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11 pages, 322 KiB

Open AccessArticle

On a Mathematical Model of a General Autoimmune Disease

by Mikhail Kolev, Nikolay Netov, Iveta Nikolova, Irina Naskinova, Velika Kuneva and Marian Milev

Axioms 2023, 12(11), 1021; https://doi.org/10.3390/axioms12111021 - 30 Oct 2023

Cited by 2 | Viewed by 1136

Abstract

The proposed paper is devoted to presenting and analyzing a kinetic model describing the development of autoimmune disorders. The proposed model is a nonlinear system of differential equations that considers the biological activity of the interacting populations. The main characteristics of autoimmune diseases [...] Read more.

The proposed paper is devoted to presenting and analyzing a kinetic model describing the development of autoimmune disorders. The proposed model is a nonlinear system of differential equations that considers the biological activity of the interacting populations. The main characteristics of autoimmune diseases are taken into account. Preliminaries to the research area are provided. The modeling problem is discretized and solved approximately. The numerical results illustrate typical outcomes of autoimmune diseases. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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13 pages, 281 KiB

Open AccessArticle

Generalizations of Hardy-Type Inequalities by Montgomery Identity and New Green Functions

by Kristina Krulić Himmelreich, Josip Pečarić, Dora Pokaz and Marjan Praljak

Axioms 2023, 12(5), 434; https://doi.org/10.3390/axioms12050434 - 27 Apr 2023

Viewed by 1113

Abstract

In this paper we extend general Hardy’s inequality by appropriately combining Montgomery’s identity and Green functions. Related Grüss and Ostrowski-type inequalities are also derived. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

13 pages, 823 KiB

Open AccessArticle

On the Construction of Quantum and LCD Codes from Cyclic Codes over the Finite Commutative Rings

by Shakir Ali, Amal S. Alali, Mohammad Jeelani, Muhammet Kurulay, Elif Segah Öztas and Pushpendra Sharma

Axioms 2023, 12(4), 367; https://doi.org/10.3390/axioms12040367 - 10 Apr 2023

Cited by 6 | Viewed by 1594

Abstract

Let

F_{q}

be a field of order q, where q is a power of an odd prime p, and

α

and

β

are two non-zero elements of

F_{q}

. The primary goal of this article is to study the [...] Read more.

Let

F_{q}

be a field of order q, where q is a power of an odd prime p, and

α

and

β

are two non-zero elements of

F_{q}

. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring

R = F_{q} [u_{1}, u_{2}] / ⟨ {u_{1}}^{2} - α^{2}, u_{2}^{2} - β^{2}, u_{1} u_{2} - u_{2} u_{1} ⟩

. We decompose the ring R by using orthogonal idempotents

Δ_{1}, Δ_{2}, Δ_{3}

, and

Δ_{4}

as

R = Δ_{1} R \oplus Δ_{2} R \oplus Δ_{3} R \oplus Δ_{4} R

, and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

13 pages, 291 KiB

Open AccessArticle

Between Soft θ-Openness and Soft ω⁰-Openness

by Samer Al Ghour

Axioms 2023, 12(3), 311; https://doi.org/10.3390/axioms12030311 - 20 Mar 2023

Cited by 4 | Viewed by 1303

Abstract

In this paper, we define and investigate soft

ω_{θ}

-open sets as a novel type of soft set. We characterize them and demonstrate that they form a soft topology that lies strictly between the soft topologies of soft

θ

-open sets and [...] Read more.

In this paper, we define and investigate soft

ω_{θ}

-open sets as a novel type of soft set. We characterize them and demonstrate that they form a soft topology that lies strictly between the soft topologies of soft

θ

-open sets and soft

ω^{0}

-open sets. Moreover, we show that soft

ω_{θ}

-open sets and soft

ω^{0}

-open sets are equivalent for soft regular spaces. Furthermore, we investigate the connections between particular types of soft sets in a given soft anti-locally countable space and the soft topological space of soft

ω_{θ}

-open sets generated by it. In addition to these, we define soft

(ω_{θ}, ω)

-sets and soft

(ω_{θ}, θ)

-sets as two classes of sets, and via these sets, we introduce two decompositions of soft

θ

-open sets and soft

ω_{θ}

-open sets, respectively. Finally, the relationships between these three new classes of soft sets and their analogs in general topology are examined. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

16 pages, 3443 KiB

Open AccessArticle

The Qualitative Analysis of Host–Parasitoid Model with Inclusion of Spatial Refuge Effect

by Muhammad Sajjad Shabbir, Qamar Din, Wafa F. Alfwzan and Manuel De la Sen

Axioms 2023, 12(3), 290; https://doi.org/10.3390/axioms12030290 - 10 Mar 2023

Cited by 2 | Viewed by 1344

Abstract

The objective of this work was to investigate the dynamics of host–parasitoid model with spatial refuge effect. For this, two discrete host–parasitoid models were considered under spatial refuge effect. Suppose that a constant population of hosts may seek refuge and protection from an [...] Read more.

