;
Fuzzy differential equations and their applications;
Fractal signal processing and applications;
Numerical evaluation of fractional differential equations;
Fractal theory and its latest developments;
Fractal spacetime and two-scale thermodynamics.

Dr. Hijaz Ahmad
Guest Editor

Manuscript Submission Information

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Published Papers (14 papers)

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Research

26 pages, 1910 KiB

Open AccessArticle

Stochastic Optimal Control Analysis of a Mathematical Model: Theory and Application to Non-Singular Kernels

by Anwarud Din and Qura Tul Ain

Fractal Fract. 2022, 6(5), 279; https://doi.org/10.3390/fractalfract6050279 - 23 May 2022

Cited by 15 | Viewed by 2129

Abstract

Some researchers believe fractional differential operators should not have a non-singular kernel, while others strongly believe that due to the complexity of nature, fractional differential operators can have either singular or non-singular kernels. This contradiction in thoughts has led to the publication of a few papers that are against differential operators with non-singular kernels, causing some negative impacts. Thus, publishers and some Editors-in-Chief are concerned about the future of fractional calculus, which has generally brought confusion among the vibrant and innovative young researchers who desire to apply fractional calculus within their respective fields. Thus, the present work aims to develop a model based on a stochastic process that could be utilized to portray the effect of arbitrary-order derivatives. A nonlinear perturbation is used to study the proposed stochastic model with the help of white noises. The required condition(s) for the existence of an ergodic stationary distribution is obtained via Lyapunov functional theory. The finding of the study indicated that the proposed noises have a remarkable impact on the dynamics of the system. To reduce the spread of a disease, we imposed some control measures on the stochastic model, and the optimal system was achieved. The models both with and without control were coded in MATLAB, and at the conclusion of the research, numerical solutions are provided. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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11 pages, 505 KiB

Open AccessArticle

New Exact Solutions of Some Important Nonlinear Fractional Partial Differential Equations with Beta Derivative

by Erdogan Mehmet Ozkan

Fractal Fract. 2022, 6(3), 173; https://doi.org/10.3390/fractalfract6030173 - 21 Mar 2022

Cited by 24 | Viewed by 2443

Abstract

In this work, the F-expansion method is used to find exact solutions of the space-time fractional modified Benjamin Bona Mahony equation and the nonlinear time fractional Schrödinger equation with beta derivative. One of the most efficient and significant methods for obtaining new exact solutions to nonlinear equations is this method. With the aid of Maple, more exact solutions defined by the Jacobi elliptic function are obtained. Hyperbolic function solutions and some exact solutions expressed by trigonometric functions are gained in the case of m modulus 1 and 0 limits of the Jacobi elliptic function. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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13 pages, 1057 KiB

Open AccessArticle

A Chebyshev Collocation Approach to Solve Fractional Fisher–Kolmogorov–Petrovskii–Piskunov Equation with Nonlocal Condition

by Dapeng Zhou, Afshin Babaei, Seddigheh Banihashemi, Hossein Jafari, Jehad Alzabut and Seithuti P. Moshokoa

Fractal Fract. 2022, 6(3), 160; https://doi.org/10.3390/fractalfract6030160 - 15 Mar 2022

Cited by 3 | Viewed by 2213

Abstract

We provide a detailed description of a numerical approach that makes use of the shifted Chebyshev polynomials of the sixth kind to approximate the solution of some fractional order differential equations. Specifically, we choose the fractional Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) to describe this method. We write our approximate solution in the product form, which consists of unknown coefficients and shifted Chebyshev polynomials. To compute the numerical values of coefficients, we use the initial and boundary conditions and the collocation technique to create a system of equations whose number matches the unknowns. We test the applicability and accuracy of this numerical approach using two examples. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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10 pages, 295 KiB

Open AccessArticle

Estimates for a Rough Fractional Integral Operator and Its Commutators on p-Adic Central Morrey Spaces

by Naqash Sarfraz and Fahd Jarad

Fractal Fract. 2022, 6(2), 117; https://doi.org/10.3390/fractalfract6020117 - 16 Feb 2022

Cited by 4 | Viewed by 1778

Abstract

In the current paper, we obtain the boundedness of a rough p-adic fractional integral operator on p-adic central Morrey spaces. Moreover, we establish the

λ

-central bounded mean oscillations estimate for commutators of a rough p-adic fractional integral operator on [...] Read more.

