Mathematics

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12 pages, 286 KiB

Open AccessFeature PaperArticle

Impulsive Memristive Cohen–Grossberg Neural Networks Modeled by Short Term Generalized Proportional Caputo Fractional Derivative and Synchronization Analysis

by Ravi Agarwal and Snezhana Hristova

Mathematics 2022, 10(13), 2355; https://doi.org/10.3390/math10132355 - 5 Jul 2022

Cited by 6 | Viewed by 1513

Abstract

The synchronization problem for impulsive fractional-order Cohen–Grossberg neural networks with generalized proportional Caputo fractional derivatives with changeable lower limit at any point of impulse is studied. We consider the cases when the control input is acting continuously as well as when it is [...] Read more.

The synchronization problem for impulsive fractional-order Cohen–Grossberg neural networks with generalized proportional Caputo fractional derivatives with changeable lower limit at any point of impulse is studied. We consider the cases when the control input is acting continuously as well as when it is acting instantaneously at the impulsive times. We defined the global Mittag–Leffler synchronization as a generalization of exponential synchronization. We obtained some sufficient conditions for Mittag–Leffler synchronization. Our results are illustrated with examples. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

13 pages, 1094 KiB

Open AccessArticle

A Reliable Way to Deal with Fractional-Order Equations That Describe the Unsteady Flow of a Polytropic Gas

by M. Mossa Al-Sawalha, Ravi P. Agarwal, Rasool Shah, Osama Y. Ababneh and Wajaree Weera

Mathematics 2022, 10(13), 2293; https://doi.org/10.3390/math10132293 - 30 Jun 2022

Cited by 17 | Viewed by 1994

Abstract

In this paper, fractional-order system gas dynamics equations are solved analytically using an appealing novel method known as the Laplace residual power series technique, which is based on the coupling of the residual power series approach with the Laplace transform operator to develop [...] Read more.

In this paper, fractional-order system gas dynamics equations are solved analytically using an appealing novel method known as the Laplace residual power series technique, which is based on the coupling of the residual power series approach with the Laplace transform operator to develop analytical and approximate solutions in quick convergent series types by utilizing the idea of the limit with less effort and time than the residual power series method. The given model is tested and simulated to confirm the proposed technique’s simplicity, performance, and viability. The results show that the above-mentioned technique is simple, reliable, and appropriate for investigating nonlinear engineering and physical problems. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

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12 pages, 714 KiB

Open AccessArticle

On Caputo–Katugampola Fractional Stochastic Differential Equation

by McSylvester Ejighikeme Omaba and Hamdan Al Sulaimani

Mathematics 2022, 10(12), 2086; https://doi.org/10.3390/math10122086 - 16 Jun 2022

Cited by 6 | Viewed by 2134

Abstract

We consider the following stochastic fractional differential equation

^{C} D_{0^{+}}^{α, ρ} φ (t) = κ ϑ (t, φ (t)) \dot{w} (t)

,

0 < t \leq T,

where [...] Read more.

We consider the following stochastic fractional differential equation

^{C} D_{0^{+}}^{α, ρ} φ (t) = κ ϑ (t, φ (t)) \dot{w} (t)

,

0 < t \leq T,

where

φ (0) = φ_{0}

is the initial function,

^{C} D_{0^{+}}^{α, ρ}

is the Caputo–Katugampola fractional differential operator of orders

0 < α \leq 1, ρ > 0

, the function

ϑ : [0, T] \times R \to R

is Lipschitz continuous on the second variable,

\dot{w} (t)

denotes the generalized derivative of the Wiener process

w (t)

and

κ > 0

represents the noise level. The main result of the paper focuses on the energy growth bound and the asymptotic behaviour of the random solution. Furthermore, we employ Banach fixed point theorem to establish the existence and uniqueness result of the mild solution. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

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15 pages, 323 KiB

Open AccessArticle

Exact Solvability Conditions for the Non-Local Initial Value Problem for Systems of Linear Fractional Functional Differential Equations

by Natalia Dilna and Michal Fečkan

Mathematics 2022, 10(10), 1759; https://doi.org/10.3390/math10101759 - 21 May 2022

Cited by 4 | Viewed by 1400

Abstract

The exact conditions sufficient for the unique solvability of the initial value problem for a system of linear fractional functional differential equations determined by isotone operators are established. In a sense, the conditions obtained are optimal. The method of the test elements intended [...] Read more.

