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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 18955

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Prof. Dr. Sameerah Jamal

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School of Mathematics, University of the Witwatersrand, Johannesburg 2001, South Africa
Interests: differential equations; symmetries; conservation laws; exact solutions; cosmology
Special Issues, Collections and Topics in MDPI journals

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Dear Colleagues,

Fractional order derivatives have had a revolutionary impact on the scientific community. Its study has grown in leaps and bounds, from analytical methods to numerical techniques. Consequently, applications of fractional order differential equations are now widespread across every possible area of research.

The focus of this Special Issue is on the advancement of research on fractional order derivatives and their multi-faceted applications. Topics that are invited for submission include (but are not limited to):

Mathematical modeling with fractional order derivatives;
Symmetry analysis of fractional order equations;
Conserved quantities related to fractional order models;
The various solution techniques for fractional order equations;
Special functions that are linked to the solution of fractional order equations;
Software to aid computations and analysis for fractional order derivatives and equations.

Prof. Dr. Sameerah Jamal
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

fractional order derivatives
fractional calculus
Caputo derivatives
Riemann–Liouville derivatives
numerical analysis
modeling
application

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Advances in Fractional Order Derivatives and Their Applications, 3rd Edition in Fractal and Fractional (1 article)

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12 pages, 1455 KiB

Open AccessArticle

Fractional Pricing Models: Transformations to a Heat Equation and Lie Symmetries

by Reginald Champala, Sameerah Jamal and Suhail Khan

Fractal Fract. 2023, 7(8), 632; https://doi.org/10.3390/fractalfract7080632 - 19 Aug 2023

Cited by 2 | Viewed by 1110

Abstract

The study of fractional partial differential equations is often plagued with complicated models and solution processes. In this paper, we tackle how to simplify a specific parabolic model to facilitate its analysis and solution process. That is, we investigate a general time-fractional pricing equation, and propose new transformations to reduce the underlying model to a different but equivalent problem that is less challenging. Our procedure leads to a conversion of the model to a fractional 1 + 1 heat transfer equation, and more importantly, all the transformations are invertible. A significant result which emerges is that we prove such transformations yield solutions under the Riemann–Liouville and Caputo derivatives. Furthermore, Lie point symmetries are necessary to construct solutions to the model that incorporate the behaviour of the underlying financial assets. In addition, various graphical explorations exemplify our results. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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34 pages, 10758 KiB

Open AccessArticle

Image Enhancement Model Based on Fractional Time-Delay and Diffusion Tensor

by Wenjuan Yao, Yi Huang, Boying Wu and Zhongxiang Zhou

Fractal Fract. 2023, 7(8), 569; https://doi.org/10.3390/fractalfract7080569 - 25 Jul 2023

Viewed by 1068

Abstract

Image enhancement is one of the bases of image processing technology, which can enhance useful features and suppress useless information of images according to the specified task. In order to ensure coherent enhancement for images with oriented flow-like structures, we propose a nonlinear diffusion system model based on time-fractional delay. By combining the nonlinear isotropic diffusion equation with fractional time-delay regularization, we construct a structure tensor. Meanwhile, the introduction of source terms enhances the contrast of the image, making it effective for denoising images with high-level noise. Based on compactness principles, the existence of weak solutions for the model is proved by using the Galerkin method. In addition, various experimental results verify the enhancement ability of the proposed model. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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23 pages, 24623 KiB

Open AccessArticle

Anomalous Thermally Induced Deformation in Kelvin–Voigt Plate with Ultrafast Double-Strip Surface Heating

by Emad Awad, Sharifah E. Alhazmi, Mohamed A. Abdou and Mohsen Fayik

Fractal Fract. 2023, 7(7), 563; https://doi.org/10.3390/fractalfract7070563 - 22 Jul 2023

Cited by 4 | Viewed by 1558

Abstract

The Jeffreys-type heat conduction equation with flux precedence describes the temperature of diffusive hot electrons during the electron–phonon interaction process in metals. In this paper, the deformation resulting from ultrafast surface heating on a “nanoscale” plate is considered. The focus is on the anomalous heat transfer mechanisms that result from anomalous diffusion of hot electrons and are characterized by retarded thermal conduction, accelerated thermal conduction, or transition from super-thermal conductivity in the short-time response to sub-thermal conductivity in the long-time response and described by the fractional Jeffreys equation with three fractional parameters. The recent double-strip problem, Awad et al., Eur. Phy. J. Plus 2022, allowing the overlap between two propagating thermal waves, is generalized from the semi-infinite heat conductor case to thermoelastic case in the finite domain. The elastic response in the material is not simultaneous (i.e., not Hookean), rather it is assumed to be of the Kelvin–Voigt type, i.e.,

