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Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications

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A special issue of Mathematical and Computational Applications (ISSN 2297-8747).

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 31664

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Dr. Mehmet Yavuz

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Guest Editor

1. Department of Mathematics and Computer Sciences, Necmettin Erbakan University, Meram Yeniyol, 42090 Meram, Konya, Turkey
2. Department of Mathematics, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Cornwall TR10 9FE, UK
Interests: differential/difference equations; dynamical systems; modeling and stability analysis of biological systems; financial mathematics; fractional calculus; mathematical modeling; fluid dynamics; optimal control
Special Issues, Collections and Topics in MDPI journals

Dr. Ioannis Dassios

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Guest Editor

School of Electrical and Electronic Engineering, University College Dublin, D04 V1W8 Dublin, Ireland
Interests: differential/difference equations; dynamical systems; modeling and stability analysis of electric power systems; mathematics of networks; fractional calculus; mathematical modeling (power systems, materials science, energy, macroeconomics, social media, etc.); optimization for the analysis of large-scale data sets; fluid mechanics; discrete calculus; Bayes control; e-learning
Special Issues, Collections and Topics in MDPI journals

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Dear Colleagues,

Mathematical modeling and systems control arise in many research problems, ranging from physical and chemical processes to biomathematics and life science. Their theoretical description is closely connected with various areas of pure and applied mathematics including nonlinear modeling, integro-differential equations, nonlinear dynamics, pattern formation, non-Markovian processes, nonlinear and anomalous transport, time-delay equations and so on.

The aim of this Special Issue is to collect original and high-quality contributions related to the mathematical theory of such processes and phenomena including the dynamical models, applied and computational algorithms, controller design and mathematical methods regarded as new and prominent for understanding the problems that arise in natural phenomena.

This Special Issue will cover new perspectives of the recent theoretical developments in mathematical modeling and/or optimal control and their illustrative applications in biology, engineering, finance, and health sciences. It aims to highlight new techniques that can be applied to the real-life problems which are modeled and to introduce new constructed effective models for the accurate prediction of infectious diseases, financial crisis, etc., into the literature by adopting suitable controls/control strategies. Moreover, it aims to provide new analytical and numerical methods to propose appropriate solutions to 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 proposes the latest developments in nonlinear dynamical modeling, optimization and solution strategies that can be applied to prominent problems in engineering and biological systems.

This Special Issue helps the reader learn new theories and new methods of nonlinear dynamical systems, modeling and controlling them. It will also help the reader find new solutions to complex engineering, biological, financial and life sciences problems. This Special Issue also provides readers with new insights for novel modeling and optimization processes and will underline the relation between theory and practice.

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

Mathematical modeling in real-world phenomena;
Optimal control strategies in biosystems;
New analytical and numerical methods for fractional differential equations;
Modeling of fractional order systems with and without nonsingular kernels;
Deterministic and stochastic differential equations arising in science;
Applications in bioengineering, biology, and health sciences;
Applications in finance and economic sciences;
Optimal control problems of fractional order;
Modeling of diffusion, heat, mass, and momentum transfer (fluid dynamics);
Biomechanical and biomedical applications of fractional calculus;
Impulsive systems;
Fuzzy differential equations and their applications.

Dr. Mehmet Yavuz
Dr. Ioannis Dassios
Guest Editors

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Published Papers (14 papers)

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Editorial

Jump to: Research

4 pages, 144 KiB

Open AccessEditorial

Significance of Mathematical Modeling and Control in Real-World Problems: New Developments and Applications

by Mehmet Yavuz and Ioannis Dassios

Math. Comput. Appl. 2024, 29(5), 82; https://doi.org/10.3390/mca29050082 - 18 Sep 2024

Viewed by 843

Abstract

Mathematical modeling and system control are employed in many research problems, ranging from physical and chemical processes to biomathematics and life sciences [...] Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

Research

Jump to: Editorial

20 pages, 352 KiB

Open AccessArticle

Fractional Hermite–Hadamard-Type Inequalities for Differentiable Preinvex Mappings and Applications to Modified Bessel and q-Digamma Functions

by Muhammad Tariq, Hijaz Ahmad, Asif Ali Shaikh, Sotiris K. Ntouyas, Evren Hınçal and Sania Qureshi