In the current paper, we obtain the boundedness of a rough p-adic fractional integral operator on p-adic central Morrey spaces. Moreover, we establish the

λ

-central bounded mean oscillations estimate for commutators of a rough p-adic fractional integral operator on p-adic central Morrey spaces. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

11 pages, 2843 KiB

Open AccessArticle

A Fractional Analysis of Hyperthermia Therapy on Breast Cancer in a Porous Medium along with Radiative Microwave Heating

by Dolat Khan, Ata ur Rahman, Poom Kumam and Wiboonsak Watthayu

Fractal Fract. 2022, 6(2), 82; https://doi.org/10.3390/fractalfract6020082 - 1 Feb 2022

Cited by 3 | Viewed by 2115

Abstract

Cancer is a prominent source of mortality and morbidity globally, but little is known about how it develops and spreads. Tumor cells are unable to thrive in high-temperature environments, according to recent research. Hyperthermia is the name for this therapy method. This study provides insights into hyperthermia therapy on breast cancer in the presence of a porous material with fractional derivative access when using radiative microwave heating. The mathematical model is formulated by PDE, while the time-fractional Caputo derivative is applied to make our equation more general as compared to the classical model. To produce a more efficient analysis of blood temperature distributions inside the tissues of the breast, the unsteady state is calculated by using the Laplace transform technique. The Laplace inversion is found by Durbin’s and Zakian’s algorithms. The treatment involves mild temperature hyperthermia, which causes cell death by enhancing cell sensitivity to radiation therapy and blood flow in the tumor. The variations of different parameters to control the temperate profile during therapy are discussed; we can also see how a fractional parameter makes our study more realistic for further experimental study. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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8 pages, 321 KiB

Open AccessArticle

A Slow Single-Species Model with Non-Symmetric Variation of the Coefficients

by Fahad M. Alharbi

Fractal Fract. 2022, 6(2), 72; https://doi.org/10.3390/fractalfract6020072 - 29 Jan 2022

Cited by 3 | Viewed by 1740

Abstract

A single-species population model exhibiting a symmetric slow variation for the carrying capacity and intrinsic growth rate is evaluated explicitly. However, it is unrealistic to eliminate the possibility of a clear separation in the evolution of the biotic environmental elements; thus, this paper considers the situation where these elements have a hierarchical variation on the time scales. Accordingly, two particular situations are recognized, which are the carrying capacity varies faster than the growth rate and vice versa. Applying the multi-time scaling technique in such a system provides a small parameter, which leads us to construct analytical approximate expressions for the population behavior, using the perturbation approach. Such approximations display very good agreement with the numerical simulations. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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15 pages, 559 KiB

Open AccessArticle

Analytical Solutions of a Class of Fluids Models with the Caputo Fractional Derivative

by Ndolane Sene

Fractal Fract. 2022, 6(1), 35; https://doi.org/10.3390/fractalfract6010035 - 11 Jan 2022

Cited by 38 | Viewed by 2391

Abstract

This paper studies the analytical solutions of the fractional fluid models described by the Caputo derivative. We combine the Fourier sine and the Laplace transforms. We analyze the influence of the order of the Caputo derivative the Prandtl number, the Grashof numbers, and the Casson parameter on the dynamics of the fractional diffusion equation with reaction term and the fractional heat equation. In this paper, we notice that the order of the Caputo fractional derivative plays the retardation effect or the acceleration. The physical interpretations of the influence of the parameters of the model have been proposed. The graphical representations illustrate the main findings of the present paper. This paper contributes to answering the open problem of finding analytical solutions to the fluid models described by the fractional operators. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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26 pages, 2841 KiB