σ = E (ε + τ_{ε} \dot{ε})

, where

σ

refers to the stress,

ε

is the strain,

E

is the Young modulus, and

τ_{ε}

refers to the strain relaxation time. The delayed strain response of the Kelvin–Voigt model eliminates the discontinuity of stresses, a hallmark of the Hookean solid. The immobilization of thermal conduction described by the ordinary Jeffreys equation of heat conduction is salient in metals when the heat flux precedence is considered. The absence of the finite speed thermal waves in the Kelvin–Voigt model results in a smooth stress surface during the heating process. The temperature contours and the displacement vector chart show that the anomalous heat transfer characterized by retardation or crossover from super- to sub-thermal conduction may disrupt the ultrafast laser heating of metals. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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13 pages, 1287 KiB

Open AccessArticle

Fractional Gradient Optimizers for PyTorch: Enhancing GAN and BERT

by Oscar Herrera-Alcántara and Josué R. Castelán-Aguilar

Fractal Fract. 2023, 7(7), 500; https://doi.org/10.3390/fractalfract7070500 - 23 Jun 2023

Cited by 3 | Viewed by 1397

Abstract

Machine learning is a branch of artificial intelligence that dates back more than 50 years. It is currently experiencing a boom in research and technological development. With the rise of machine learning, the need to propose improved optimizers has become more acute, leading to the search for new gradient-based optimizers. In this paper, the ancient concept of fractional derivatives has been applied to some optimizers available in PyTorch. A comparative study is presented to show how the fractional versions of gradient optimizers could improve their performance on generative adversarial networks (GAN) and natural language applications with Bidirectional Encoder Representations from Transformers (BERT). The results are encouraging for both state-of-the art algorithms, GAN and BERT, and open up the possibility of exploring further applications of fractional calculus in machine learning. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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15 pages, 5332 KiB

Open AccessArticle

The Effect of Fractional Derivatives on Thermo-Mechanical Interaction in Biological Tissues during Hyperthermia Treatment Using Eigenvalues Approach

by Aatef Hobiny and Ibrahim Abbas

Fractal Fract. 2023, 7(6), 432; https://doi.org/10.3390/fractalfract7060432 - 26 May 2023

Cited by 6 | Viewed by 1268

Abstract

This article studies the effects of fractional time derivatives on thermo-mechanical interaction in living tissue during hyperthermia treatment by using the eigenvalues approach. A comprehensive understanding of the heat transfer mechanism and the related thermo-mechanical interactions with the patient’s living tissues is crucial for the effective implementation of thermal treatment procedures. The surface of living tissues is traction-free and is exposed to a pulse boundary heat flux that decays exponentially. The Laplace transforms and their associated techniques are applied to the generalized bio-thermo-elastic model, and analytical procedures are then implemented. The eigenvalue approach is utilized to obtain the solution of governing equations. Graphical representations are given for the temperature, the displacement, and the thermal stress results. Afterward, a parametric study was carried out to determine the best method for selecting crucial design parameters that can improve the precision of hyperthermia therapies. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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15 pages, 739 KiB

Open AccessArticle

Combined Liouville–Caputo Fractional Differential Equation

by McSylvester Ejighikeme Omaba, Hamdan Al Sulaimani, Soh Edwin Mukiawa, Cyril Dennis Enyi and Tijani Abdul-Aziz Apalara

Fractal Fract. 2023, 7(5), 366; https://doi.org/10.3390/fractalfract7050366 - 28 Apr 2023

Cited by 1 | Viewed by 1190

Abstract

This paper studies a fractional differential equation combined with a Liouville–Caputo fractional differential operator, namely, [...] Read more.