Math. Comput. Appl. 2023, 28(6), 108; https://doi.org/10.3390/mca28060108 - 9 Nov 2023

Cited by 1 | Viewed by 1461

Abstract

The theory of convexity pertaining to fractional calculus is a well-established concept that has attracted significant attention in mathematics and various scientific disciplines for over a century. In the realm of applied mathematics, convexity, particularly in relation to fractional analysis, finds extensive and remarkable applications. In this manuscript, we establish new fractional identities. Employing these identities, some extensions of the fractional H-H type inequality via generalized preinvexities are explored. Finally, we discuss some applications to the q-digamma and Bessel functions via the established results. We believe that the methodologies and approaches presented in this work will intrigue and spark the researcher’s interest even more. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

29 pages, 1311 KiB

Open AccessArticle

Preconditioning Technique for an Image Deblurring Problem with the Total Fractional-Order Variation Model

by Adel M. Al-Mahdi

Math. Comput. Appl. 2023, 28(5), 97; https://doi.org/10.3390/mca28050097 - 22 Sep 2023

Cited by 1 | Viewed by 1657

Abstract

Total fractional-order variation (TFOV) in image deblurring problems can reduce/remove the staircase problems observed with the image deblurring technique by using the standard total variation (TV) model. However, the discretization of the Euler–Lagrange equations associated with the TFOV model generates a saddle point system of equations where the coefficient matrix of this system is dense and ill conditioned (it has a huge condition number). The ill-conditioned property leads to slowing of the convergence of any iterative method, such as Krylov subspace methods. One treatment for the slowness property is to apply the preconditioning technique. In this paper, we propose a block triangular preconditioner because we know that using the exact triangular preconditioner leads to a preconditioned matrix with exactly two distinct eigenvalues. This means that we need at most two iterations to converge to the exact solution. However, we cannot use the exact preconditioner because the Shur complement of our system is of the form

S = K^{*} K + λ L_{α}

which is a huge and dense matrix. The first matrix,

K^{*} K

, comes from the blurred operator, while the second one is from the TFOV regularization model. To overcome this difficulty, we propose two preconditioners based on the circulant and standard TV matrices. In our algorithm, we use the flexible preconditioned GMRES method for the outer iterations, the preconditioned conjugate gradient (PCG) method for the inner iterations, and the fixed point iteration (FPI) method to handle the nonlinearity. Fast convergence was found in the numerical results by using the proposed preconditioners. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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21 pages, 636 KiB

Open AccessArticle

Numerical Computation of Ag/Al₂O₃ Nanofluid over a Riga Plate with Heat Sink/Source and Non-Fourier Heat Flux Model

by S. Divya, S. Eswaramoorthi and Karuppusamy Loganathan

Math. Comput. Appl. 2023, 28(1), 20; https://doi.org/10.3390/mca28010020 - 3 Feb 2023

Cited by 7 | Viewed by 2065

Abstract

The main goal of the current research is to investigate the numerical computation of

Ag / {Al}_{2} O_{3}

nanofluid over a Riga plate with injection/suction. The energy equation is formulated using the Cattaneo–Christov heat flux, non-linear thermal radiation, and heat sink/source. [...] Read more.

The main goal of the current research is to investigate the numerical computation of

Ag / {Al}_{2} O_{3}

nanofluid over a Riga plate with injection/suction. The energy equation is formulated using the Cattaneo–Christov heat flux, non-linear thermal radiation, and heat sink/source. The leading equations are non-dimensionalized by employing the suitable transformations, and the numerical results are achieved by using the MATLAB bvp4c technique. The fluctuations of fluid flow and heat transfer on porosity, Forchheimer number, radiation, suction/injection, velocity slip, and nanoparticle volume fraction are investigated. Furthermore, the local skin friction coefficient (SFC), and local Nusselt number (LNN) are also addressed. Compared to previously reported studies, our computational results exactly coincided with the outcomes of the previous reports. We noticed that the Forchheimer number, suction/injection, slip, and nanoparticle volume fraction factors slow the velocity profile. We also noted that with improving rates of thermal radiation and convective heating, the heat transfer gradient decreases. The