The objective of this work was to investigate the dynamics of host–parasitoid model with spatial refuge effect. For this, two discrete host–parasitoid models were considered under spatial refuge effect. Suppose that a constant population of hosts may seek refuge and protection from an attack of parasitoids. We found the parametric factors affecting the existence of the equilibrium points and uniqueness of equilibrium points. A local stability analysis of host–parasitoid models was also carried out. Bifurcation theory was used to observe that the host–parasitoid models undergo Neimark–Sacker bifurcation. The effect of the existence of constant refuge effect on the local stability and bifurcation of models was also explored. Hybrid chaos control methodology was used to control the chaotic behavior of model. In addition, numerical simulations, bifurcation diagrams, and phase portraits of the models are also presented. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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19 pages, 860 KiB

Open AccessArticle

Stabilization of the Moving Front Solution of the Reaction-Diffusion-Advection Problem

by Nikolay Nefedov, Elena Polezhaeva and Natalia Levashova

Axioms 2023, 12(3), 253; https://doi.org/10.3390/axioms12030253 - 1 Mar 2023

Cited by 1 | Viewed by 1427

Abstract

We consider the initial-boundary value problem of reaction-diffusion-advection that has a solution of a front form. The statement comes from the theory of wave physics. We study the question of the solution stabilizing to the stationary one. Proof of the stabilization theorem is [...] Read more.

We consider the initial-boundary value problem of reaction-diffusion-advection that has a solution of a front form. The statement comes from the theory of wave physics. We study the question of the solution stabilizing to the stationary one. Proof of the stabilization theorem is based on the concepts of upper and lower solutions and corollaries from comparison theorems. The upper and lower solutions with large gradients are constructed as modifications of the formal moving front asymptotic approximation in a small parameter. The main idea of the proof is to show that the upper and lower solutions of the initial-boundary value problem get into the attraction domain of the asymptotically stable stationary solution on a sufficiently large time interval. The study conducted in this work gives an answer about the non-local attraction domain of the stationary solution and can give some stationing criteria. The results are illustrated by computational examples. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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21 pages, 3662 KiB

Open AccessArticle

An h-Adaptive Poly-Sinc-Based Local Discontinuous Galerkin Method for Elliptic Partial Differential Equations

by Omar A. Khalil and Gerd Baumann

Axioms 2023, 12(3), 227; https://doi.org/10.3390/axioms12030227 - 21 Feb 2023

Viewed by 1499

Abstract

For the purpose of solving elliptic partial differential equations, we suggest a new approach using an h-adaptive local discontinuous Galerkin approximation based on Sinc points. The adaptive approach, which uses Poly-Sinc interpolation to achieve a predetermined level of approximation accuracy, is a [...] Read more.

For the purpose of solving elliptic partial differential equations, we suggest a new approach using an h-adaptive local discontinuous Galerkin approximation based on Sinc points. The adaptive approach, which uses Poly-Sinc interpolation to achieve a predetermined level of approximation accuracy, is a local discontinuous Galerkin method. We developed an a priori error estimate and demonstrated the exponential convergence of the Poly-Sinc-based discontinuous Galerkin technique, as well as the adaptive piecewise Poly-Sinc method, for function approximation and ordinary differential equations. In this paper, we demonstrate the exponential convergence in the number of iterations of the a priori error estimate derived for the local discontinuous Galerkin technique under the condition that a reliable estimate of the precise solution of the partial differential equation at the Sinc points exists. For the purpose of refining the computational domain, we employ a statistical strategy. The numerical results for elliptic PDEs with Dirichlet and mixed Neumann-Dirichlet boundary conditions are demonstrated to validate the adaptive greedy Poly-Sinc approach. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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30 pages, 2696 KiB

Open AccessArticle

On the Application of the Block Hybrid Methods to Solve Linear and Non-Linear First Order Differential Equations

by Stanford Shateyi

Axioms 2023, 12(2), 189; https://doi.org/10.3390/axioms12020189 - 11 Feb 2023

Cited by 2 | Viewed by 1572

Abstract

Block hybrid methods with intra-step points are considered in this study. These methods are implemented to solve linear and nonlinear single and systems of first order differential equations. The stability, convergence, and accuracy of the proposed methods are qualitatively investigated through the absolute [...] Read more.

