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Open AccessArticle

New Conditions for Testing the Asymptotic Behavior of Solutions of Odd-Order Neutral Differential Equations with Multiple Delays

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Axioms 2023, 12(7), 658; https://doi.org/10.3390/axioms12070658 - 02 Jul 2023

Cited by 1 | Viewed by 389

Abstract

The purpose of this research is to investigate the asymptotic and oscillatory characteristics of odd-order neutral differential equation solutions with multiple delays. The relationship between the solution and its derivatives of different orders, as well as their related functions, must be understood in [...] Read more.

The purpose of this research is to investigate the asymptotic and oscillatory characteristics of odd-order neutral differential equation solutions with multiple delays. The relationship between the solution and its derivatives of different orders, as well as their related functions, must be understood in order to determine the oscillation terms of the studied equation. In order to contribute to this subject, we create new and significant relationships and inequalities. We use these relationships to create conditions in which positive and N-Kneser solutions of the considered equation are excluded. To obtain these terms, we employ the comparison method and the Riccati technique. Furthermore, we use the relationships obtained to create new criteria, so expanding the existing literature on the field. Finally, we provide an example from the general case to demonstrate the results’ significance. The findings given in this work provide light on the behavior of odd-order neutral differential equation solutions with multiple delays. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

Open AccessArticle

The Regional Enlarged Observability for Hilfer Fractional Differential Equations

by

Abu Bakr Elbukhari

,

Zhenbin Fan

and

Gang Li

Axioms 2023, 12(7), 648; https://doi.org/10.3390/axioms12070648 - 29 Jun 2023

Viewed by 410

Abstract

In this paper, we investigate the concept of regional enlarged observability (ReEnOb) for fractional differential equations (FDEs) with the Hilfer derivative. To proceed this, we develop an approach based on the Hilbert uniqueness method (HUM). We mainly reconstruct the initial state [...] Read more.

In this paper, we investigate the concept of regional enlarged observability (ReEnOb) for fractional differential equations (FDEs) with the Hilfer derivative. To proceed this, we develop an approach based on the Hilbert uniqueness method (HUM). We mainly reconstruct the initial state

ν_{0}^{1}

on an internal subregion

ω

from the whole domain

Ω

with knowledge of the initial information of the system and some given measurements. This approach shows that it is possible to obtain the desired state between two profiles in some selective internal subregions. Our findings develop and generalize some known results. Finally, we give two examples to support our theoretical results. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

Open AccessArticle

Stability Results for the Darboux Problem of Conformable Partial Differential Equations

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Abdellatif Ben Makhlouf

Axioms 2023, 12(7), 640; https://doi.org/10.3390/axioms12070640 - 28 Jun 2023

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Abstract

In this paper, we investigate the Darboux problem of conformable partial differential equations (DPCDEs) using fixed point theory. We focus on the existence and Ulam–Hyers–Rassias stability (UHRS) of the solutions to the problem, which requires finding solutions to nonlinear partial differential equations that [...] Read more.

In this paper, we investigate the Darboux problem of conformable partial differential equations (DPCDEs) using fixed point theory. We focus on the existence and Ulam–Hyers–Rassias stability (UHRS) of the solutions to the problem, which requires finding solutions to nonlinear partial differential equations that satisfy certain boundary conditions. Using fixed point theory, we establish the existence and uniqueness of solutions to the DPCDEs. We then explore the UHRS of the solutions, which measures the sensitivity of the solutions to small perturbations in the equations. We provide three illustrative examples to demonstrate the effectiveness of our approach. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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Open AccessArticle

Hyers–Ulam and Hyers–Ulam–Rassias Stability for Linear Fractional Systems with Riemann–Liouville Derivatives and Distributed Delays

by

Hristo Kiskinov

,

Ekaterina Madamlieva

and

Andrey Zahariev

Axioms 2023, 12(7), 637; https://doi.org/10.3390/axioms12070637 - 27 Jun 2023

Viewed by 377

Abstract

The aim of the present paper is to study the asymptotic properties of the solutions of linear fractional system with Riemann–Liouville-type derivatives and distributed delays. We prove under natural assumptions (similar to those used in the case when the derivatives are first (integer) [...] Read more.

