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21 pages, 344 KiB

Open AccessArticle

Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus

by Surang Sitho, Muhammad Aamir Ali, Hüseyin Budak, Sotiris K. Ntouyas and Jessada Tariboon

Mathematics 2021, 9(14), 1666; https://doi.org/10.3390/math9141666 - 15 Jul 2021

Cited by 5 | Viewed by 1437

In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the

q^{ϰ_{2}}

-quantum integral to show midpoint and trapezoidal inequalities for

q^{ϰ_{2}}

-differentiable preinvex functions. Furthermore, [...] Read more.

In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the

q^{ϰ_{2}}

-quantum integral to show midpoint and trapezoidal inequalities for

q^{ϰ_{2}}

-differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via

q_{ϰ_{1}}

-quantum integral is not valid for preinvex functions, and we present its proper form. We use

q_{ϰ_{1}}

-quantum integrals to show midpoint inequalities for

q_{ϰ_{1}}

-differentiable preinvex functions. It is also demonstrated that by considering the limit

q \to 1^{-}

and

η (ϰ_{2}, ϰ_{1}) = - η (ϰ_{1}, ϰ_{2}) = ϰ_{2} - ϰ_{1}

in the newly derived results, the newly proved findings can be turned into certain known results. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

12 pages, 319 KiB

Open AccessArticle

Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients

by Elsayed I. Mahmoud and Viktor N. Orlov

Mathematics 2021, 9(11), 1260; https://doi.org/10.3390/math9111260 - 31 May 2021

Cited by 5 | Viewed by 2745

Abstract

This paper presents a practical numerical method, an implicit finite-difference scheme for solving a two-dimensional time-space fractional Fokker–Planck equation with space–time depending on variable coefficients and source term, which represents a model of a Brownian particle in a periodic potential. The Caputo derivative [...] Read more.

This paper presents a practical numerical method, an implicit finite-difference scheme for solving a two-dimensional time-space fractional Fokker–Planck equation with space–time depending on variable coefficients and source term, which represents a model of a Brownian particle in a periodic potential. The Caputo derivative and the Riemann–Liouville derivative are considered in the temporal and spatial directions, respectively. The Riemann–Liouville derivative is approximated by the standard Grünwald approximation and the shifted Grünwald approximation. The stability and convergence of the numerical scheme are discussed. Finally, we provide a numerical example to test the theoretical analysis. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

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18 pages, 313 KiB

Open AccessArticle

Common Fixed Points Technique for Existence of a Solution of Urysohn Type Integral Equations System in Complex Valued b-Metric Spaces

by Muhammad Suhail Aslam, Monica Felicia Bota, Mohammad S. R. Chowdhury, Liliana Guran and Naeem Saleem

Mathematics 2021, 9(4), 400; https://doi.org/10.3390/math9040400 - 18 Feb 2021

Cited by 14 | Viewed by 2085

Abstract

In this paper we give some common fixed point theorems for Ćirić type operators in complex valued b-metric spaces. Also, some corollaries under this contraction condition are obtained. Our results extend and generalize the results of Hammad et al. In the second [...] Read more.

In this paper we give some common fixed point theorems for Ćirić type operators in complex valued b-metric spaces. Also, some corollaries under this contraction condition are obtained. Our results extend and generalize the results of Hammad et al. In the second part of the paper, in order to strengthen our main results, an illustrative example and some applications are given. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

14 pages, 1550 KiB

Open AccessArticle

Barrier Lyapunov Function-Based Adaptive Back-Stepping Control for Electronic Throttle Control System

by Dapeng Wang, Shaogang Liu, Youguo He and Jie Shen

Mathematics 2021, 9(4), 326; https://doi.org/10.3390/math9040326 - 6 Feb 2021

Cited by 7 | Viewed by 2056

Abstract

This paper presents an adaptive constraint control approach for Electronic Throttle Control System (ETCS) with asymmetric throttle angle constraints. The adaptive constraint control method, which is based on barrier Lyapunov function (BLF), is designed not only to track the desired throttle angle but [...] Read more.

