Special Issue "New Challenges Arising in Engineering Problems with Fractional and Integer Order"

A special issue of Fractal and Fractional (ISSN 2504-3110).

Deadline for manuscript submissions: closed (1 December 2020).

Special Issue Editors

Prof. Dr. Haci Mehmet Baskonus
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Guest Editor
Department of Mathematics and Science Education, Faculty of Education, Harran University, 63100 Sanliurfa, Turkey
Interests: partial differential equations; analytical methods; numerical methods; ordinary differential equations; fractional calculus; epidemiological models
Prof. Dr. Luis Manuel Sánchez Ruiz
E-Mail
Guest Editor
ETSID-Departamento de Matemática Aplicada & CITG, Universitat Politecnica de Valencia, E-46022 Valencia, Spain
Interests: applied mathematics; engineering education; functional analysis with theoretical and applied
Prof. Dr. Armando Ciancio
E-Mail
Guest Editor
Department of Biomedical and Dental Sciences and Morphofunctional Imaging, University of Messina, 98125 Messina, Italy
Interests: time series based on wavelets; analysis of solutions in the field of physical-mathematical models of rheological media; fractional calculus; mathematical models in economics and finance; physical-mathematical models for biological media and applications to biotechnological and medical sciences

Special Issue Information

Dear Colleagues,

Recently, many new models have been developed that deal with real-world problems that are seen as serious threats to the future of human kind. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done on fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world.

The focus of this Special Issue will be on reviewing new developments based on fractional differentiation and integration, both with respect to theoretical and numerical aspects.

This Special Issue is a place for experts to share new ideas on theories, applications, numerical and analytical methods and simulations of fractional calculus and fractional differential equations, as well as integer order. Topics of interest are defined below, and submissions relating to relevant fields are welcome.

  • New analytical and numerical methods to solve partial differential equations
  • Computational methods for fractional differential equations
  • Analysis, modeling, and control of phenomena in the following:
    • Electrical engineering;
    • Fluids dynamics and thermal engineering;
    • Mechanics;
    • Biology;
    • Physics;
    • Applied sciences;
    • Computer science.
  • Engineering problems
  • Deterministic and stochastic fractional order models

This Special Issue is organized together with the 5th International Conference on Computational Mathematics and Engineering Sciences (CMES-2020) (1–3 July 2020, Van, Turkey); hence, participants in CMES-2020 are especially welcome to submit their contributions. However, this Special Issue will accept contributions from all authors, not just conference participants.

Prof. Dr. Haci Mehmet Baskonus
Prof. Dr. Luis Manuel Sánchez Ruiz
Prof. Dr. Armando Ciancio
Guest Editors

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 papers will be 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 quarterly 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 1400 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.

Published Papers (13 papers)

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Editorial

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Editorial
New Challenges Arising in Engineering Problems with Fractional and Integer Order
Fractal Fract. 2021, 5(2), 35; https://doi.org/10.3390/fractalfract5020035 - 19 Apr 2021
Viewed by 390
Abstract
Mathematical models have been frequently studied in recent decades in order to obtain the deeper properties of real-world problems [...] Full article

