Mathematics

Research

18 pages, 3915 KiB

Open AccessArticle

Charged Cavitation Multibubbles Dynamics Model: Growth Process

by Ahmed K. Abu-Nab, Amerah M. Hakami and Ali F. Abu-Bakr

Mathematics 2024, 12(4), 569; https://doi.org/10.3390/math12040569 - 14 Feb 2024

Cited by 6 | Viewed by 1415

The nonlinear dynamics of charged cavitation bubbles are investigated theoretically and analytically in this study through the Rayleigh–Plesset model in dielectric liquids. The physical and mathematical situations consist of two models: the first one is noninteracting charged cavitation bubbles (like single cavitation bubble) [...] Read more.

The nonlinear dynamics of charged cavitation bubbles are investigated theoretically and analytically in this study through the Rayleigh–Plesset model in dielectric liquids. The physical and mathematical situations consist of two models: the first one is noninteracting charged cavitation bubbles (like single cavitation bubble) and the second one is interacting charged cavitation bubbles. The proposed models are formulated and solved analytically based on the Plesset–Zwick technique. The study examines the behaviour of charged cavitation bubble growth processes under the influence of the polytropic exponent, the number of bubbles

N

, and the distance between the bubbles. From our analysis, it is observed that the radius of charged cavitation bubbles increases with increases in the distance between the bubbles, dimensionless phase transition criteria, and thermal diffusivity, and is inversely proportional to the polytropic exponent and the number of bubbles

N

. Additionally, it is evident that the growth process of charged cavitation bubbles is enhanced significantly when the number of bubbles is reduced. The electric charges and polytropic exponent weakens the growth process of charged bubbles in dielectric liquids. The obtained results are compared with experimental and theoretical previous works to validate the given solutions of the presented models of noninteraction and interparticle interaction of charged cavitation bubbles. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

► Show Figures

Figure 1

11 pages, 255 KiB

Open AccessArticle

Qualitative Properties of the Solutions to the Lane–Emden Equation in the Cylindrical Setup

by Arsen Palestini and Simone Recchi

Mathematics 2024, 12(4), 542; https://doi.org/10.3390/math12040542 - 9 Feb 2024

Viewed by 1191

Abstract

We analyze the Lane–Emden equations in the cylindrical framework. Although the explicit forms of the solutions (which are also called polytropes) are not known, we identify some of their qualitative properties. In particular, possible critical points and zeros of the polytropes are investigated [...] Read more.

We analyze the Lane–Emden equations in the cylindrical framework. Although the explicit forms of the solutions (which are also called polytropes) are not known, we identify some of their qualitative properties. In particular, possible critical points and zeros of the polytropes are investigated and discussed, leading to possible improvements in the approximation methods which are currently employed. The cases when the critical parameter is odd and even are separately analyzed. Furthermore, we propose a technique to evaluate the distance between a pair of polytropes in small intervals. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

16 pages, 317 KiB

Open AccessArticle

Compactness of Commutators for Riesz Potential on Generalized Morrey Spaces

by Nurzhan Bokayev, Dauren Matin, Talgat Akhazhanov and Aidos Adilkhanov

Mathematics 2024, 12(2), 304; https://doi.org/10.3390/math12020304 - 17 Jan 2024

Cited by 2 | Viewed by 1158

Abstract

In this paper, we give the sufficient conditions for the compactness of sets in generalized Morrey spaces

M_{p}^{w (\cdot)}

. This result is an analogue of the well-known Fréchet–Kolmogorov theorem on the compactness of a set in Lebesgue spaces [...] Read more.

In this paper, we give the sufficient conditions for the compactness of sets in generalized Morrey spaces

M_{p}^{w (\cdot)}

. This result is an analogue of the well-known Fréchet–Kolmogorov theorem on the compactness of a set in Lebesgue spaces

L_{p}, p > 0 .

As an application, we prove the compactness of the commutator of the Riesz potential

[b, I_{α}]

in generalized Morrey spaces, where

b \in V M O

(

V M O (R^{n})

denote the

B M O

-closure of

C_{0}^{\infty} (R^{n})

). We prove auxiliary statements regarding the connection between the norm of average functions and the norm of the difference of functions in the generalized Morrey spaces. Such results are also of independent interest. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

26 pages, 811 KiB

Open AccessArticle

A Bimodal Model for Oil Prices

by Joanna Goard and Mohammed AbaOud

Mathematics 2023, 11(10), 2222; https://doi.org/10.3390/math11102222 - 9 May 2023

Cited by 1 | Viewed by 2268

Abstract

Oil price behaviour over the last 10 years has shown to be bimodal in character, displaying a strong tendency to congregate around one range of high oil prices and one range of low prices, indicating two distinct peaks in its frequency distribution. In [...] Read more.