This paper presents an adaptive constraint control approach for Electronic Throttle Control System (ETCS) with asymmetric throttle angle constraints. The adaptive constraint control method, which is based on barrier Lyapunov function (BLF), is designed not only to track the desired throttle angle but also to guarantee no violation on the throttle angle constraints. An ETC mathematic model with complex non-linear system is considered and the asymmetric barrier Lyapunov function (ABLF) is introduced into the design of the controller. Based on Lyapunov stability theory, it can be concluded that the proposed controller can guarantee the stability of the whole system and uniformly converge the state error to track the desired throttle angle. The results of simulations show that the proposed controller can ensure that there is no violation on the throttle angle constraints. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

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15 pages, 512 KiB

Open AccessArticle

An Optimal Control Perspective on Weather and Climate Modification

by Sergei Soldatenko and Rafael Yusupov

Mathematics 2021, 9(4), 305; https://doi.org/10.3390/math9040305 - 4 Feb 2021

Cited by 3 | Viewed by 2044

Abstract

Intentionally altering natural atmospheric processes using various techniques and technologies for changing weather patterns is one of the appropriate human responses to climate change and can be considered a rather drastic adaptation measure. A fundamental understanding of the human ability to modify weather [...] Read more.

Intentionally altering natural atmospheric processes using various techniques and technologies for changing weather patterns is one of the appropriate human responses to climate change and can be considered a rather drastic adaptation measure. A fundamental understanding of the human ability to modify weather conditions requires collaborative research in various scientific fields, including, but not limited to, atmospheric sciences and different branches of mathematics. This article being theoretical and methodological in nature, generalizes and, to some extent, summarizes our previous and current research in the field of climate and weather modification and control. By analyzing the deliberate change in weather and climate from an optimal control and dynamical systems perspective, we get the ability to consider the modification of natural atmospheric processes as a dynamic optimization problem with an emphasis on the optimal control problem. Within this conceptual and unified theoretical framework for developing and synthesizing an optimal control for natural weather phenomena, the atmospheric process in question represents a closed-loop dynamical system described by an appropriate mathematical model or, in other words, by a set of differential equations. In this context, the human control actions can be described by variations of the model parameters selected on the basis of sensitivity analysis as control variables. Application of the proposed approach to the problem of weather and climate modification is illustrated using a low-order conceptual model of the Earth’s climate system. For the sake of convenient interpretation, we provide some weather and climate basics, as well as we give a brief glance at control theory and sensitivity analysis of dynamical systems. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

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17 pages, 3184 KiB

Open AccessArticle

Improving Convergence in Therapy Scheduling Optimization: A Simulation Study

by Juan C. Chimal-Eguia, Julio C. Rangel-Reyes and Ricardo T. Paez-Hernandez

Mathematics 2020, 8(12), 2114; https://doi.org/10.3390/math8122114 - 26 Nov 2020

Cited by 1 | Viewed by 1496

Abstract

The infusion times and drug quantities are two primary variables to optimize when designing a therapeutic schedule. In this work, we test and analyze several extensions to the gradient descent equations in an optimal control algorithm conceived for therapy scheduling optimization. The goal [...] Read more.

The infusion times and drug quantities are two primary variables to optimize when designing a therapeutic schedule. In this work, we test and analyze several extensions to the gradient descent equations in an optimal control algorithm conceived for therapy scheduling optimization. The goal is to provide insights into the best strategies to follow in terms of convergence speed when implementing our method in models for dendritic cell immunotherapy. The method gives a pulsed-like control that models a series of bolus injections and aims to minimize a cost a function, which minimizes tumor size and to keep the tumor under a threshold. Additionally, we introduce a stochastic iteration step in the algorithm, which serves to reduce the number of gradient computations, similar to a stochastic gradient descent scheme in machine learning. Finally, we employ the algorithm to two therapy schedule optimization problems in dendritic cell immunotherapy and contrast our method’s stochastic and non-stochastic optimizations. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

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15 pages, 1402 KiB

Open AccessArticle

A Nonlinear Model Predictive Control with Enlarged Region of Attraction via the Union of Invariant Sets

by Ismi Rosyiana Fitri and Jung-Su Kim

Mathematics 2020, 8(11), 2087; https://doi.org/10.3390/math8112087 - 22 Nov 2020

Cited by 3 | Viewed by 2690

Abstract

In the dual-mode model predictive control (MPC) framework, the size of the stabilizable set, which is also the region of attraction, depends on the terminal constraint set. This paper aims to formulate a larger terminal set for enlarging the region of attraction in [...] Read more.

In the dual-mode model predictive control (MPC) framework, the size of the stabilizable set, which is also the region of attraction, depends on the terminal constraint set. This paper aims to formulate a larger terminal set for enlarging the region of attraction in a nonlinear MPC. Given several control laws and their corresponding terminal invariant sets, a convex combination of the given sets is used to construct a time-varying terminal set. The resulting region of attraction is the union of the regions of attraction from each invariant set. Simulation results show that the proposed MPC has a larger stabilizable initial set than the one obtained when a fixed terminal set is used. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

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27 pages, 670 KiB

Open AccessArticle

Solving the Boundary Value Problems for Differential Equations with Fractional Derivatives by the Method of Separation of Variables

by Temirkhan Aleroev

Mathematics 2020, 8(11), 1877; https://doi.org/10.3390/math8111877 - 29 Oct 2020

Cited by 6 | Viewed by 3269

Abstract

This paper is devoted to solving boundary value problems for differential equations with fractional derivatives by the Fourier method. The necessary information is given (in particular, theorems on the completeness of the eigenfunctions and associated functions, multiplicity of eigenvalues, and questions of the [...] Read more.