This paper studies a fractional differential equation combined with a Liouville–Caputo fractional differential operator, namely,

^{L C} D_{η}^{β, γ} Q (t) = λ ϑ (t, Q (t)), t \in [c, d], β, γ \in (0, 1], η \in [0, 1],

where

Q (c) = q_{c}

is a bounded and non-negative initial value. The function

ϑ : [c, d] \times R \to R

is Lipschitz continuous in the second variable,

λ > 0

is a constant and the operator

^{L C} D_{η}^{β, γ}

is a convex combination of the left and the right Liouville–Caputo fractional derivatives. We study the well-posedness using the fixed-point theorem, estimate the growth bounds of the solution and examine the asymptotic behaviours of the solutions. Our findings are illustrated with some analytical and numerical examples. Furthermore, we investigate the effect of noise on the growth behaviour of the solution to the combined Liouville–Caputo fractional differential equation. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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15 pages, 436 KiB

Open AccessArticle

Lie Symmetries and Third- and Fifth-Order Time-Fractional Polynomial Evolution Equations

by Jollet Truth Kubayi and Sameerah Jamal

Fractal Fract. 2023, 7(2), 125; https://doi.org/10.3390/fractalfract7020125 - 29 Jan 2023

Cited by 6 | Viewed by 1304

Abstract

This paper is concerned with a class of ten time-fractional polynomial evolution equations. The one-parameter Lie point symmetries of these equations are found and the symmetry reductions are provided. These reduced equations are transformed into nonlinear ordinary differential equations, which are challenging to [...] Read more.

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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12 pages, 14347 KiB

Open AccessArticle

A New Perspective on the Exact Solutions of the Local Fractional Modified Benjamin–Bona–Mahony Equation on Cantor Sets

by Kang-Jia Wang and Feng Shi

Fractal Fract. 2023, 7(1), 72; https://doi.org/10.3390/fractalfract7010072 - 9 Jan 2023

Cited by 20 | Viewed by 1550

Abstract

A new local fractional modified Benjamin–Bona–Mahony equation is proposed within the local fractional derivative in this study for the first time. By defining some elementary functions via the Mittag–Leffler function (MLF) on the Cantor sets (CSs), a set of nonlinear local fractional ordinary differential equations (NLFODEs) is constructed. Then, a fast algorithm namely Yang’s special function method is employed to find the non-differentiable (ND) exact solutions. By this method, we can extract abundant exact solutions in just one step. Finally, the obtained solutions on the CS are outlined in the form of the 3-D plot. The whole calculation process clearly shows that Yang’s special function method is simple and effective, and can be applied to investigate the exact ND solutions of the other local fractional PDEs. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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12 pages, 339 KiB

Open AccessArticle

New Results for Homoclinic Fractional Hamiltonian Systems of Order α∈(1/2,1]

by Abdelkader Moumen, Hamid Boulares, Jehad Alzabut, Fathi Khelifi and Moheddine Imsatfia

Fractal Fract. 2023, 7(1), 39; https://doi.org/10.3390/fractalfract7010039 - 29 Dec 2022

Viewed by 1150

Abstract

In this manuscript, we are interested in studying the homoclinic solutions of fractional Hamiltonian system of the form [...] Read more.

In this manuscript, we are interested in studying the homoclinic solutions of fractional Hamiltonian system of the form

- D_{\infty}^{α}_{ς} (D_{ς}^{α}_{- \infty} Z (ς)) - A (ς) Z (ς) + \nabla ω (ς, Z (ς)) = 0,

where

α \in (\frac{1}{2}, 1]

Z \in H^{α} (R, R^{N})

and

ω \in C^{1} (R \times R^{N}, R)

are not periodic in

ς

. The characteristics of the critical point theory are used to illustrate the primary findings. Our results substantially improve and generalize the most recent results of the proposed system. We conclude our study by providing an example to highlight the significance of the theoretical results. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

14 pages, 922 KiB

Open AccessArticle

Soliton Solution of the Peyrard–Bishop–Dauxois Model of DNA Dynamics with M-Truncated and β-Fractional Derivatives Using Kudryashov’s R Function Method

by Xiaoming Wang, Ghazala Akram, Maasoomah Sadaf, Hajra Mariyam and Muhammad Abbas

Fractal Fract. 2022, 6(10), 616; https://doi.org/10.3390/fractalfract6100616 - 21 Oct 2022

Cited by 13 | Viewed by 1408

Abstract

In this paper, the Peyrard–Bishop–Dauxois model of DNA dynamics is discussed along with the fractional effects of the M-truncated derivative and

β

-derivative. The Kudryashov’s R method was applied to the model in order to obtain a solitary wave solution. The obtained solution [...] Read more.