Open AccessArticle

Efficient Approaches for Solving Systems of Nonlinear Time-Fractional Partial Differential Equations

by Hegagi Mohamed Ali, Hijaz Ahmad, Sameh Askar and Ismail Gad Ameen

Fractal Fract. 2022, 6(1), 32; https://doi.org/10.3390/fractalfract6010032 - 10 Jan 2022

Cited by 10 | Viewed by 2212

Abstract

In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fractional-order in the Caputo derivative. We apply the MGMLFM and LADM on systems of nonlinear time-fractional PDEs. Precisely, we consider some important fractional-order nonlinear systems, namely Broer–Kaup (BK) and Burgers, which have found major significance because they arise in many physical applications such as shock wave, wave processes, vorticity transport, dispersal in porous media, and hydrodynamic turbulence. The analysis of these methods is implemented on the BK, Burgers systems and solutions have been offered in a simple formula. We show our results in figures and tables to demonstrate the efficiency and reliability of the used methods. Furthermore, our outcome converges rapidly to the given exact solutions. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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23 pages, 384 KiB

Open AccessArticle

On Starlike Functions of Negative Order Defined by q-Fractional Derivative

by Sadia Riaz, Ubaid Ahmed Nisar, Qin Xin, Sarfraz Nawaz Malik and Abdul Raheem

Fractal Fract. 2022, 6(1), 30; https://doi.org/10.3390/fractalfract6010030 - 6 Jan 2022

Cited by 15 | Viewed by 1764

Abstract

In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class

\tilde{S_{q}^{*}} (α)

α \in (- 3, 1]

[...] Read more.

In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class

\tilde{S_{q}^{*}} (α)

α \in (- 3, 1]

q \in (0, 1)

generalizes the class

S_{q}^{*}

of q-starlike functions and the class

\tilde{T_{q}^{*}} (α)

α \in [- 1, 1]

q \in (0, 1)

comprises the q-starlike univalent functions with negative coefficients. Some basic properties and the behavior of the functions in these classes are examined. The order of starlikeness in the class of convex function is investigated. It provides some interesting connections of newly defined classes with known classes. The mapping property of these classes under the family of q-Bernardi integral operator and its radius of univalence are studied. Additionally, certain coefficient inequalities, the radius of q-convexity, growth and distortion theorem, the covering theorem and some applications of fractional q-calculus for these new classes are investigated, and some interesting special cases are also included. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

13 pages, 1086 KiB

Open AccessArticle

Legendre Spectral Collocation Technique for Advection Dispersion Equations Included Riesz Fractional

by Mohamed M. Al-Shomrani and Mohamed A. Abdelkawy

Fractal Fract. 2022, 6(1), 9; https://doi.org/10.3390/fractalfract6010009 - 25 Dec 2021

Cited by 2 | Viewed by 2243

Abstract

The advection–dispersion equations have gotten a lot of theoretical attention. The difficulty in dealing with these problems stems from the fact that there is no perfect answer and that tackling them using local numerical methods is tough. The Riesz fractional advection–dispersion equations are quantitatively studied in this research. The numerical methodology is based on the collocation approach and a simple numerical algorithm. To show the technique’s performance and competency, a comprehensive theoretical formulation is provided, along with numerical examples. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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14 pages, 376 KiB

Open AccessArticle

Solutions of General Fractional-Order Differential Equations by Using the Spectral Tau Method

by Hari Mohan Srivastava, Daba Meshesha Gusu, Pshtiwan Othman Mohammed, Gidisa Wedajo, Kamsing Nonlaopon and Y. S. Hamed

Fractal Fract. 2022, 6(1), 7; https://doi.org/10.3390/fractalfract6010007 - 24 Dec 2021