Block hybrid methods with intra-step points are considered in this study. These methods are implemented to solve linear and nonlinear single and systems of first order differential equations. The stability, convergence, and accuracy of the proposed methods are qualitatively investigated through the absolute and residual error analysis in some selected cases. A number of different numerical examples are tested to demonstrate the efficiency and applicability of the proposed methods. In this study we also implement the proposed methods to solve chaotic systems such as the Glukhvsky–Dolzhansky system, producing very comparable results to those already in the literature. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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8 pages, 278 KiB

Open AccessArticle

Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility

by Gianni Bosi and Magalì Zuanon

Axioms 2023, 12(2), 123; https://doi.org/10.3390/axioms12020123 - 27 Jan 2023

Cited by 1 | Viewed by 1160

Abstract

We present some lifting theorems for continuous order-preserving functions on locally and

σ

-compact Hausdorff preordered topological spaces. In particular, we show that a preorder on a locally and

σ

-compact Hausdorff topological space has a continuous multi-utility representation if, and only if, [...] Read more.

We present some lifting theorems for continuous order-preserving functions on locally and

σ

-compact Hausdorff preordered topological spaces. In particular, we show that a preorder on a locally and

σ

-compact Hausdorff topological space has a continuous multi-utility representation if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. Such a characterization is also presented by introducing a lifting property of ≾-C-compatible continuous order-preserving functions on closed subspaces. The assumption of paracompactness is also used in connection to lifting conditions. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

19 pages, 347 KiB

Open AccessArticle

Hypersingular Integral Equations of Prandtl’s Type: Theory, Numerical Methods, and Applications

by Ilya Boykov, Vladimir Roudnev and Alla Boykova

Axioms 2022, 11(12), 705; https://doi.org/10.3390/axioms11120705 - 7 Dec 2022

Viewed by 1333

Abstract

In this paper, we propose and justify a spline-collocation method with first-order splines for approximate solution of nonlinear hypersingular integral equations of Prandtl’s type. We obtained the estimates of the convergence rate and the method error. The constructed computational scheme includes a continuous [...] Read more.

In this paper, we propose and justify a spline-collocation method with first-order splines for approximate solution of nonlinear hypersingular integral equations of Prandtl’s type. We obtained the estimates of the convergence rate and the method error. The constructed computational scheme includes a continuous method for solving nonlinear operator equations, which is stable for perturbations of the coefficients and the right-hand sides of equations. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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20 pages, 907 KiB

Open AccessArticle

Computational Framework of the SVIR Epidemic Model with a Non-Linear Saturation Incidence Rate

by Attaullah, Adil Khurshaid, Zeeshan, Sultan Alyobi, Mansour F. Yassen and Din Prathumwan

Axioms 2022, 11(11), 651; https://doi.org/10.3390/axioms11110651 - 17 Nov 2022

Cited by 3 | Viewed by 3111

Abstract

In this study, we developed an autonomous non-linear epidemic model for the transmission dynamics of susceptible, vaccinated, infected, and recovered individuals (SVIR model) with non-linear saturation incidence and vaccination rates. The non-linear saturation incidence rate significantly reduces the death ratio of infected individuals [...] Read more.

In this study, we developed an autonomous non-linear epidemic model for the transmission dynamics of susceptible, vaccinated, infected, and recovered individuals (SVIR model) with non-linear saturation incidence and vaccination rates. The non-linear saturation incidence rate significantly reduces the death ratio of infected individuals by increasing human immunity. We discuss a detailed explanation of the model equilibrium, its basic reproduction number

R_{0},

local stability, and global stability. The disease-free equilibrium is observed to be stable if

R_{0} < 1

, while the endemic equilibrium exists and the disease exists permanently in the population if

R_{0} > 1

. To approximate the solution of the model, the well-known Runge–Kutta (RK4) methodology is utilized. The implications of numerous parameters on the population dynamics of susceptible, vaccinated, infected, and recovered individuals are addressed. We discovered that increasing the value of the disease-included death rate

