Research

12 pages, 268 KiB

Open AccessArticle

Distributed-Order Non-Local Optimal Control

by Faïçal Ndaïrou and Delfim F. M. Torres

Axioms 2020, 9(4), 124; https://doi.org/10.3390/axioms9040124 - 25 Oct 2020

Cited by 5 | Viewed by 2170

Distributed-order fractional non-local operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. [...] Read more.

Distributed-order fractional non-local operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional, subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical system constraints depending on distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin’s maximum principle and a sufficient optimality condition under appropriate convexity assumptions. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

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23 pages, 1431 KiB

Open AccessArticle

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

by Hasan S. Panigoro, Agus Suryanto, Wuryansari Muharini Kusumawinahyu and Isnani Darti

Axioms 2020, 9(4), 122; https://doi.org/10.3390/axioms9040122 - 22 Oct 2020

Cited by 15 | Viewed by 2882

Abstract

The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The [...] Read more.

The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The THP is applied to maintain the existence of the population in the prey–predator mechanism. We first consider the Rosenzweig–MacArthur model using the Caputo fractional-order derivative (that is, the operator with the power-law kernel) and perform some dynamical analysis such as the existence and uniqueness, non-negativity, boundedness, local stability, global stability, and the existence of Hopf bifurcation. We then reconsider the same model involving the Atangana–Baleanu fractional derivative with the Mittag–Leffler kernel in the Caputo sense (ABC). The existence and uniqueness of the solution of the model with ABC operator are established. We also explore the dynamics of the model with both fractional derivative operators numerically and confirm the theoretical findings. In particular, it is shown that models with both Caputo operator and ABC operator undergo a Hopf bifurcation that can be controlled by the conversion rate of consumed prey into the predator birth rate or by the order of fractional derivative. However, the bifurcation point of the model with the Caputo operator is different from that of the model with the ABC operator. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

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24 pages, 879 KiB

Open AccessArticle

Inverse Problem for a Mixed Type Integro-Differential Equation with Fractional Order Caputo Operators and Spectral Parameters

by Tursun K. Yuldashev and Erkinjon T. Karimov

Axioms 2020, 9(4), 121; https://doi.org/10.3390/axioms9040121 - 20 Oct 2020

Cited by 31 | Viewed by 2414

Abstract

The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is [...] Read more.

The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differential equation of fractional order in both positive and negative parts of the multidimensional rectangular domain under consideration. This mixed type of equation, with respect to redefinition functions, is a nonlinear Fredholm type integral equation. The fractional Caputo operators’ orders are smaller in the positive part of the domain than the orders of Caputo operators in the negative part of the domain under consideration. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels and different orders of integro-differentation are obtained. Furthermore, a method of degenerate kernels is used. In order to determine arbitrary integration constants, a linear system of functional algebraic equations is obtained. From the solvability condition of this system are calculated the regular and irregular values of the spectral parameters. The solution of the inverse problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. During the proof of the convergence of the Fourier series, certain properties of the Mittag–Leffler function of two variables, the Cauchy–Schwarz inequality and Bessel inequality, are used. We also studied the continuous dependence of the solution of the problem on small parameters for regular values of spectral parameters. The existence and uniqueness of redefined functions have been justified by solving the systems of two countable systems of nonlinear integral equations. The results are formulated as a theorem. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

12 pages, 764 KiB

Open AccessArticle

Finite Series of Distributional Solutions for Certain Linear Differential Equations

by Nipon Waiyaworn, Kamsing Nonlaopon and Somsak Orankitjaroen

Axioms 2020, 9(4), 116; https://doi.org/10.3390/axioms9040116 - 13 Oct 2020

Cited by 1 | Viewed by 1507

Abstract

In this paper, we present the distributional solutions of the modified spherical Bessel differential equations [...] Read more.

In this paper, we present the distributional solutions of the modified spherical Bessel differential equations

t^{2} y^{″} (t) + 2 t y^{'} (t) - [t^{2} + ν (ν + 1)] y (t) = 0

and the linear differential equations of the forms

t^{2} y^{″} (t) + 3 t y^{'} (t) - (t^{2} + ν^{2} - 1) y (t) = 0,

where

ν \in N \cup {0}

and

t \in R

. We find that the distributional solutions, in the form of a finite series of the Dirac delta function and its derivatives, depend on the values of

ν

. The results of several examples are also presented. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

11 pages, 743 KiB

Open AccessArticle

Asymptotic Properties of Neutral Differential Equations with Variable Coefficients

by Omar Bazighifan, Rami Ahmad El-Nabulsi and Osama Moaaz

Axioms 2020, 9(3), 96; https://doi.org/10.3390/axioms9030096 - 12 Aug 2020

Cited by 2 | Viewed by 1811

Abstract

The aim of this work is to study oscillatory behavior of solutions for even-order neutral nonlinear differential equations. By using the Riccati substitution, a new oscillation conditions is obtained which insures that all solutions to the studied equation are oscillatory. The obtained results [...] Read more.

