New Challenges Arising in Engineering Problems with Fractional and Integer Order III

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

Deadline for manuscript submissions: closed (30 September 2023) | Viewed by 10184

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ETSID-Department of Applied Mathematics, Universitat Politecnica de Valencia, 46022 Valencia, Spain
Interests: fractional calculus; analytical and computational methods; differential and difference equations; real and complex analysis; applied and computational mathematics; mathematical physics
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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 Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Recently, many new models have been developed to address real-world problems which represent serious threats to the future of humankind. These result from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research conducted 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, with respect to both theoretical and numerical aspects.

This Special Issue is a place for experts to share new ideas on theories, applications, and 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 areas:
    • 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 has been organized together with the 7th International Conference on Computational Mathematics and Engineering Sciences (CMES-2023) (20–21 May 2023, Elazig, Turkey); hence, participants in CMES-2023 are especially welcome to submit their contributions. However, we will accept contributions from all authors, not just conference participants, for Special Issue.

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

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Published Papers (7 papers)

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Research

12 pages, 890 KiB  
Article
A New Hybrid Block Method for Solving First-Order Differential System Models in Applied Sciences and Engineering
by Mufutau Ajani Rufai, Bruno Carpentieri and Higinio Ramos
Fractal Fract. 2023, 7(10), 703; https://doi.org/10.3390/fractalfract7100703 - 24 Sep 2023
Viewed by 993
Abstract
This paper presents a new hybrid block method formulated in variable stepsize mode to solve some first-order initial value problems of ODEs and time-dependent partial differential equations in applied sciences and engineering. The proposed method is implemented considering an adaptive stepsize strategy to [...] Read more.
This paper presents a new hybrid block method formulated in variable stepsize mode to solve some first-order initial value problems of ODEs and time-dependent partial differential equations in applied sciences and engineering. The proposed method is implemented considering an adaptive stepsize strategy to maintain the estimated error in each step within a specified tolerance. In order to evaluate the performance and usefulness of the proposed technique in real-world applications, several differential problems from applied sciences and engineering, such as the SIR model, Jacobi elliptic function problem, and chemical reactions problems, are solved numerically. The results of numerical simulations in this work demonstrate that the proposed method is more efficient than other existing numerical methods used for comparisons. Full article
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28 pages, 910 KiB  
Article
Solving General Fractional Lane-Emden-Fowler Differential Equations Using Haar Wavelet Collocation Method
by Kholoud Saad Albalawi, Ashish Kumar, Badr Saad Alkahtani and Pranay Goswami
Fractal Fract. 2023, 7(8), 628; https://doi.org/10.3390/fractalfract7080628 - 17 Aug 2023
Viewed by 1183
Abstract
This paper aims to solve general fractional Lane-Emden-Fowler differential equations using the Haar wavelet collocation method. This method transforms the fractional differential equation into a nonlinear system of equations, which is further solved for Haar coefficients using Newton’s method. We have constructed the [...] Read more.
This paper aims to solve general fractional Lane-Emden-Fowler differential equations using the Haar wavelet collocation method. This method transforms the fractional differential equation into a nonlinear system of equations, which is further solved for Haar coefficients using Newton’s method. We have constructed the higher-order Lane-Emden-Fowler equations. We have also discussed the convergence rate and stability analysis of our technique. We have explained the applications and numerically simulated the examples graphically and in tabular format to elaborate on the accuracy and efficiency of this approach. Full article
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16 pages, 323 KiB  
Article
Analysis of Cauchy Problems and Diffusion Equations Associated with the Hilfer–Prabhakar Fractional Derivative via Kharrat–Toma Transform
by Ved Prakash Dubey, Jagdev Singh, Sarvesh Dubey and Devendra Kumar
Fractal Fract. 2023, 7(5), 413; https://doi.org/10.3390/fractalfract7050413 - 20 May 2023
Cited by 9 | Viewed by 872
Abstract
In this paper, the Kharrat–Toma transforms of the Prabhakar integral, a Hilfer–Prabhakar (HP) fractional derivative, and the regularized version of the HP fractional derivative are derived. Moreover, we also compute the solution of some Cauchy problems and diffusion equations modeled with the HP [...] Read more.
In this paper, the Kharrat–Toma transforms of the Prabhakar integral, a Hilfer–Prabhakar (HP) fractional derivative, and the regularized version of the HP fractional derivative are derived. Moreover, we also compute the solution of some Cauchy problems and diffusion equations modeled with the HP fractional derivative via Kharrat–Toma transform. The solutions of Cauchy problems and the diffusion equations modeled with the HP fractional derivative are computed in the form of the generalized Mittag–Leffler function. Full article
39 pages, 669 KiB  
Article
Exploring Dynamics and Hopf Bifurcation of a Fractional-Order Bertrand Duopoly Game Model Incorporating Both Nonidentical Time Delays
by Ying Li, Peiluan Li, Changjin Xu and Yuke Xie
Fractal Fract. 2023, 7(5), 352; https://doi.org/10.3390/fractalfract7050352 - 26 Apr 2023
Cited by 3 | Viewed by 2284
Abstract
In order to maximize benefits, oligopolistic competition often occurs in contemporary society. Establishing the mathematical models to reveal the law of market competition has become a vital topic. In the current study, on the basis of the earlier publications, we propose a new [...] Read more.