ψ

has a negative impact on those affected, while it has a positive impact on other populations. Furthermore, the value of interaction between vaccinated and infected

λ_{2}

has a decreasing impact on vulnerable and vaccinated people, while increasing in other populations. On the other hand, the model is solved using Euler and Euler-modified techniques, and the results are compared numerically and graphically. The quantitative computations demonstrate that the RK4 method provides very precise solutions compared to the other approaches. The results show that the suggested SVIR model that approximates the solution method is accurate and useful. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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17 pages, 337 KiB

Open AccessArticle

Joint Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by the Gram Function II

by Antanas Laurinčikas

Axioms 2022, 11(11), 613; https://doi.org/10.3390/axioms11110613 - 4 Nov 2022

Cited by 1 | Viewed by 1174

Abstract

Let

t_{τ}

be a solution to the equation

θ (t) = (τ - 1) π

,

τ > 0

, where

θ (t)

is the increment of the argument of the function [...] Read more.

Let

t_{τ}

be a solution to the equation

θ (t) = (τ - 1) π

,

τ > 0

, where

θ (t)

is the increment of the argument of the function

π^{- s / 2} Γ (s / 2)

along the segment connecting points

s = 1 / 2

and

s = 1 / 2 + i t

.

t_{τ}

is called the Gram function. In the paper, we consider the approximation of collections of analytic functions by shifts of the Riemann zeta-function

(ζ (s + i t_{τ}^{α_{1}}), \dots, ζ (s + i t_{τ}^{α_{r}}))

, where

α_{1}, \dots, α_{r}

are different positive numbers, in the interval

[T, T + H]

with

H = o (T)

,

T \to \infty

, and obtain the positivity of the density of the set of such shifts. Moreover, a similar result is obtained for shifts of a certain absolutely convergent Dirichlet series connected to

ζ (s)

. Finally, an example of the approximation of analytic functions by a composition of the above shifts is given. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

12 pages, 4023 KiB

Open AccessArticle

Hyperbolic B-Spline Function-Based Differential Quadrature Method for the Approximation of 3D Wave Equations

by Mohammad Tamsir, Mutum Zico Meetei and Ahmed H. Msmali

Axioms 2022, 11(11), 597; https://doi.org/10.3390/axioms11110597 - 28 Oct 2022

Cited by 4 | Viewed by 1930

Abstract

We propose a differential quadrature method (DQM) based on cubic hyperbolic B-spline basis functions for computing 3D wave equations. This method converts the problem into a system of ODEs. We use an optimum five-stage and order four SSP Runge-Kutta (SSPRK-(5,4)) scheme to solve [...] Read more.

We propose a differential quadrature method (DQM) based on cubic hyperbolic B-spline basis functions for computing 3D wave equations. This method converts the problem into a system of ODEs. We use an optimum five-stage and order four SSP Runge-Kutta (SSPRK-(5,4)) scheme to solve the obtained system of ODEs. The matrix stability analysis is also investigated. The accuracy and efficiency of the proposed method are demonstrated via three numerical examples. It has been found that the proposed method gives more accurate results than the existing methods. The main purpose of this work is to present an accurate, economically easy-to-implement, and stable technique for solving hyperbolic partial differential equations. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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24 pages, 4982 KiB

Open AccessArticle

A Computational Approach to a Model for HIV and the Immune System Interaction

by Attaullah, Zeeshan, Muhammad Tufail Khan, Sultan Alyobi, Mansour F. Yassen and Din Prathumwan

Axioms 2022, 11(10), 578; https://doi.org/10.3390/axioms11100578 - 21 Oct 2022

Cited by 5 | Viewed by 2211

Abstract

This study deals with the numerical solution of the human immunodeficiency virus (HIV) infection model, which is a significant problem for global public health. Acquired immunodeficiency syndrome (AIDS) is a communicable disease, and HIV is the causative agent for AIDS, which damages the [...] Read more.