The exact conditions sufficient for the unique solvability of the initial value problem for a system of linear fractional functional differential equations determined by isotone operators are established. In a sense, the conditions obtained are optimal. The method of the test elements intended for the estimation of the spectral radius of a linear operator is used. The unique solution is presented by the Neumann’s series. All theoretical investigations are shown in the examples. A pantograph-type model from electrodynamics is studied. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

16 pages, 318 KiB

Open AccessArticle

Controllability of Second Order Functional Random Differential Equations with Delay

by Mouffak Benchohra, Fatima Bouazzaoui, Erdal Karapinar and Abdelkrim Salim

Mathematics 2022, 10(7), 1120; https://doi.org/10.3390/math10071120 - 31 Mar 2022

Cited by 33 | Viewed by 2041

Abstract

In this article, we study some existence and controllability results for two classes of second order functional differential equations with delay and random effects. To begin, we employ a random fixed point theorem with a stochastic domain to demonstrate the existence of mild [...] Read more.

In this article, we study some existence and controllability results for two classes of second order functional differential equations with delay and random effects. To begin, we employ a random fixed point theorem with a stochastic domain to demonstrate the existence of mild random solutions. Next, we prove that our problems are controllable. Finally, an example is given to validate the theory part. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

13 pages, 279 KiB

Open AccessArticle

Infinite Interval Problems for Fractional Evolution Equations

by Yong Zhou

Mathematics 2022, 10(6), 900; https://doi.org/10.3390/math10060900 - 11 Mar 2022

Cited by 23 | Viewed by 2443

Abstract

In this paper, we investigate infinite interval problems for the fractional evolution equations with Hilfer fractional derivative. By using the generalized Ascoli–Arzelà theorem and some new techniques, we prove the existence of mild solutions of Hilfer fractional evolution equations when the semigroup is [...] Read more.

In this paper, we investigate infinite interval problems for the fractional evolution equations with Hilfer fractional derivative. By using the generalized Ascoli–Arzelà theorem and some new techniques, we prove the existence of mild solutions of Hilfer fractional evolution equations when the semigroup is compact as well as noncompact. In addition, an example is provided to illustrate the results. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

21 pages, 331 KiB

Open AccessArticle

Stability Results of Mixed Type Quadratic-Additive Functional Equation in β-Banach Modules by Using Fixed-Point Technique

by Kandhasamy Tamilvanan, Rubayyi T. Alqahtani and Syed Abdul Mohiuddine

Mathematics 2022, 10(3), 493; https://doi.org/10.3390/math10030493 - 3 Feb 2022

Cited by 5 | Viewed by 1535

Abstract

We aim to introduce the quadratic-additive functional equation (shortly, QA-functional equation) and find its general solution. Then, we study the stability of the kind of Hyers-Ulam result with a view of the aforementioned functional equation by utilizing the technique based on a fixed [...] Read more.

We aim to introduce the quadratic-additive functional equation (shortly, QA-functional equation) and find its general solution. Then, we study the stability of the kind of Hyers-Ulam result with a view of the aforementioned functional equation by utilizing the technique based on a fixed point in the framework of

β

-Banach modules. We here discuss our results for odd and even mappings as well as discuss the stability of mixed cases. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

18 pages, 327 KiB

Open AccessArticle

Anticipated Backward Doubly Stochastic Differential Equations with Non-Lipschitz Coefficients

by Tie Wang and Siyu Cui

Mathematics 2022, 10(3), 396; https://doi.org/10.3390/math10030396 - 27 Jan 2022

Cited by 2 | Viewed by 2019

Abstract

The work presented in this paper focuses on a type of differential equations called anticipated backward doubly stochastic differential equations (ABDSDEs) whose generators not only depend on the anticipated terms of the solution (

Y_{\cdot}, Z_{\cdot}

) but also satisfy [...] Read more.

The work presented in this paper focuses on a type of differential equations called anticipated backward doubly stochastic differential equations (ABDSDEs) whose generators not only depend on the anticipated terms of the solution (

Y_{\cdot}, Z_{\cdot}

) but also satisfy one kind of non-Lipschitz assumption. Firstly, we give the existence and uniqueness theorem. Further, two comparison theorems for the solutions of these equations are obtained after finding a new comparison theorem for backward doubly stochastic differential equations (BDSDEs) with non-Lipschitz coefficients. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

9 pages, 279 KiB

Open AccessArticle

Boundary Value Problem for ψ-Caputo Fractional Differential Equations in Banach Spaces via Densifiability Techniques

by Choukri Derbazi, Zidane Baitiche, Mouffak Benchohra and Yong Zhou

Mathematics 2022, 10(1), 153; https://doi.org/10.3390/math10010153 - 5 Jan 2022

Cited by 6 | Viewed by 1925

Abstract

A novel fixed-point theorem based on the degree of nondensifiability (DND) is used in this article to examine the existence of solutions to a boundary value problem containing the

ψ

-Caputo fractional derivative in Banach spaces. Besides that, an example is included to [...] Read more.