Cited by 12 | Viewed by 3689

Abstract

Here, in this article, we investigate the solution of a general family of fractional-order differential equations by using the spectral Tau method in the sense of Liouville–Caputo type fractional derivatives with a linear functional argument. We use the Chebyshev polynomials of the second kind to develop a recurrence relation subjected to a certain initial condition. The behavior of the approximate series solutions are tabulated and plotted at different values of the fractional orders

ν

and

α

. The method provides an efficient convergent series solution form with easily computable coefficients. The obtained results show that the method is remarkably effective and convenient in finding solutions of fractional-order differential equations. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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20 pages, 353 KiB

Open AccessArticle

Weighted Midpoint Hermite-Hadamard-Fejér Type Inequalities in Fractional Calculus for Harmonically Convex Functions

by Humaira Kalsoom, Miguel Vivas-Cortez, Muhammad Amer Latif and Hijaz Ahmad

Fractal Fract. 2021, 5(4), 252; https://doi.org/10.3390/fractalfract5040252 - 2 Dec 2021

Cited by 16 | Viewed by 2268

Abstract

In this paper, we establish a new version of Hermite-Hadamard-Fejér type inequality for harmonically convex functions in the form of weighted fractional integral. Secondly, an integral identity and some weighted midpoint fractional Hermite-Hadamard-Fejér type integral inequalities for harmonically convex functions by involving a positive weighted symmetric functions have been obtained. As shown, all of the resulting inequalities generalize several well-known inequalities, including classical and Riemann–Liouville fractional integral inequalities. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

17 pages, 364 KiB

Open AccessArticle

LR-Preinvex Interval-Valued Functions and Riemann–Liouville Fractional Integral Inequalities

by Muhammad Bilal Khan, Muhammad Aslam Noor, Thabet Abdeljawad, Abd Allah A. Mousa, Bahaaeldin Abdalla and Safar M. Alghamdi

Fractal Fract. 2021, 5(4), 243; https://doi.org/10.3390/fractalfract5040243 - 29 Nov 2021

Cited by 29 | Viewed by 1962

Abstract

Convexity is crucial in obtaining many forms of inequalities. As a result, there is a significant link between convexity and integral inequality. Due to the significance of these concepts, the purpose of this study is to introduce a new class of generalized convex interval-valued functions called LR-preinvex interval-valued functions (LR-preinvex I-V-Fs) and to establish Hermite–Hadamard type inequalities for LR-preinvex I-V-Fs using partial order relation (

\leq_{p}

). Furthermore, we demonstrate that our results include a large class of new and known inequalities for LR-preinvex interval-valued functions and their variant forms as special instances. Further, we give useful examples that demonstrate usefulness of the theory produced in this study. These findings and diverse approaches may pave the way for future research in fuzzy optimization, modeling, and interval-valued functions. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

10 pages, 392 KiB

Open AccessArticle

A Special Study of the Mixed Weighted Fractional Brownian Motion

by Anas D. Khalaf, Anwar Zeb, Tareq Saeed, Mahmoud Abouagwa, Salih Djilali and Hashim M. Alshehri

Fractal Fract. 2021, 5(4), 192; https://doi.org/10.3390/fractalfract5040192 - 31 Oct 2021

Cited by 8 | Viewed by 2079

Abstract

In this work, we present the analysis of a mixed weighted fractional Brownian motion, defined by

η_{t} : = B_{t} + ξ_{t}

, where B is a Brownian motion and

ξ

is an independent weighted fractional Brownian motion. We also [...] Read more.

In this work, we present the analysis of a mixed weighted fractional Brownian motion, defined by

η_{t} : = B_{t} + ξ_{t}

, where B is a Brownian motion and

ξ

is an independent weighted fractional Brownian motion. We also consider the parameter estimation problem for the drift parameter

θ > 0

in the mixed weighted fractional Ornstein–Uhlenbeck model of the form

X_{0} = 0; X_{t} = θ X_{t} d t + d η_{t}

. Moreover, a simulation is given of sample paths of the mixed weighted fractional Ornstein–Uhlenbeck process. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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