The aim of this work is to study oscillatory behavior of solutions for even-order neutral nonlinear differential equations. By using the Riccati substitution, a new oscillation conditions is obtained which insures that all solutions to the studied equation are oscillatory. The obtained results complement the well-known oscillation results present in the literature. Some example are illustrated to show the applicability of the obtained results. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

8 pages, 226 KiB

Open AccessArticle

On the Uniqueness Classes of Solutions of Boundary Value Problems for Third-Order Equations of the Pseudo-Elliptic Type

by Abdukomil Risbekovich Khashimov and Dana Smetanová

Axioms 2020, 9(3), 80; https://doi.org/10.3390/axioms9030080 - 16 Jul 2020

Cited by 1 | Viewed by 1517

Abstract

The paper is devoted to solutions of the third order pseudo-elliptic type equations. An energy estimates for solutions of the equations considering transformation’s character of the body form were established by using of an analog of the Saint-Venant principle. In consequence of this [...] Read more.

The paper is devoted to solutions of the third order pseudo-elliptic type equations. An energy estimates for solutions of the equations considering transformation’s character of the body form were established by using of an analog of the Saint-Venant principle. In consequence of this estimate, the uniqueness theorems were obtained for solutions of the first boundary value problem for third order equations in unlimited domains. The energy estimates are illustrated on two examples. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

19 pages, 350 KiB

Open AccessArticle

Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator

by Tursun K. Yuldashev and Bakhtiyor J. Kadirkulov

Axioms 2020, 9(2), 68; https://doi.org/10.3390/axioms9020068 - 17 Jun 2020

Cited by 35 | Viewed by 3370

Abstract

In this paper, we consider a boundary value problem for a nonlinear partial differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. With respect to the first [...] Read more.

In this paper, we consider a boundary value problem for a nonlinear partial differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. With respect to the first variable, this equation is a nonlinear fractional differential equation in the positive part of the considering segment and is a second-order nonlinear differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of nonlinear boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the classical solution of the problem are proved for regular values of the spectral parameter. For irregular values of the spectral parameter, an infinite number of solutions of the mixed equation in the form of a Fourier series are constructed. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

9 pages, 263 KiB

Open AccessArticle

On the Solvability of Nonlinear Third-Order Two-Point Boundary Value Problems

by Ravi P. Agarwal, Petio S. Kelevedjiev and Todor Z. Todorov

Axioms 2020, 9(2), 62; https://doi.org/10.3390/axioms9020062 - 31 May 2020

Cited by 1 | Viewed by 1791

Abstract

Under barrier strips type assumptions we study the existence of

C^{3} [0, 1]

—solutions to various two-point boundary value problems for the equation [...] Read more.

Under barrier strips type assumptions we study the existence of

C^{3} [0, 1]

—solutions to various two-point boundary value problems for the equation

x^{‴} = f (t, x, x^{'}, x^{″}) .

We give also some results guaranteeing positive or non-negative, monotone, convex or concave solutions. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

13 pages, 346 KiB

Open AccessEditor’s ChoiceArticle

Initial Value Problem For Nonlinear Fractional Differential Equations With ψ-Caputo Derivative Via Monotone Iterative Technique

by Choukri Derbazi, Zidane Baitiche, Mouffak Benchohra and Alberto Cabada

Axioms 2020, 9(2), 57; https://doi.org/10.3390/axioms9020057 - 21 May 2020

Cited by 34 | Viewed by 3404

Abstract

In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the

ψ

-Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More [...] Read more.

In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the

ψ

-Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More precisely we apply the monotone iterative technique combined with the method of upper and lower solutions to establish sufficient conditions for existence as well as the uniqueness of extremal solutions to the initial value problem. An illustrative example is presented to point out the applicability of our main results. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

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11 pages, 774 KiB

Open AccessArticle

Sufficient Conditions for Oscillation of Fourth-Order Neutral Differential Equations with Distributed Deviating Arguments

by Omar Bazighifan, Feliz Minhos and Osama Moaaz

Axioms 2020, 9(2), 39; https://doi.org/10.3390/axioms9020039 - 11 Apr 2020

Cited by 8 | Viewed by 1788

Abstract

Some new sufficient conditions are established for the oscillation of fourth order neutral differential equations with continuously distributed delay of the form [...] Read more.

Some new sufficient conditions are established for the oscillation of fourth order neutral differential equations with continuously distributed delay of the form

{(r (t) {(N_{x}^{‴} (t))}^{α})}^{'} + \int_{a}^{b} q (t, ϑ) x^{β} (δ (t, ϑ)) d ϑ = 0,

where

t \geq t_{0}

and

N_{x} (t) : = x (t) + p (t) x (φ (t))

. An example is provided to show the importance of these results. Full article

(This article belongs to the Special Issue Nonlinear Differential Equations and Dynamical Systems: Theory and Applications)

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