The aim of the present paper is to study the asymptotic properties of the solutions of linear fractional system with Riemann–Liouville-type derivatives and distributed delays. We prove under natural assumptions (similar to those used in the case when the derivatives are first (integer) order) the existence and uniqueness of the solutions in the initial problem for these systems with discontinuous initial functions. As a consequence, we also prove the existence of a unique fundamental matrix for the homogeneous system, which allows us to establish an integral representation of the solutions to the initial problem for the corresponding inhomogeneous system. Then, we introduce for the studied systems a concept for Hyers–Ulam in time stability and Hyers–Ulam–Rassias in time stability. As an application of the obtained results, we propose a new approach (instead of the standard fixed point approach) based on the obtained integral representation and establish sufficient conditions, which guarantee Hyers–Ulam-type stability in time. Finally, it is proved that the Hyers–Ulam-type stability in time leads to Lyapunov stability in time for the investigated homogeneous systems. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

Open AccessArticle

New Wave Solutions for the Two-Mode Caudrey–Dodd–Gibbon Equation

by

Rodica Cimpoiasu

and

Radu Constantinescu

Axioms 2023, 12(7), 619; https://doi.org/10.3390/axioms12070619 - 21 Jun 2023

Viewed by 477

Abstract

In this paper, we present new dynamical properties of the two-mode Caudrey–Dodd–Gibbon (TMCDG) equation. This equation describes the propagation of dual waves in the same direction with different phase velocities, dispersion parameters, and nonlinearity. This study takes a full advantage of the Kudryashov [...] Read more.

In this paper, we present new dynamical properties of the two-mode Caudrey–Dodd–Gibbon (TMCDG) equation. This equation describes the propagation of dual waves in the same direction with different phase velocities, dispersion parameters, and nonlinearity. This study takes a full advantage of the Kudryashov method and of the exponential expansion method. For the first time, dual-wave solutions are obtained for arbitrary values of the nonlinearity and dispersive factors. Graphs of the novel solutions are included in order to show the waves’ propagation, as well as the influence of the involved parameters. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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Open AccessArticle

The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches

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Khadijah M. Abualnaja

Axioms 2023, 12(6), 599; https://doi.org/10.3390/axioms12060599 - 16 Jun 2023

Viewed by 469

Abstract

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the [...] Read more.

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the

β

-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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Open AccessArticle

Study on the Nonlinear Dynamics of the (3+1)-Dimensional Jimbo-Miwa Equation in Plasma Physics

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Axioms 2023, 12(6), 592; https://doi.org/10.3390/axioms12060592 - 15 Jun 2023

Viewed by 512

Abstract

The Jimbo-Miwa equation (JME) that describes certain interesting (3+1)-dimensional waves in plasma physics is studied in this work. The Hirota bilinear equation is developed via the Cole-Hopf transform. Then, the symbolic computation, together with the ansatz function schemes, are utilized to seek exact [...] Read more.

The Jimbo-Miwa equation (JME) that describes certain interesting (3+1)-dimensional waves in plasma physics is studied in this work. The Hirota bilinear equation is developed via the Cole-Hopf transform. Then, the symbolic computation, together with the ansatz function schemes, are utilized to seek exact solutions. Some new solutions, such as the multi-wave complexiton solution (MWCS), multi-wave solution (MWS) and periodic lump solution (PLS), are successfully constructed. Additionally, different types of travelling wave solutions (TWS), including the dark, bright-dark and singular periodic wave solutions, are disclosed by employing the sub-equation method. Finally, the physical characteristics and interaction behaviors of the extracted solutions are depicted graphically by assigning appropriate parameters. The obtained outcomes in this paper are more general and newer. Additionally, they reveal that the used methods are concise, direct, and can be employed to study other partial differential equations (PDEs) in physics. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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Open AccessArticle

Mittag-Leffler-Type Stability of BAM Neural Networks Modeled by the Generalized Proportional Riemann–Liouville Fractional Derivative

by

Ravi P. Agarwal

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Snezhana Hristova

and

Donal O’Regan

Axioms 2023, 12(6), 588; https://doi.org/10.3390/axioms12060588 - 14 Jun 2023

Cited by 1 | Viewed by 439

Abstract

The main goal of the paper is to use a generalized proportional Riemann–Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved, such as the fractional derivative [...] Read more.

The main goal of the paper is to use a generalized proportional Riemann–Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved, such as the fractional derivative of a squared function. Additionally, some comparison results for GPRLFD are provided. Two types of equilibrium of the BAM model with GPRLFD are defined. In connection with the applied fractional derivative and its singularity at the initial time, the Mittag-Leffler exponential stability in time of the equilibrium is introduced and studied. An example is given, illustrating the meaning of the equilibrium as well as its stability properties. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

Open AccessArticle

Solvability of Iterative Classes of Nonlinear Elliptic Equations on an Exterior Domain

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Axioms 2023, 12(5), 474; https://doi.org/10.3390/axioms12050474 - 14 May 2023

Viewed by 481

Abstract

This work explores the possibility that iterative classes of elliptic equations have both single and coupled positive radial solutions. Our approach is based on using the well-known Guo–Krasnoselskii and Avery–Henderson fixed-point theorems in a Banach space. Furthermore, we utilize Rus’ theorem in a [...] Read more.