Research

Jump to: Editorial

Article
A New Approach for the Fractional Integral Operator in Time Scales with Variable Exponent Lebesgue Spaces
Fractal Fract. 2021, 5(1), 7; https://doi.org/10.3390/fractalfract5010007 - 08 Jan 2021
Cited by 2 | Viewed by 744
Abstract
Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the [...] Read more.
Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the integral operators are examined. However, the norm of integral operators on time scales has been a matter of curiosity to us. In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-α integral operator Ma,δc to the norm of the centered fractional maximal diamond-α integral operator Mac on time scales with variable exponent Lebesgue spaces. This study will lead to the study of problems such as the boundedness and compactness of integral operators on time scales. Full article
Article
Extraction Complex Properties of the Nonlinear Modified Alpha Equation
Fractal Fract. 2021, 5(1), 6; https://doi.org/10.3390/fractalfract5010006 - 07 Jan 2021
Cited by 1 | Viewed by 464
Abstract
This paper applies one of the special cases of auxiliary method, which is named as the Bernoulli sub-equation function method, to the nonlinear modified alpha equation. The characteristic properties of these solutions, such as complex and soliton solutions, are extracted. Moreover, the strain [...] Read more.
This paper applies one of the special cases of auxiliary method, which is named as the Bernoulli sub-equation function method, to the nonlinear modified alpha equation. The characteristic properties of these solutions, such as complex and soliton solutions, are extracted. Moreover, the strain conditions of solutions are also reported in detail. Observing the figures plotted by considering various values of parameters of these solutions confirms the effectiveness of the approximation method used for the governing model. Full article
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Article
A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems
Fractal Fract. 2020, 4(4), 53; https://doi.org/10.3390/fractalfract4040053 - 24 Nov 2020
Cited by 2 | Viewed by 487
Abstract
In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, [...] Read more.
In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to α and β are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms. Full article
Article
Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact
Fractal Fract. 2020, 4(3), 43; https://doi.org/10.3390/fractalfract4030043 - 21 Aug 2020
Cited by 7 | Viewed by 847
Abstract
The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly finance system. In the developed system, we add the critical minimum interest rate d parameter in order to develop [...] Read more.
The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly finance system. In the developed system, we add the critical minimum interest rate d parameter in order to develop a new stable financial model. The new emerging paradigm increases the demand for innovation, which is the gateway to the knowledge economy. The derivatives are characterized in the Caputo fractional order derivative and Atangana-Baleanu derivative. We prove the existence and uniqueness of the solutions with fixed point theorem and an iterative scheme. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the financial system’s actual macroeconomic behavior. Specifically component of its application to the large scale and smaller scale forms, just as the utilization of specific strategies and instruments such fractal stochastic procedures and expectation. Full article
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Article
Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation
Fractal Fract. 2020, 4(3), 41; https://doi.org/10.3390/fractalfract4030041 - 16 Aug 2020
Cited by 24 | Viewed by 1133
Abstract
This manuscript focuses on the application of the (m+1/G)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are derived. Various figures such as 3D and 2D surfaces with the selecting the suitable of parameter values are plotted. Full article
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Article
Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate
Fractal Fract. 2020, 4(3), 35; https://doi.org/10.3390/fractalfract4030035 - 16 Jul 2020
Cited by 29 | Viewed by 1378
Abstract
In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral [...] Read more.
In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense. Full article
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Article
On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions
Fractal Fract. 2020, 4(3), 33; https://doi.org/10.3390/fractalfract4030033 - 09 Jul 2020
Cited by 2 | Viewed by 687
Abstract
In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce [...] Read more.
In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here. Full article
Article
Laplace Transform Method for Economic Models with Constant Proportional Caputo Derivative
Fractal Fract. 2020, 4(3), 30; https://doi.org/10.3390/fractalfract4030030 - 03 Jul 2020
Cited by 7 | Viewed by 791
Abstract
In this study, we solved the economic models based on market equilibrium with constant proportional Caputo derivative using the Laplace transform. We proved the accuracy and efficiency of the method. We constructed the relations between the solutions of the problems and bivariate Mittag–Leffler [...] Read more.
In this study, we solved the economic models based on market equilibrium with constant proportional Caputo derivative using the Laplace transform. We proved the accuracy and efficiency of the method. We constructed the relations between the solutions of the problems and bivariate Mittag–Leffler functions. Full article
Article
Numerical Solution of Fractional Order Burgers’ Equation with Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method
Fractal Fract. 2020, 4(2), 27; https://doi.org/10.3390/fractalfract4020027 - 19 Jun 2020
Cited by 3 | Viewed by 697
Abstract
In this research, obtaining of approximate solution for fractional-order Burgers’ equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. [...] Read more.
In this research, obtaining of approximate solution for fractional-order Burgers’ equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective. Full article
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Article
On the Fractional Maximal Delta Integral Type Inequalities on Time Scales
Fractal Fract. 2020, 4(2), 26; https://doi.org/10.3390/fractalfract4020026 - 17 Jun 2020
Cited by 3 | Viewed by 642
Abstract
Time scales have been the target of work of many mathematicians for more than a quarter century. Some of these studies are of inequalities and dynamic integrals. Inequalities and fractional maximal integrals have an important place in these studies. For example, inequalities and [...] Read more.
Time scales have been the target of work of many mathematicians for more than a quarter century. Some of these studies are of inequalities and dynamic integrals. Inequalities and fractional maximal integrals have an important place in these studies. For example, inequalities and integrals contributed to the solution of many problems in various branches of science. In this paper, we will use fractional maximal integrals to establish integral inequalities on time scales. Moreover, our findings show that inequality is valid for discrete and continuous conditions. Full article
Article
Exact Solution of Two-Dimensional Fractional Partial Differential Equations
Fractal Fract. 2020, 4(2), 21; https://doi.org/10.3390/fractalfract4020021 - 12 May 2020
Cited by 5 | Viewed by 798
Abstract
In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the [...] Read more.
In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the Sumudu transform method and decomposition method. The fractional derivative is described in the Caputo sense. The results obtained show that the approach is easy to implement and accurate when applied to various fractional differential equations. Full article
Article
Fractional Kinetic Equations Associated with Incomplete I-Functions
Fractal Fract. 2020, 4(2), 19; https://doi.org/10.3390/fractalfract4020019 - 04 May 2020
Cited by 6 | Viewed by 714
Abstract
In this paper, we investigate the solution of fractional kinetic equation (FKE) associated with the incomplete I-function (IIF) by using the well-known integral transform (Laplace transform). The FKE plays a great role in solving astrophysical problems. The solutions are represented in terms [...] Read more.
In this paper, we investigate the solution of fractional kinetic equation (FKE) associated with the incomplete I-function (IIF) by using the well-known integral transform (Laplace transform). The FKE plays a great role in solving astrophysical problems. The solutions are represented in terms of IIF. Next, we present some interesting corollaries by specializing the parameters of IIF in the form of simpler special functions and also mention a few known results, which are very useful in solving physical or real-life problems. Finally, some graphical results are presented to demonstrate the influence of the order of the fractional integral operator on the reaction rate. Full article
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