40 %

presence of the Hartmann number leads to improved drag force by

14 %

and heat transfer gradient by

0.5 %

. The

20 %

presence of nanoparticle volume fraction leads to a decrement in heat transfer gradient for

21 %

of Ag nanoparticles and

18 %

{Al}_{2} O_{3}

nanoparticles. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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16 pages, 360 KiB

Open AccessArticle

Controllability Criteria for Nonlinear Impulsive Fractional Differential Systems with Distributed Delays in Controls

by Amar Debbouche, Bhaskar Sundara Vadivoo, Vladimir E. Fedorov and Valery Antonov

Math. Comput. Appl. 2023, 28(1), 13; https://doi.org/10.3390/mca28010013 - 15 Jan 2023

Cited by 4 | Viewed by 1580

Abstract

We establish a class of nonlinear fractional differential systems with distributed time delays in the controls and impulse effects. We discuss the controllability criteria for both linear and nonlinear systems. The main results required a suitable Gramian matrix defined by the Mittag–Leffler function, using the standard Laplace transform and Schauder fixed-point techniques. Further, we provide an illustrative example supported by graphical representations to show the validity of the obtained abstract results. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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16 pages, 642 KiB

Open AccessArticle

A SARS-CoV-2 Fractional-Order Mathematical Model via the Modified Euler Method

by Ihtisham Ul Haq, Mehmet Yavuz, Nigar Ali and Ali Akgül

Math. Comput. Appl. 2022, 27(5), 82; https://doi.org/10.3390/mca27050082 - 26 Sep 2022

Cited by 18 | Viewed by 2620

Abstract

This article develops a within-host viral kinetics model of SARS-CoV-2 under the Caputo fractional-order operator. We prove the results of the solution’s existence and uniqueness by using the Banach mapping contraction principle. Using the next-generation matrix method, we obtain the basic reproduction number. We analyze the model’s endemic and disease-free equilibrium points for local and global stability. Furthermore, we find approximate solutions for the non-linear fractional model using the Modified Euler Method (MEM). To support analytical findings, numerical simulations are carried out. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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17 pages, 6300 KiB

Open AccessArticle

Numerical Study of the Effect of a Heated Cylinder on Natural Convection in a Square Cavity in the Presence of a Magnetic Field

by Muhammad Sajjad Hossain, Muhammad Fayz-Al-Asad, Muhammad Saiful Islam Mallik, Mehmet Yavuz, Md. Abdul Alim and Kazi Md. Khairul Basher

Math. Comput. Appl. 2022, 27(4), 58; https://doi.org/10.3390/mca27040058 - 11 Jul 2022

Cited by 5 | Viewed by 2787

Abstract

The present research was developed to find out the effect of heated cylinder configurations in accordance with the magnetic field on the natural convective flow within a square cavity. In the cavity, four types of configurations—left bottom heated cylinder (LBC), right bottom heated cylinder (RBC), left top heated cylinder (LTC) and right top heated cylinder (RTC)—were considered in the investigation. The current mathematical problem was formulated using the non-linear governing equations and then solved by engaging the process of Galerkin weighted residuals based on the finite element scheme (FES). The investigation of the present problem was conducted using numerous parameters: the Rayleigh number (Ra = 10³–10⁵), the Hartmann number (Ha = 0–200) at Pr = 0.71 on the flow field, thermal pattern and the variation of heat inside the enclosure. The clarifications of the numerical result were exhibited in the form of streamlines, isotherms, velocity profiles and temperature profiles, local and mean Nusselt number, along with heated cylinder configurations. From the obtained outcomes, it was observed that the rate of heat transport, as well as the local Nusselt number, decreased for the LBC and LTC configurations, but increased for the RBC and RTC configurations with the increase of the Hartmann number within the square cavity. In addition, the mean Nusselt number for the LBC, RBC, LTC and RTC configurations increased when the Hartmann number was absent, but decreased when the Hartmann number increased in the cavity. The computational results were verified in relation to a published work and were found to be in good agreement. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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13 pages, 971 KiB

Open AccessArticle

Double-Diffusive Convection in Bidispersive Porous Medium with Coriolis Effect

by Chirnam Ramchandraiah, Naikoti Kishan, Gundlapally Shiva Kumar Reddy, Kiran Kumar Paidipati and Christophe Chesneau