This work explores the possibility that iterative classes of elliptic equations have both single and coupled positive radial solutions. Our approach is based on using the well-known Guo–Krasnoselskii and Avery–Henderson fixed-point theorems in a Banach space. Furthermore, we utilize Rus’ theorem in a metric space, to prove the uniqueness of solutions for the problem. Examples are constructed for the sake of verification. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

Open AccessArticle

The Influence of White Noise and the Beta Derivative on the Solutions of the BBM Equation

by

Farah M. Al-Askar

,

Clemente Cesarano

and

Wael W. Mohammed

Axioms 2023, 12(5), 447; https://doi.org/10.3390/axioms12050447 - 30 Apr 2023

Cited by 5 | Viewed by 628

Abstract

In the current study, we investigate the stochastic Benjamin–Bona–Mahony equation with beta derivative (SBBME-BD). The considered stochastic term is the multiplicative noise in the Itô sense. By combining the

F

-expansion approach with two separate equations, such as the Riccati and elliptic equations, [...] Read more.

In the current study, we investigate the stochastic Benjamin–Bona–Mahony equation with beta derivative (SBBME-BD). The considered stochastic term is the multiplicative noise in the Itô sense. By combining the

F

-expansion approach with two separate equations, such as the Riccati and elliptic equations, new hyperbolic, trigonometric, rational, and Jacobi elliptic solutions for SBBME-BD can be generated. The solutions to the Benjamin–Bona–Mahony equation are useful in understanding various scientific phenomena, including Rossby waves in spinning fluids and drift waves in plasma. Our results are presented using MATLAB, with numerous 3D and 2D figures illustrating the impacts of white noise and the beta derivative on the obtained solutions of SBBME-BD. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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Open AccessArticle

Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation

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Axioms 2023, 12(4), 371; https://doi.org/10.3390/axioms12040371 - 12 Apr 2023

Cited by 1 | Viewed by 502

Abstract

We consider the possibility of constructing a hierarchy of the complex extension of the Korteweg–de Vries equation (cKdV), which under the assumption that the function is real passes into the KdV hierarchy. A hierarchy is understood here as a family of nonlinear partial [...] Read more.

We consider the possibility of constructing a hierarchy of the complex extension of the Korteweg–de Vries equation (cKdV), which under the assumption that the function is real passes into the KdV hierarchy. A hierarchy is understood here as a family of nonlinear partial differential equations with a Lax pair with a common scattering operator. The cKdV hierarchy is obtained by examining the equation on the eigenvalues of the fourth-order Hermitian self-conjugate operator on the invariant transformations of the eigenvector-functions. It is proved that for an operator

{\hat{H}}_{n}

to transform a solution of the equation on eigenvalues

(\hat{M} - λ E) V = 0

into a solution of the same equation, it is necessary and sufficient that the complex function

u (x, t)

of the operator

\hat{M}

satisfies special conditions that are the complexifications of the KdV hierarchy equations. The operators

{\hat{H}}_{n}

are constructed as differential operators of order 2n + 1. We also construct a hierarchy of perturbed KdV equations (pKdV) with a special perturbation function, the dynamics of which is described by a linear equation. It is based on the system of operator equations obtained by Bogoyavlensky. Since the elements of the hierarchies are united by a common scattering operator, it remains unchanged in the derivation of the equations. The second differential operator of the Lax pair has increasing odd derivatives while retaining a skew-symmetric form. It is shown that when perturbation tends to zero, all hierarchy equations are converted to higher KdV equations. It is proved that the pKdV hierarchy equations are a necessary and sufficient condition for the solutions of the equation on eigenvalues to have invariant transformations. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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Open AccessArticle

A Fractional Rheological Model of Viscoanelastic Media

by

Armando Ciancio

,

Vincenzo Ciancio

and

Bruno Felice Filippo Flora

Axioms 2023, 12(3), 243; https://doi.org/10.3390/axioms12030243 - 27 Feb 2023

Cited by 1 | Viewed by 579

Abstract

The mechanical behaviour of materials can be described by a phenomenological relationship that binds strain to stress, by the complex modulus function:

M (ω)

, which represents the frequency response of the medium in which a transverse mechanical wave is propagated. [...] Read more.