Oil price behaviour over the last 10 years has shown to be bimodal in character, displaying a strong tendency to congregate around one range of high oil prices and one range of low prices, indicating two distinct peaks in its frequency distribution. In this paper, we propose a new, single nonlinear stochastic process to model the bimodal behaviour, namely,

d p = α (p_{1} - p) (p_{2} - p_{(} p_{3} - p) d t + σ p^{γ} d Z, γ = 0, 0.5

. Further, we find analytic approximations of oil price futures under this model in the cases where the stable fixed points of the corresponding deterministic model are (a) evenly spaced about the unstable fixed point and (b) are spaced in the ratio 1:2 about the unstable fixed point. The solutions are shown to produce accurate prices when compared to numerical solutions. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

► Show Figures

Figure 1

12 pages, 1965 KiB

Open AccessArticle

Three-Dimensional Unsteady Mixed Convection Flow of Non-Newtonian Nanofluid with Consideration of Retardation Time Effects

by Badreddine Ayadi, Kaouther Ghachem, Kamel Al-Khaled, Sami Ullah Khan, Karim Kriaa, Chemseddine Maatki, Nesrine Zahi and Lioua Kolsi

Mathematics 2023, 11(8), 1892; https://doi.org/10.3390/math11081892 - 17 Apr 2023

Cited by 1 | Viewed by 1600

Abstract

The advances in nanotechnology led to the development of new kinds of engineered fluids called nanofluids. Nanofluids have several industrial and engineering applications, such as solar energy systems, heat conduction processes, nuclear systems, chemical processes, etc. The motivation of the present work is [...] Read more.

The advances in nanotechnology led to the development of new kinds of engineered fluids called nanofluids. Nanofluids have several industrial and engineering applications, such as solar energy systems, heat conduction processes, nuclear systems, chemical processes, etc. The motivation of the present work is to analyze and explore the thermal and dynamic behaviors of a non-Newtonian fluid flow under time retardation effects. The flow is unsteady and caused by a bidirectional, periodically moving surface. In addition to the convective heat transfer and fluid flow, the radiation and chemical reactions have also been considered. The governing equations are established based on the modified Cattaneo–Christov heat flux formulation. It was found that the bidirectional velocities oscillate periodically, and that the magnitude of the oscillation increases with the retardation time. Higher temperatures occur when the porosity parameter is increased, and lower concentrations are encountered for higher values of the concentration relaxation parameter. The current results can be applied in thermal systems, heat transfer enhancement, chemical synthesis, solar systems, power generation, medical applications, the automotive industry, process industries, refrigeration, etc. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

► Show Figures

Figure 1

16 pages, 7288 KiB

Open AccessArticle

ADI Method for Pseudoparabolic Equation with Nonlocal Boundary Conditions

by Mifodijus Sapagovas, Artūras Štikonas and Olga Štikonienė

Mathematics 2023, 11(6), 1303; https://doi.org/10.3390/math11061303 - 8 Mar 2023

Cited by 6 | Viewed by 1899

Abstract

This paper deals with the numerical solution of nonlocal boundary-value problem for two-dimensional pseudoparabolic equation which arise in many physical phenomena. A three-layer alternating direction implicit method is investigated for the solution of this problem. This method generalizes Peaceman–Rachford’s ADI method for the [...] Read more.

This paper deals with the numerical solution of nonlocal boundary-value problem for two-dimensional pseudoparabolic equation which arise in many physical phenomena. A three-layer alternating direction implicit method is investigated for the solution of this problem. This method generalizes Peaceman–Rachford’s ADI method for the 2D parabolic equation. The stability of the proposed method is proved in the special norm. We investigate algebraic eigenvalue problem with nonsymmetric matrices to prove this stability. Numerical results are presented. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

► Show Figures

Figure 1

12 pages, 298 KiB

Open AccessArticle

An Alternative Numerical Scheme to Approximate the Early Exercise Boundary of American Options

by Denis Veliu, Roberto De Marchis, Mario Marino and Antonio Luciano Martire

Mathematics 2023, 11(1), 187; https://doi.org/10.3390/math11010187 - 29 Dec 2022

Viewed by 1821

Abstract

This paper deals with a new numerical method for the approximation of the early exercise boundary in the American option pricing problem. In more detail, using the mean-value theorem for integrals, we provide a flexible algorithm that allows for reaching a more accurate [...] Read more.

This paper deals with a new numerical method for the approximation of the early exercise boundary in the American option pricing problem. In more detail, using the mean-value theorem for integrals, we provide a flexible algorithm that allows for reaching a more accurate numerical solution with fewer calculations rather than other previously described methods. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

► Show Figures

Figure 1

15 pages, 580 KiB

Open AccessFeature PaperArticle

Reclamation of a Resource Extraction Site Model with Random Components

by Ekaterina Gromova, Anastasiia Zaremba and Nahid Masoudi

Mathematics 2022, 10(24), 4805; https://doi.org/10.3390/math10244805 - 17 Dec 2022

Viewed by 1490

Abstract

We compute the cooperative and the Nash equilibrium solutions for the discounted optimal control problem in a two-player differential game of reclamation of a resource extraction site, where each firm’s planning horizon presents the period that extraction of the resources from their site [...] Read more.

