Research

Open AccessArticle

Some Relations on the _rR_s(P,Q,z) Matrix Function

by

Ayman Shehata

,

Ghazi S. Khammash

and

Carlo Cattani

Axioms 2023, 12(9), 817; https://doi.org/10.3390/axioms12090817 - 25 Aug 2023

Viewed by 259

Abstract

In this paper, we derive some classical and fractional properties of the

_{r} R_{s}

matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as [...] Read more.

In this paper, we derive some classical and fractional properties of the

_{r} R_{s}

matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss hypergeometric matrix functions. We will also show the relationship with other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

Open AccessArticle

Effects of the Wiener Process and Beta Derivative on the Exact Solutions of the Kadomtsev–Petviashvili Equation

by

Farah M. Al-Askar

,

Clemente Cesarano

and

Wael W. Mohammed

Axioms 2023, 12(8), 748; https://doi.org/10.3390/axioms12080748 - 29 Jul 2023

Viewed by 315

Abstract

We take into account the (2 + 1)-dimensional stochastic Kadomtsev–Petviashvili equation with beta-derivative (SKPE-BD) in this paper. To develop new hyperbolic, trigonometric, elliptic, and rational solutions, the Riccati equation and Jacobi elliptic function methods are employed. Because the KP equation is required for [...] Read more.

We take into account the (2 + 1)-dimensional stochastic Kadomtsev–Petviashvili equation with beta-derivative (SKPE-BD) in this paper. To develop new hyperbolic, trigonometric, elliptic, and rational solutions, the Riccati equation and Jacobi elliptic function methods are employed. Because the KP equation is required for explaining the development of quasi-one-dimensional shallow-water waves, the solutions obtained can be used to interpret various attractive physical phenomena. To display how the multiplicative white noise and beta-derivative impact the exact solutions of the SKPE-BD, we plot a few graphs in MATLAB and display different 3D and 2D figures. We deduce how multiplicative noise stabilizes the solutions of SKPE-BD at zero. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

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Open AccessArticle

Study of Time-Fractional Nonlinear Model Governing Unsteady Flow of Polytropic Gas

by

,

,

,

and

Axioms 2023, 12(3), 285; https://doi.org/10.3390/axioms12030285 - 08 Mar 2023

Viewed by 578

Abstract

The present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis

J

-transform [...] Read more.

The present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis

J

-transform method (

_{O}

HA

J

TM) and

J

-variational iteration transform method (

J

-VITM) have been adopted. The _OHA

J

TM is the hybrid method, where optimal-homotopy analysis method (

_{O}

HAM) is utilized after implementing the properties of

J

-transform (

J

T), and in

J

-VITM is the

J

-transform-based variational iteration method. Banach’s fixed point approach is adopted to analyze the convergence of these methods. It is demonstrated that

J

-VITM is

T

-stable, and the evaluated dynamics of pGas are described in terms of Mittag–Leffler functions. The proposed evaluation confirms that the implemented methods perform better for the referred model equation of pGas. In addition, for a given iteration, the proposed behavior via

_{O}

HA

J

TM performs better in producing more accurate behavior in comparison to

J

-VITM and the methods introduced recently. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

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Open AccessArticle

Electrothermal Monte Carlo Simulation of a GaAs Resonant Tunneling Diode

by

Orazio Muscato

Axioms 2023, 12(2), 216; https://doi.org/10.3390/axioms12020216 - 19 Feb 2023

Viewed by 672

Abstract

This paper deals with the electron transport and heat generation in a Resonant Tunneling Diode semiconductor device. A new electrothermal Monte Carlo method is introduced. The method couples a Monte Carlo solver of the Boltzmann–Wigner transport equation with a steady-state solution of the [...] Read more.

This paper deals with the electron transport and heat generation in a Resonant Tunneling Diode semiconductor device. A new electrothermal Monte Carlo method is introduced. The method couples a Monte Carlo solver of the Boltzmann–Wigner transport equation with a steady-state solution of the heat diffusion equation. This methodology provides an accurate microscopic description of the spatial distribution of self-heating and its effect on the detailed nonequilibrium carrier dynamics. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

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Open AccessArticle

Dynamic Behaviors of a COVID-19 and Influenza Co-Infection Model with Time Delays and Humoral Immunity

by

Ahmed M. Elaiw

,

Raghad S. Alsulami

and

Aatef D. Hobiny

Axioms 2023, 12(2), 151; https://doi.org/10.3390/axioms12020151 - 01 Feb 2023

Cited by 1 | Viewed by 2928

Abstract

Co-infections with respiratory viruses were reported in hospitalized patients in several cases. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and influenza A virus (IAV) are two respiratory viruses and are similar in terms of their seasonal occurrence, clinical manifestations, transmission routes, and related [...] Read more.