In order to maximize benefits, oligopolistic competition often occurs in contemporary society. Establishing the mathematical models to reveal the law of market competition has become a vital topic. In the current study, on the basis of the earlier publications, we propose a new fractional-order Bertrand duopoly game model incorporating both nonidentical time delays. The dynamics involving existence and uniqueness, non-negativeness, and boundedness of solution to the considered fractional-order Bertrand duopoly game model are systematacially analyzed via the Banach fixed point theorem, mathematical analysis technique, and construction of an appropriate function. Making use of different delays as bifurcation parameters, several sets of new stability and bifurcation conditions ensuring the stability and the creation of Hopf bifurcation of the established fractional-order Bertrand duopoly game model are acquired. By virtue of a proper definite function, we set up a new sufficient condition that ensures globally asymptotically stability of the considered fractional-order Bertrand duopoly game model. The work reveals the impact of different types of delays on the stability and Hopf bifurcation of the proposed fractional-order Bertrand duopoly game model. The study shows that we can adjust the delay to achieve price balance of different products. To confirm the validity of the derived criteria, we put computer simulation into effect. The derived conclusions in this article are wholly new and have great theoretical value in administering companies. Full article
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12 pages, 879 KiB  
Article
New Fractional Cancer Mathematical Model via IL-10 Cytokine and Anti-PD-L1 Inhibitor
by Esmehan Uçar and Necati Özdemir
Fractal Fract. 2023, 7(2), 151; https://doi.org/10.3390/fractalfract7020151 - 03 Feb 2023
Cited by 5 | Viewed by 1395
Abstract
In this study, we explore a recent biological model created to analyze the behavior of cancer cells by administering a dose of a drug containing anti-PD-L1 and IL-10 with the Caputo and Atangana–Baleanu derivative in the Caputo sense (ABC). Using the Caputo derivative [...] Read more.
In this study, we explore a recent biological model created to analyze the behavior of cancer cells by administering a dose of a drug containing anti-PD-L1 and IL-10 with the Caputo and Atangana–Baleanu derivative in the Caputo sense (ABC). Using the Caputo derivative in order to examine the stability of the non-linear system, we are able to demonstrate that it is existent and unique, and to introduce several numeric data obtained for the fractional values in MATLAB by using the Adams–Bashforth–Moulton (ABM) method. Additionally, by using the predictor–corrector approach, the numerical results from the system with ABC derivative will be produced. As a result, it has been observed that immune system cells that are exposed to single-dose drug with fractional order effectively combat cancer cells. The tumor cells decrease by 70.44% and 80.16% for the system generalized by the Caputo and ABC derivative, respectively, for the order α=0.42. Full article
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38 pages, 1795 KiB  
Article
Comprehensive Investigation of Thermal and Flow Features of Alloy Based Nanofluid Considering Shape and Newtonian Heating Effects via New Fractional Approach
by Shah Muhammad, Talha Anwar, Asifa and Mehmet Yavuz
Fractal Fract. 2023, 7(2), 150; https://doi.org/10.3390/fractalfract7020150 - 03 Feb 2023
Cited by 4 | Viewed by 952
Abstract
The core purpose of this work is the formulation of a mathematical model by dint of a new fractional modeling approach to study the dynamics of flow and heat transfer phenomena. This approach involves the incorporation of the Prabhakar fractional operator in mathematical [...] Read more.
The core purpose of this work is the formulation of a mathematical model by dint of a new fractional modeling approach to study the dynamics of flow and heat transfer phenomena. This approach involves the incorporation of the Prabhakar fractional operator in mathematical analysis to transform the governing system from a conventional framework to a generalized one. This generalized model evaluates the improvement in thermal efficacy of vacuum pump oil because of the inclusion of aluminum alloy nanoparticles. The flow of the under-observation nanofluid starts due to the combined effects of natural convection and the ramped velocity function at the boundary. Meanwhile, an analysis of the energy equation is conducted by taking the Newtonian heating mechanism into consideration. The characteristics of platelet-, brick-, cylinder-, and blade-shaped alloy nanoparticles are incorporated into the primary system using shape-dependent relations for thermal conductivity and viscosity. Both the classical and generalized models are solved to derive the exact solutions by first inserting some dimension-independent quantities and then operating the Laplace transform on the succeeding equations. These solutions are utilized for the development of graphical illustrations to serve the purpose of covering all features of the problem under consideration. Furthermore, changes in energy and flow functions due to the dominant influences of the relevant contributing factors are delineated with appropriate physical arguments. In addition, the numerical results of the skin friction coefficient and Nusselt number are displayed via multiple tables to analyze the disturbance in shear stress and discuss the contribution of the fractional parameters, the volume concentration of the considered nanoparticles, and the shape factor in the boost of the thermal potential of the considered nanofluid. The findings imply that aluminum alloy nanoparticles have the ability to produce a 44% enhancement in the thermal effectiveness of vacuum pump oil. Moreover, the flow velocity is reduced as the loading range of the nanoparticles rises. Full article
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13 pages, 442 KiB  
Article
Analysis of the Multi-Dimensional Navier–Stokes Equation by Caputo Fractional Operator
by Kholoud Saad Albalawi, Manvendra Narayan Mishra and Pranay Goswami
Fractal Fract. 2022, 6(12), 743; https://doi.org/10.3390/fractalfract6120743 - 15 Dec 2022
Cited by 9 | Viewed by 1376
Abstract
In this article, we investigate the solution of the fractional multidimensional Navier–Stokes equation based on the Caputo fractional derivative operator. The behavior of the solution regarding the Navier–Stokes equation system using the Sumudu transform approach is discussed analytically and further discussed graphically. Full article
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