The mechanical behaviour of materials can be described by a phenomenological relationship that binds strain to stress, by the complex modulus function:

M (ω)

, which represents the frequency response of the medium in which a transverse mechanical wave is propagated. From the experimental measurements of the internal friction obtained when varying the frequency of a transverse mechanical wave, the parameters that characterize the complex module are determined. The internal friction or loss tangent is bound to the dissipation of the specific mechanical energy. The non-equilibrium thermodynamics theory leads to a general description of irreversible phenomena such as relaxation and viscosity that can coexist in a material. Through the state variables introduced by Ciancio and Kluitenberg, and applying the fractional calculation due to a particular memory mechanism, a model of a viscoanelastic medium is obtained in good agreement with the experimental results. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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Open AccessArticle

Periodic Solutions of Quasi-Monotone Semilinear Multidimensional Hyperbolic Systems

by

Corrado Mascia

Axioms 2023, 12(2), 208; https://doi.org/10.3390/axioms12020208 - 16 Feb 2023

Viewed by 630

Abstract

This paper deals with the Cauchy problem for a class of first-order semilinear hyperbolic equations of the form

\partial_{t} f_{i} + \sum_{j = 1}^{d} λ_{i j} \partial_{x_{j}} f_{i} = Q_{i} (f) .

[...] Read more.

This paper deals with the Cauchy problem for a class of first-order semilinear hyperbolic equations of the form

\partial_{t} f_{i} + \sum_{j = 1}^{d} λ_{i j} \partial_{x_{j}} f_{i} = Q_{i} (f) .

where

f_{i} = f_{i} (x, t)

(

i = 1, \dots, n

) and

x = (x_{1}, \dots, x_{d}) \in {I R}^{d}

(

n \geq 2, d \geq 1

). Under assumption of the existence of a conserved quantity

\sum_{i} α_{i} f_{i}

for some

α_{1}, \dots, α_{n} > 0

, of (strong) quasimonotonicity and an additional assumption on the speed vectors

Λ_{i} = (λ_{i 1}, \dots, λ_{i d}) \in {I R}^{d}

—namely,

span {Λ_{j} - Λ_{k} : j = 1, \dots, n} = {I R}^{d}

for any k—it is proved that the set of constant steady state

{\bar{f} \in {I R}^{n} : Q (\bar{f}) = 0}

is asymptotically stable with respect to periodic perturbations, i.e., any initial data given by an periodic

L^{1}

–perturbations of a constant steady state

\bar{f}

leads to a solution converging to another constant steady state

\bar{g}

(uniquely determined by the initial condition) as

t \to + \infty

. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

Open AccessArticle

Two Reliable Computational Techniques for Solving the MRLW Equation

by

Kamel Al-Khaled

and

Haneen Jafer

Axioms 2023, 12(2), 174; https://doi.org/10.3390/axioms12020174 - 08 Feb 2023

Cited by 1 | Viewed by 708

Abstract

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained using the Sinc-collocation method. This approach approximates the space dimension of the solution with a cardinal expansion of Sinc functions. First, discretizing the time derivative of the [...] Read more.

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained using the Sinc-collocation method. This approach approximates the space dimension of the solution with a cardinal expansion of Sinc functions. First, discretizing the time derivative of the MRLW equation by a classic finite difference formula, while the space derivatives are approximated by a

θ —

weighted scheme. For comparison purposes, we also find a soliton solution using the Adomian decomposition method (ADM). The Sinc-collocation method was were found to be more accurate and efficient than the ADM schemes. Furthermore, we show that the number of solitons generated can be approximated using the Maxwellian initial condition. The proposed methods’ results, analytical solutions, and numerical methods are compared. Finally, a variety of graphical representations for the obtained solutions makes the dynamics of the MRLW equation visible and provides the mathematical foundation for physical and engineering applications. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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Open AccessArticle

Study of the Hypergeometric Equation via Data Driven Koopman-EDMD Theory

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Axioms 2023, 12(2), 134; https://doi.org/10.3390/axioms12020134 - 29 Jan 2023

Viewed by 757

Abstract

We consider a data-driven method, which combines Koopman operator theory with Extended Dynamic Mode Decomposition. We apply this method to the hypergeometric equation which is the Fuchsian equation with three regular singular points. The space of solutions at any of its singular points [...] Read more.

We consider a data-driven method, which combines Koopman operator theory with Extended Dynamic Mode Decomposition. We apply this method to the hypergeometric equation which is the Fuchsian equation with three regular singular points. The space of solutions at any of its singular points is a two-dimensional linear vector space on the field of reals when the independent variable is restricted to take values in the real axis and the unknown function is restricted to be a real-valued function of a real variable. A basis of the linear vector space of solutions is spanned by the hypergeometric function and its products with appropriate powers of the independent variable or the logarithmic function depending on the roots of the indicial equation of the hypergeometric equation. With our work, we obtain a new representation of the fundamental solutions of the hypergeometric equation and relate them to the spectral analysis of the finite approximation of the Koopman operator associated with the hypergeometric equation. We expect that the usefulness of our results will come more to the fore when we extend our study into the complex domain. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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