This paper is devoted to solving boundary value problems for differential equations with fractional derivatives by the Fourier method. The necessary information is given (in particular, theorems on the completeness of the eigenfunctions and associated functions, multiplicity of eigenvalues, and questions of the localization of root functions and eigenvalues are discussed) from the spectral theory of non-self-adjoint operators generated by differential equations with fractional derivatives and boundary conditions of the Sturm–Liouville type, obtained by the author during implementation of the method of separation of variables (Fourier). Solutions of boundary value problems for a fractional diffusion equation and wave equation with a fractional derivative are presented with respect to a spatial variable. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

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10 pages, 2216 KiB

Open AccessArticle

Modeling of Strength Characteristics of Polymer Concrete Via the Wave Equation with a Fractional Derivative

by Ludmila Kirianova

Mathematics 2020, 8(10), 1843; https://doi.org/10.3390/math8101843 - 20 Oct 2020

Cited by 7 | Viewed by 2134

Abstract

The article presents a solution to a boundary value problem for a wave equation containing a fractional derivative with respect to a spatial variable. This model is used to describe oscillation processes in a viscoelastic medium, in particular changes in the deformation-strength characteristics [...] Read more.

The article presents a solution to a boundary value problem for a wave equation containing a fractional derivative with respect to a spatial variable. This model is used to describe oscillation processes in a viscoelastic medium, in particular changes in the deformation-strength characteristics of polymer concrete (dian and dichloroanhydride-1,1-dichloro-2,2-diethylene) under the influence of the gravity force. Based on the obtained solution to the boundary value problem, the article presents four numerical examples corresponding to homogeneous boundary conditions and various initial conditions. The graphs of the found solutions were constructed and the calculation accuracy in the considered examples was estimated. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

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9 pages, 240 KiB

Open AccessArticle

Analytical and Approximate Solution for Solving the Vibration String Equation with a Fractional Derivative

by Temirkhan S. Aleroev and Asmaa M. Elsayed

Mathematics 2020, 8(7), 1154; https://doi.org/10.3390/math8071154 - 14 Jul 2020

Cited by 6 | Viewed by 2359

Abstract

This paper is proposed for solving a partial differential equation of second order with a fractional derivative with respect to time (the vibration string equation), where the fractional derivative order is in the range from zero to two. We propose a numerical solution [...] Read more.

This paper is proposed for solving a partial differential equation of second order with a fractional derivative with respect to time (the vibration string equation), where the fractional derivative order is in the range from zero to two. We propose a numerical solution that is based on the Laplace transform method with the homotopy perturbation method. The method of the separation of variables (the Fourier method) is constructed for the analytic solution. The derived solutions are represented by Mittag–LefLeffler type functions. Orthogonality and convergence of the solution are discussed. Finally, we present an example to illustrate the methods. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

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10 pages, 302 KiB

Open AccessArticle

Numerical Scheme for Solving Time–Space Vibration String Equation of Fractional Derivative

by Asmaa M. Elsayed and Viktor N. Orlov

Mathematics 2020, 8(7), 1069; https://doi.org/10.3390/math8071069 - 2 Jul 2020

Cited by 9 | Viewed by 2146

Abstract

In this paper, we present a numerical scheme and alternating direction implicit scheme for the one-dimensional time–space fractional vibration equation. Firstly, the considered time–space fractional vibration equation is equivalently transformed into their partial integro-differential forms by using the integral operator. Secondly, we use [...] Read more.

In this paper, we present a numerical scheme and alternating direction implicit scheme for the one-dimensional time–space fractional vibration equation. Firstly, the considered time–space fractional vibration equation is equivalently transformed into their partial integro-differential forms by using the integral operator. Secondly, we use the Crank–Nicholson scheme based on the weighted and shifted Grünwald–difference formula to discretize the Riemann–Liouville and Caputo derivative, also use the midpoint formula to discretize the first order derivative. Meanwhile, the classical central difference formula is applied to approximate the second order derivative. The convergence and unconditional stability of the suggested scheme are obtained. Finally, we present an example to illustrate the method. Full article

(This article belongs to the Special Issue Dynamical Systems and Optimal Control)

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