Math. Comput. Appl. 2022, 27(4), 56; https://doi.org/10.3390/mca27040056 - 30 Jun 2022

Cited by 3 | Viewed by 2094

Abstract

In this paper, the thermal instability of rotating convection in a bidispersive porous layer is analyzed. The linear stability analysis is employed to examine the stability of the system. The neutral curves for different values of the physical parameters are shown graphically. The critical Rayleigh number is evaluated for appropriate values of the other governing parameters. Among the obtained results, we find: the Taylor number has a stabilizing effect on the onset of convection; the Soret number does not show any effect on oscillatory convection, as the oscillatory Rayleigh number is independent of the Soret number; there exists a threshold,

R_{c}^{*}

∈ (0.45, 0.46), for the solute Rayleigh number, such that, if

R_{C}

R_{c}^{*}

, then the convection arises via an oscillatory mode; and the oscillatory convection sets in and as soon as the value of the Soret number reaches a critical value, (∈(0.6, 0.7)), and the convection arises via stationary convection. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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15 pages, 1526 KiB

Open AccessArticle

Dissolution-Driven Convection in a Porous Medium Due to Vertical Axis of Rotation and Magnetic Field

by Gundlapally Shiva Kumar Reddy, Nilam Venkata Koteswararao, Ragoju Ravi, Kiran Kumar Paidipati and Christophe Chesneau

Math. Comput. Appl. 2022, 27(3), 53; https://doi.org/10.3390/mca27030053 - 20 Jun 2022

Cited by 2 | Viewed by 1746

Abstract

This article aims to study the effect of the vertical rotation and magnetic field on the dissolution-driven convection in a saturated porous layer with a first-order chemical reaction. The system’s physical parameters depend on the Vadasz number, the Hartmann number, the Taylor number, and the Damkohler number. We analyze them in an in-depth manner. On the other hand, based on an artificial neural network (ANN) technique, the Levenberg–Marquardt backpropagation algorithm is adopted to predict the distribution of the critical Rayleigh number and for the linear stability analysis. The simulated critical Rayleigh numbers obtained by the numerical study and the predicted critical Rayleigh numbers by the ANN are compared and are in good agreement. The system becomes more stable by increasing the Damkohler and Taylor numbers. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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17 pages, 2528 KiB

Open AccessArticle

Exploration of Darcy–Forchheimer Flows of Non-Newtonian Casson and Williamson Conveying Tiny Particles Experiencing Binary Chemical Reaction and Thermal Radiation: Comparative Analysis

by Sheniyappan Eswaramoorthi, S. Thamaraiselvi and Karuppusamy Loganathan

Math. Comput. Appl. 2022, 27(3), 52; https://doi.org/10.3390/mca27030052 - 20 Jun 2022

Cited by 11 | Viewed by 2322

Abstract

This discussion intends to scrutinize the Darcy–Forchheimer flow of Casson–Williamson nanofluid in a stretching surface with non-linear thermal radiation, suction and heat consumption. In addition, this investigation assimilates the influence of the Brownian motion, thermophoresis, activation energy and binary chemical reaction effects. Cattaneo–Christov heat-mass flux theory is used to frame the energy and nanoparticle concentration equations. The suitable transformation is used to remodel the governing PDE model into an ODE model. The remodeled flow problems are numerically solved via the BVP4C scheme. The effects of various material characteristics on nanofluid velocity, nanofluid temperature and nanofluid concentration, as well as connected engineering aspects such as drag force, heat, and mass transfer gradients, are also calculated and displayed through tables, charts and figures. It is noticed that the nanofluid velocity upsurges when improving the quantity of Richardson number, and it downfalls for larger magnitudes of magnetic field and porosity parameters. The nanofluid temperature grows when enhancing the radiation parameter and Eckert number. The nanoparticle concentration upgrades for larger values of activation energy parameter while it slumps against the reaction rate parameter. The surface shear stress for the Williamson nanofluid is greater than the Casson nanofluid. There are more heat transfer gradient losses the greater the heat generation/absorption parameter and Eckert number. In addition, the local Sherwood number grows when strengthening the Forchheimer number and fitted rate parameter. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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20 pages, 8706 KiB