This study deals with the numerical solution of the human immunodeficiency virus (HIV) infection model, which is a significant problem for global public health. Acquired immunodeficiency syndrome (AIDS) is a communicable disease, and HIV is the causative agent for AIDS, which damages the ability of the body to fight against disease and easily usual innocuous infections attack the body. On entering the body, HIV infects a large amount of CD4+ T-cells and disturbs the supply rate of these cells from the thymus. Herein, we consider the model with variable source terms in which the production of these cells is a monotonically decreasing function of viral load. Based on the reproduction number, we describe the stability of free equilibrium. The continuous Galerkin–Petrov method, in particular the cGP(2)-method, is implemented to determine the numerical solutions of the model. The influence of different parameters on the population dynamics of healthy/infected CD4+ T-cells and free HIV particles are examined, and the results are presented graphically. On the other hand, the model is solved using the fourth-order Runge–Kutta method, and briefly, the RK4-method, and the results of the proposed schemes are compared with those obtained from other classical schemes such as the Bessel collocation method (BCM), Laplace Adomian decomposition method (LADM), perturbation iteration algorithm (PIA), modified variational iteration method (MVIM), differential transform method (DTM), and exponential Galerkin method (EGM), numerically. Furthermore, absolute errors relative to the RK4 method are computed to describe the accuracy of the proposed scheme. It is presented that the cGP(2)-method gains accurate results at larger time step sizes in comparison with the results of the aforementioned methods. The numerical and graphical comparison reveals that the proposed scheme yields more accurate results relative to other traditional schemes from the literature. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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13 pages, 10352 KiB

Open AccessArticle

Simulation of Marine Debris Path Using Mathematical Model in the Gulf of Thailand

by Jettapol Phiphit, Angkool Wangwongchai and Usa Wannasingha Humphries

Axioms 2022, 11(10), 571; https://doi.org/10.3390/axioms11100571 - 20 Oct 2022

Cited by 2 | Viewed by 2064

Abstract

Marine debris is an important environmental problem that affects aquatic animals, ecosystems, economy, society, and humans. This research aims to simulate the path of marine debris in the Gulf of Thailand using a mathematical model that includes two models: the Oceanic Model (OCM), [...] Read more.

Marine debris is an important environmental problem that affects aquatic animals, ecosystems, economy, society, and humans. This research aims to simulate the path of marine debris in the Gulf of Thailand using a mathematical model that includes two models: the Oceanic Model (OCM), which is based on the Shallow Water Equations (SWE), and the Lagrangian Particle Tracking (LPT) model. The OCM is the partial derivative equation system solved by the finite difference method to satisfy the Arakawa C-grid and the splitting method. The LPT model includes the current velocity, wind velocity at 10 m above sea level, random walk term, and the buoyancy ratio of marine debris with six cases, which are 100:1, 10:1, 1:1, 0:1, 1:10, and 1:100. The current velocity from OCM is applied to the LPT model. This research uses a garbage boat that capsized near Koh Samui on 1 August 2020 as a case study. The simulated current velocity of OCM is compared with Ocean Surface Current Analyses Real-time (OSCAR) data. The Root Mean Square Error (RMSE) of u-velocity is 0.070 m/s, and that of v-velocity is 0.058 m/s. The simulation of the marine debris’s path from the LPT model demonstrates the movement to Koh Samui, Koh Taen, Koh Wang Nai, Koh Wang Nok, Koh Rap, the east coast of Nakorn Si Thammarat province, Phu Quoc Island of Vietnam and the middle of the Gulf of Thailand with the different buoyancy ratios and time durations. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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10 pages, 2768 KiB

Open AccessArticle

Variational Principle and Diverse Wave Structures of the Modified Benjamin-Bona-Mahony Equation Arising in the Optical Illusions Field

by Kang-Jia Wang

Axioms 2022, 11(9), 445; https://doi.org/10.3390/axioms11090445 - 31 Aug 2022

Cited by 30 | Viewed by 2046

Abstract

This study focuses on investigating the modified Benjamin-Bona-Mahony equation that is used to model the long wave in nonlinear dispersive media of the optical illusion field. Two effective techniques, the variational direct method and He’s frequency formulation method, are employed to seek the [...] Read more.

This study focuses on investigating the modified Benjamin-Bona-Mahony equation that is used to model the long wave in nonlinear dispersive media of the optical illusion field. Two effective techniques, the variational direct method and He’s frequency formulation method, are employed to seek the travelling wave solutions. Using these two techniques, abundant exact solutions such as the bright wave, bright-dark wave, bright-like wave, kinky-bright wave and periodic wave solutions, are obtained. The 3-D contours and 2-D curves are drawn to present the dynamic physical behaviors of the solutions by assigning the proper parameters. It shows that the proposed methods are effective but simple and only need one or two steps to construct the exact solutions, which are expected to provide some new insights to study the travelling wave solutions of the PDEs arising in physics. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applications)

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