Co-infections with respiratory viruses were reported in hospitalized patients in several cases. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and influenza A virus (IAV) are two respiratory viruses and are similar in terms of their seasonal occurrence, clinical manifestations, transmission routes, and related immune responses. SARS-CoV-2 is the cause of coronavirus disease 2019 (COVID-19). In this paper, we study the dynamic behaviors of an influenza and COVID-19 co-infection model in vivo. The role of humoral (antibody) immunity in controlling the co-infection is modeled. The model considers the interactions among uninfected epithelial cells (ECs), SARS-CoV-2-infected ECs, IAV-infected ECs, SARS-CoV-2 particles, IAV particles, SARS-CoV-2 antibodies, and IAV antibodies. The model is given by a system of delayed ordinary differential equations (DODEs), which include four time delays: (i) a delay in the SARS-CoV-2 infection of ECs, (ii) a delay in the IAV infection of ECs, (iii) a maturation delay of newly released SARS-CoV-2 virions, and (iv) a maturation delay of newly released IAV virions. We establish the non-negativity and boundedness of the solutions. We examine the existence and stability of all equilibria. The Lyapunov method is used to prove the global stability of all equilibria. The theoretical results are supported by performing numerical simulations. We discuss the effects of antiviral drugs and time delays on the dynamics of influenza and COVID-19 co-infection. It is noted that increasing the delay length has a similar influence to that of antiviral therapies in eradicating co-infection from the body. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

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Open AccessArticle

Analysis of Finite Solution Spaces of Second-Order ODE with Dirac Delta Periodic Forcing

by

Susmit Bagchi

Axioms 2023, 12(1), 85; https://doi.org/10.3390/axioms12010085 - 13 Jan 2023

Viewed by 928

Abstract

Second-order Ordinary Differential Equations (ODEs) with discontinuous forcing have numerous applications in engineering and computational sciences. The analysis of the solution spaces of non-homogeneous ODEs is difficult due to the complexities in multidimensional systems, with multiple discontinuous variables present in forcing functions. Numerical [...] Read more.

Second-order Ordinary Differential Equations (ODEs) with discontinuous forcing have numerous applications in engineering and computational sciences. The analysis of the solution spaces of non-homogeneous ODEs is difficult due to the complexities in multidimensional systems, with multiple discontinuous variables present in forcing functions. Numerical solutions are often prone to failures in the presence of discontinuities. Algebraic decompositions are employed for analysis in such cases, assuming that regularities exist, operators are present in Banach (solution) spaces, and there is finite measurability. This paper proposes a generalized, finite-dimensional algebraic analysis of the solution spaces of second-order ODEs equipped with periodic Dirac delta forcing. The proposed algebraic analysis establishes the conditions for the convergence of responses within the solution spaces without requiring relative smoothness of the forcing functions. The Lipschitz regularizations and Lebesgue measurability are not considered as preconditions maintaining generality. The analysis shows that smooth and locally finite responses can be admitted in an exponentially stable solution space. The numerical analysis of the solution spaces is computed based on combinatorial changes in coefficients. It exhibits a set of locally uniform responses in the solution spaces. In contrast, the global response profiles show localized as well as oriented instabilities at specific neighborhoods in the solution spaces. Furthermore, the bands of the expansions–contractions of the stable response profiles are observable within the solution spaces depending upon the values of the coefficients and time intervals. The application aspects and distinguishing properties of the proposed approaches are outlined in brief. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

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Open AccessArticle

Boundary-Value Problem for Nonlinear Fractional Differential Equations of Variable Order with Finite Delay via Kuratowski Measure of Noncompactness

by

Benoumran Telli

,

Mohammed Said Souid

and

Ivanka Stamova

Axioms 2023, 12(1), 80; https://doi.org/10.3390/axioms12010080 - 12 Jan 2023

Cited by 3 | Viewed by 741

Abstract

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. [...] Read more.

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. All of the results in this study are established with the help of generalized intervals and piecewise constant functions. We convert the Riemann–Liouville fractional variable-order problem to equivalent standard Riemann–Liouville problems of fractional-constant orders. Finally, two examples are constructed to illustrate the validity of the observed results. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

Open AccessArticle

Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation

by

Essam Roshdy El-Zahar

and

Abdelhalim Ebaid

Axioms 2022, 11(12), 741; https://doi.org/10.3390/axioms11120741 - 17 Dec 2022

Cited by 1 | Viewed by 770

Abstract

In this paper, the pantograph delay differential equation

y^{'} (t) = a y (t) + b y (c t)

subject to the condition

y (0) = λ

is reanalyzed for the real constants a, b [...] Read more.

In this paper, the pantograph delay differential equation

y^{'} (t) = a y (t) + b y (c t)

subject to the condition

y (0) = λ

is reanalyzed for the real constants a, b, and c. In the literature, it has been shown that the pantograph delay differential equation, for

λ = 1

, is well-posed if

c < 1

, but not if

c > 1

. In addition, the solution is available in the form of a standard power series when

λ = 1

. In the present research, we are able to determine the solution of the pantograph delay differential equation in a closed series form in terms of exponential functions. The convergence of such a series is analysed. It is found that the solution converges for

c \in (- 1, 1)

such that

|\frac{b}{a}| < 1

and it also converges for

c > 1

when

a < 0

. For

c = - 1

, the exact solution is obtained in terms of trigonometric functions, i.e., a periodic solution with periodicity

\frac{2 π}{\sqrt{b^{2} - a^{2}}}

when

b > a

. The current results are introduced for the first time and have not been reported in the relevant literature. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

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