A novel fixed-point theorem based on the degree of nondensifiability (DND) is used in this article to examine the existence of solutions to a boundary value problem containing the

ψ

-Caputo fractional derivative in Banach spaces. Besides that, an example is included to verify our main results. Moreover, the outcomes obtained in this research paper ameliorate and expand some previous findings in this area. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

31 pages, 1283 KiB

Open AccessArticle

A Study of ψ-Hilfer Fractional Boundary Value Problem via Nonlinear Integral Conditions Describing Navier Model

by Songkran Pleumpreedaporn, Weerawat Sudsutad, Chatthai Thaiprayoon, Juan E. Nápoles and Jutarat Kongson

Mathematics 2021, 9(24), 3292; https://doi.org/10.3390/math9243292 - 17 Dec 2021

Cited by 1 | Viewed by 2112

Abstract

This paper investigates existence, uniqueness, and Ulam’s stability results for a nonlinear implicit

ψ

-Hilfer

FBVP

describing Navier model with

NIBC

s. By Banach’s fixed point theorem, the unique property is established. Meanwhile, existence results are proved by using the fixed point theory [...] Read more.

This paper investigates existence, uniqueness, and Ulam’s stability results for a nonlinear implicit

ψ

-Hilfer

FBVP

describing Navier model with

NIBC

s. By Banach’s fixed point theorem, the unique property is established. Meanwhile, existence results are proved by using the fixed point theory of Leray-Schauder’s and Krasnoselskii’s types. In addition, Ulam’s stability results are analyzed. Furthermore, several instances are provided to demonstrate the efficacy of the main results. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

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23 pages, 403 KiB

Open AccessArticle

Existence of Positive Solutions for a Higher-Order Fractional Differential Equation with Multi-Term Lower-Order Derivatives

by Weiwei Liu and Lishan Liu

Mathematics 2021, 9(23), 3031; https://doi.org/10.3390/math9233031 - 26 Nov 2021

Viewed by 1359

Abstract

This paper deals with the study of the existence of positive solutions for a class of nonlinear higher-order fractional differential equations in which the nonlinear term contains multi-term lower-order derivatives. By reducing the order of the highest derivative, the higher-order fractional differential equation [...] Read more.

This paper deals with the study of the existence of positive solutions for a class of nonlinear higher-order fractional differential equations in which the nonlinear term contains multi-term lower-order derivatives. By reducing the order of the highest derivative, the higher-order fractional differential equation is transformed into a lower-order fractional differential equation. Then, combining with the properties of left-sided Riemann–Liouville integral operators, we obtain the existence of the positive solutions of fractional differential equations utilizing some weaker conditions. Furthermore, some examples are given to demonstrate the validity of our main results. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

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10 pages, 257 KiB

Open AccessArticle

Some Inequalities of Extended Hypergeometric Functions

by Shilpi Jain, Rahul Goyal, Praveen Agarwal and Juan L. G. Guirao

Mathematics 2021, 9(21), 2702; https://doi.org/10.3390/math9212702 - 25 Oct 2021

Cited by 2 | Viewed by 2209

Abstract

Hypergeometric functions and their inequalities have found frequent applications in various fields of mathematical sciences. Motivated by the above, we set up certain inequalities including extended type Gauss hypergeometric function and confluent hypergeometric function, respectively, by virtue of Hölder integral inequality and Chebyshev’s [...] Read more.

Hypergeometric functions and their inequalities have found frequent applications in various fields of mathematical sciences. Motivated by the above, we set up certain inequalities including extended type Gauss hypergeometric function and confluent hypergeometric function, respectively, by virtue of Hölder integral inequality and Chebyshev’s integral inequality. We also studied the monotonicity, log-concavity, and log-convexity of extended hypergeometric functions, which are derived by using the inequalities on an extended beta function. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

10 pages, 272 KiB

Open AccessArticle

An Investigation of an Integral Equation Involving Convex–Concave Nonlinearities

by Ravi P. Agarwal, Mohamed Jleli and Bessem Samet

Mathematics 2021, 9(19), 2372; https://doi.org/10.3390/math9192372 - 24 Sep 2021

Cited by 2 | Viewed by 1483

Abstract

We investigate the existence and uniqueness of positive solutions to an integral equation involving convex or concave nonlinearities. A numerical algorithm based on Picard iterations is provided to obtain an approximation of the unique solution. The main tools used in this work are [...] Read more.