In this paper, the Peyrard–Bishop–Dauxois model of DNA dynamics is discussed along with the fractional effects of the M-truncated derivative and

β

-derivative. The Kudryashov’s R method was applied to the model in order to obtain a solitary wave solution. The obtained solution is explained graphically and the fractional effects of the

β

and M-truncated derivatives are also shown for a better understanding of the model. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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13 pages, 361 KiB

Open AccessArticle

Collocation Method for Optimal Control of a Fractional Distributed System

by Wen Cao and Yufeng Xu

Fractal Fract. 2022, 6(10), 594; https://doi.org/10.3390/fractalfract6100594 - 14 Oct 2022

Cited by 1 | Viewed by 1259

Abstract

In this paper, a collocation method based on the Jacobi polynomial is proposed for a class of optimal-control problems of a fractional distributed system. By using the Lagrange multiplier technique and fractional variational principle, the stated problem is reduced to a system of fractional partial differential equations about control and state functions. The uniqueness of this fractional coupled system is discussed. For spatial second-order derivatives, the proposed method takes advantage of Jacobi polynomials with different parameters to approximate solutions. For a temporal fractional derivative in the Caputo sense, choosing appropriate basis functions allows the collocation method to be implemented easily and efficiently. Exponential convergence is verified numerically under continuous initial conditions. As a particular example, the relation between the state function and the order of the fractional derivative is analyzed with a discontinuous initial condition. Moreover, the numerical results show that the integration of the state function will decay as the order of the fractional derivative decreases. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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24 pages, 6947 KiB

Open AccessArticle

The Nonlinear Dynamics Characteristics and Snap-Through of an SD Oscillator with Nonlinear Fractional Damping

by Minghao Wang, Enli Chen, Ruilan Tian and Cuiyan Wang

Fractal Fract. 2022, 6(9), 493; https://doi.org/10.3390/fractalfract6090493 - 2 Sep 2022

Cited by 5 | Viewed by 1727

Abstract

A smooth and discontinuous (SD) oscillator is a typical multi-stable state system with strong nonlinear properties and has been widely used in many fields. The nonlinear dynamic characteristics of the system have not been thoroughly investigated because the nonlinear restoring force cannot be integrated. In this paper, the nonlinear restoring force is represented by a piecewise nonlinear function. The equivalent coefficients of fractional damping are obtained with an orthogonal function. The influence of fractional damping on the transition set, the amplitude–frequency response and the snap-through of the SD oscillator are analyzed. The conclusions are as follows: The nonlinear piecewise function accurately mimics the nonlinear restoring force and maintains a nonlinearity property. Fractional damping can significantly affect the stiffness and damping property simultaneously. The equivalent coefficients of the fractional damping are variable with regard to the fractional-order power of the excitation frequency. A hysteresis point, a bifurcation point, a frequency island, pitchfork bifurcations and transcritical bifurcations were discovered in the small-amplitude resonant region. In the non-resonant region, the increase in the fractional parameters leads to the probability of snap-through declining by increasing the symmetry of the attraction domain or reducing the number of stable states. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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9 pages, 294 KiB

Open AccessArticle

Symmetry Analysis and PT-Symmetric Extension of the Fifth-Order Korteweg-de Vries-Like Equation

by Gangwei Wang, Bo Shen, Mengyue He, Fei Guan and Lihua Zhang

Fractal Fract. 2022, 6(9), 468; https://doi.org/10.3390/fractalfract6090468 - 26 Aug 2022

Cited by 2 | Viewed by 1368

Abstract

In the present paper, PT-symmetric extension of the fifth-order Korteweg-de Vries-like equation are investigated. Several special equations with PT symmetry are obtained by choosing different values, for which their symmetries are obtained simultaneously. In particular, for the particular equation, its conservation laws are [...] Read more.

\ddot{a}

cklund transformations of conservation laws of momentum and energy are presented for the first time. The important thing is that for the special case of

ϵ = 3

, the corresponding time fractional case are studied by Lie group method. And what is interesting is that the symmetry of the time fractional equation is obtained, and based on the symmetry, this equation is reduced to a fractional ordinary differential equation. Finally, for the general case, the symmetry of this equation is obtained, and based on the symmetry, the reduced equation is presented. Through the results obtained in this paper, it can be found that the Lie group method is a very effective method, which can be used to deal with many models in natural phenomena. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications)

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