Open AccessArticle

Magneto Mixed Convection of Williamson Nanofluid Flow through a Double Stratified Porous Medium in Attendance of Activation Energy

by B. M. Tamilzharasan, S. Karthikeyan, Mohammed K. A. Kaabar, Mehmet Yavuz and Fatma Özköse

Math. Comput. Appl. 2022, 27(3), 46; https://doi.org/10.3390/mca27030046 - 26 May 2022

Cited by 9 | Viewed by 2967

Abstract

This article aims to develop a mathematical simulation of the steady mixed convective Darcy–Forchheimer flow of Williamson nanofluid over a linear stretchable surface. In addition, the effects of Cattaneo–Christov heat and mass flux, Brownian motion, activation energy, and thermophoresis are also studied. The novel aspect of this study is that it incorporates thermal radiation to investigate the physical effects of thermal and solutal stratification on mixed convection flow and heat transfer. First, the profiles of velocity and energy equations were transformed toward the ordinary differential equation using the appropriate similarity transformation. Then, the system of equations was modified by first-order ODEs in MATLAB and solved using the bvp4c approach. Graphs and tables imply the impact of physical parameters on concentration, temperature, velocity, skin friction coefficient, mass, and heat transfer rate. The outcomes show that the nanofluid temperature and concentration are reduced with the more significant thermal and mass stratification parameters estimation. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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14 pages, 896 KiB

Open AccessArticle

Solution of a Complex Nonlinear Fractional Biochemical Reaction Model

by Fatima Rabah, Marwan Abukhaled and Suheil A. Khuri

Math. Comput. Appl. 2022, 27(3), 45; https://doi.org/10.3390/mca27030045 - 26 May 2022

Cited by 7 | Viewed by 2117

Abstract

This paper discusses a complex nonlinear fractional model of enzyme inhibitor reaction where reaction memory is taken into account. Analytical expressions of the concentrations of enzyme, substrate, inhibitor, product, and other complex intermediate species are derived using Laplace decomposition and differential transformation methods. Since different rate constants, large initial concentrations, and large time domains are unavoidable in biochemical reactions, different dynamics will result; hence, the convergence of the approximate concentrations may be lost. In this case, the proposed analytical methods will be coupled with Padé approximation. The validity and accuracy of the derived analytical solutions will be established by direct comparison with numerical simulations. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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29 pages, 22708 KiB

Open AccessArticle

Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk

by Dominic P. Clemence-Mkhope and Gregory A. Gibson

Math. Comput. Appl. 2022, 27(1), 4; https://doi.org/10.3390/mca27010004 - 10 Jan 2022

Cited by 2 | Viewed by 2260

Abstract

Four discrete models, using the exact spectral derivative discretization finite difference (ESDDFD) method, are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) [...] Read more.

α = 1

, the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is, in part, an artifact of the CEFD method, as it is absent from the ESDDFD models. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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11 pages, 5530 KiB

Open AccessArticle

The Limited Validity of the Conformable Euler Finite Difference Method and an Alternate Definition of the Conformable Fractional Derivative to Justify Modification of the Method

by Dominic P. Clemence-Mkhope and Belinda G. B. Clemence-Mkhope

Math. Comput. Appl. 2021, 26(4), 66; https://doi.org/10.3390/mca26040066 - 23 Sep 2021

Cited by 4 | Viewed by 2606

Abstract

A method recently advanced as the conformable Euler method (CEM) for the finite difference discretization of fractional initial value problem [...] Read more.

A method recently advanced as the conformable Euler method (CEM) for the finite difference discretization of fractional initial value problem

D_{t}^{α} y (t) = f (t; y (t)), y (t_{0}) = y_{0}, a \leq t \leq b

, and used to describe hyperchaos in a financial market model, is shown to be valid only for

α = 1

. The property of the conformable fractional derivative (CFD) used to show this limitation of the method is used, together with the integer definition of the derivative, to derive a modified conformable Euler method for the initial value problem considered. A method of constructing generalized derivatives from the solution of the non-integer relaxation equation is used to motivate an alternate definition of the CFD and justify alternative generalizations of the Euler method to the CFD. The conformable relaxation equation is used in numerical experiments to assess the performance of the CEM in comparison to that of the alternative methods. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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