We investigate the existence and uniqueness of positive solutions to an integral equation involving convex or concave nonlinearities. A numerical algorithm based on Picard iterations is provided to obtain an approximation of the unique solution. The main tools used in this work are based on partial-ordering methods and fixed-point theory. Our results are supported by examples. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

24 pages, 1542 KiB

Open AccessArticle

Analysis of the Time Fractional-Order Coupled Burgers Equations with Non-Singular Kernel Operators

by Noufe H. Aljahdaly, Ravi P. Agarwal, Rasool Shah and Thongchai Botmart

Mathematics 2021, 9(18), 2326; https://doi.org/10.3390/math9182326 - 19 Sep 2021

Cited by 43 | Viewed by 2628

Abstract

In this article, we have investigated the fractional-order Burgers equation via Natural decomposition method with nonsingular kernel derivatives. The two types of fractional derivatives are used in the article of Caputo–Fabrizio and Atangana–Baleanu derivative. We employed Natural transform on fractional-order Burgers equation followed [...] Read more.

In this article, we have investigated the fractional-order Burgers equation via Natural decomposition method with nonsingular kernel derivatives. The two types of fractional derivatives are used in the article of Caputo–Fabrizio and Atangana–Baleanu derivative. We employed Natural transform on fractional-order Burgers equation followed by inverse Natural transform, to achieve the result of the equations. To validate the method, we have considered a two examples and compared with the exact results. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

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21 pages, 835 KiB

Open AccessArticle

Investigation of the Fractional Strongly Singular Thermostat Model via Fixed Point Techniques

by Mohammed K. A. Kaabar, Mehdi Shabibi, Jehad Alzabut, Sina Etemad, Weerawat Sudsutad, Francisco Martínez and Shahram Rezapour

Mathematics 2021, 9(18), 2298; https://doi.org/10.3390/math9182298 - 17 Sep 2021

Cited by 17 | Viewed by 1893

Abstract

Our main purpose in this paper is to prove the existence of solutions for the fractional strongly singular thermostat model under some generalized boundary conditions. In this way, we use some recent nonlinear fixed-point techniques involving

α

-

ψ

-contractions and

α

-admissible [...] Read more.

Our main purpose in this paper is to prove the existence of solutions for the fractional strongly singular thermostat model under some generalized boundary conditions. In this way, we use some recent nonlinear fixed-point techniques involving

α

-

ψ

-contractions and

α

-admissible maps. Further, we establish the similar results for the hybrid version of the given fractional strongly singular thermostat control model. Some examples are studied to illustrate the consistency of our results. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

39 pages, 3635 KiB

Open AccessArticle

A Multispecies Cross-Diffusion Model for Territorial Development

by Abdulaziz Alsenafi and Alethea B. T. Barbaro

Mathematics 2021, 9(12), 1428; https://doi.org/10.3390/math9121428 - 19 Jun 2021

Cited by 4 | Viewed by 2176

Abstract

We develop an agent-based model on a lattice to investigate territorial development motivated by markings such as graffiti, generalizing a previously-published model to account for K groups instead of two groups. We then analyze this model and present two novel variations. Our model [...] Read more.

We develop an agent-based model on a lattice to investigate territorial development motivated by markings such as graffiti, generalizing a previously-published model to account for K groups instead of two groups. We then analyze this model and present two novel variations. Our model assumes that agents’ movement is a biased random walk away from rival groups’ markings. All interactions between agents are indirect, mediated through the markings. We numerically demonstrate that in a system of three groups, the groups segregate in certain parameter regimes. Starting from the discrete model, we formally derive the continuum system of

2 K

convection–diffusion equations for our model. These equations exhibit cross-diffusion due to the avoidance of the rival groups’ markings. Both through numerical simulations and through a linear stability analysis of the continuum system, we find that many of the same properties hold for the K-group model as for the two-group model. We then introduce two novel variations of the agent-based model, one corresponding to some groups being more timid than others, and the other corresponding to some groups being more threatening than others. These variations present different territorial patterns than those found in the original model. We derive corresponding systems of convection–diffusion equations for each of these variations, finding both numerically and through linear stability analysis that each variation exhibits a phase transition. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)

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