We compute the cooperative and the Nash equilibrium solutions for the discounted optimal control problem in a two-player differential game of reclamation of a resource extraction site, where each firm’s planning horizon presents the period that extraction of the resources from their site is economically viable. Hence, the planning horizon is defined by a random duration determined on the infinite time horizon. The comparison of the cooperative and Nash solutions and also the comparative statics are provided numerically. We also define the concept of “normalized value of cooperation” and explain how this concept could help us to better characterize the losses the players will face if they continue to refrain from cooperation. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

► Show Figures

Figure 1

18 pages, 296 KiB

Open AccessArticle

On the Fundamental Analyses of Solutions to Nonlinear Integro-Differential Equations of the Second Order

by Cemil Tunç and Osman Tunç

Mathematics 2022, 10(22), 4235; https://doi.org/10.3390/math10224235 - 13 Nov 2022

Cited by 15 | Viewed by 1724

Abstract

In this article, a scalar nonlinear integro-differential equation of second order and a non-linear system of integro-differential equations with infinite delays are considered. Qualitative properties of solutions called the global asymptotic stability, integrability and boundedness of solutions of the second-order scalar nonlinear integro-differential [...] Read more.

In this article, a scalar nonlinear integro-differential equation of second order and a non-linear system of integro-differential equations with infinite delays are considered. Qualitative properties of solutions called the global asymptotic stability, integrability and boundedness of solutions of the second-order scalar nonlinear integro-differential equation and the nonlinear system of nonlinear integro-differential equations with infinite delays are discussed. In the article, new explicit qualitative conditions are presented for solutions of both the second-order scalar nonlinear integro-differential equations with infinite delay and the nonlinear system of integro-differential equations with infinite delay. The proofs of the main results of the article are based on two new Lyapunov–Krasovskiĭ functionals. In particular cases, the results of the article are illustrated with three numerical examples, and connections to known tests are discussed. The main novelty and originality of this article are that the considered integro-differential equation and system of integro-differential equations with infinite delays are new mathematical models, the main six qualitative results given are also new. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

20 pages, 620 KiB

Open AccessArticle

Higher-Order Asymptotic Numerical Solutions for Singularly Perturbed Problems with Variable Coefficients

by Chein-Shan Liu, Essam R. El-Zahar and Chih-Wen Chang

Mathematics 2022, 10(15), 2791; https://doi.org/10.3390/math10152791 - 5 Aug 2022

Cited by 3 | Viewed by 1751

Abstract

For the purpose of solving a second-order singularly perturbed problem (SPP) with variable coefficients, a mth-order asymptotic-numerical method was developed, which decomposes the solutions into two independent sub-problems: a reduced first-order linear problem with a left-end boundary condition; and a linear second-order [...] Read more.

For the purpose of solving a second-order singularly perturbed problem (SPP) with variable coefficients, a mth-order asymptotic-numerical method was developed, which decomposes the solutions into two independent sub-problems: a reduced first-order linear problem with a left-end boundary condition; and a linear second-order problem with the boundary conditions given at two ends. These are coupled through a left-end boundary condition. Traditionally, the asymptotic solution within the boundary layer is carried out in the stretched coordinates by either analytic or numerical method. The present paper executes the mth-order asymptotic series solution in terms of the original coordinates. After introducing

2 (m + 1)

new variables, the outer and inner problems are transformed together to a set of

3 (m + 1)

first-order initial value problems with the given zero initial conditions; then, the Runge–Kutta method is applied to integrate the differential equations to determine the

2 (m + 1)

unknown terminal values of the new variables until they are convergent. The asymptotic-numerical solution exactly satisfies the boundary conditions, which are different from the conventional asymptotic solution. Several examples demonstrated that the newly proposed method can achieve a better asymptotic solution. For all values of the perturbing parameter, the method not only preserves the inherent asymptotic property within the boundary layer but also improves the accuracy of the solution in the entire domain. We derive the sufficient conditions, which terminate the series of asymptotic solutions for inner and outer problems of the SPP without having the spring term. For a specific case, we can derive a closed-form asymptotic solution, which is also the exact solution of the considered SPP. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

► Show Figures

Figure 1

20 pages, 450 KiB

Open AccessFeature PaperArticle

An Extended Theory of Rational Addiction

by Federico Perali and Luca Piccoli

Mathematics 2022, 10(15), 2652; https://doi.org/10.3390/math10152652 - 28 Jul 2022

Viewed by 3176

Abstract

This study extends the rational addiction theory by introducing an endogenous discounting of future utilities. The discount rate depends on habits accumulating over time because of the repeated consumption of an addictive good. The endogeneity of the discount rate affects consumption decisions via [...] Read more.

This study extends the rational addiction theory by introducing an endogenous discounting of future utilities. The discount rate depends on habits accumulating over time because of the repeated consumption of an addictive good. The endogeneity of the discount rate affects consumption decisions via a habit-dependent rate of time preference and discloses a patience-dependence trade-off. The existence of a steady state in which habits do not grow and its optimality are proven. The local stability properties of the steady state reveal that the equilibrium can be a saddle node, implying smooth convergence to the steady state, but also a stable or unstable focus, potentially predicting real-world behaviors such as binge drinking or extreme addiction states that may drive to death. The stability of the steady state mostly depends on the habit formation process, suggesting that heterogeneity in habit formation may be a key component to explain heterogeneity in time preferences. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

► Show Figures