Editorial

2 pages, 156 KiB

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Preface to the Special Issue “Mathematical Modeling with Differential Equations in Physics, Chemistry, Biology, and Economics”

by Arsen Palestini

Mathematics 2022, 10(10), 1633; https://doi.org/10.3390/math10101633 - 11 May 2022

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Abstract

First of all, I would like to express my warmest thanks to all the scholars who participated by submitting their papers to this Special Issue [...] Full article

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

Research

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16 pages, 310 KiB

Open AccessEditor’s ChoiceArticle

Stability of Solutions to Systems of Nonlinear Differential Equations with Discontinuous Right-Hand Sides: Applications to Hopfield Artificial Neural Networks

by Ilya Boykov, Vladimir Roudnev and Alla Boykova

Mathematics 2022, 10(9), 1524; https://doi.org/10.3390/math10091524 - 02 May 2022

Cited by 6 | Viewed by 2046

Abstract

In this paper, we study the stability of solutions to systems of differential equations with discontinuous right-hand sides. We have investigated nonlinear and linear equations. Stability sufficient conditions for linear equations are expressed as a logarithmic norm for coefficients of systems of equations. [...] Read more.

In this paper, we study the stability of solutions to systems of differential equations with discontinuous right-hand sides. We have investigated nonlinear and linear equations. Stability sufficient conditions for linear equations are expressed as a logarithmic norm for coefficients of systems of equations. Stability sufficient conditions for nonlinear equations are expressed as the logarithmic norm of the Jacobian of the right-hand side of the system of equations. Sufficient conditions for the stability of solutions of systems of differential equations expressed in terms of logarithmic norms of the right-hand sides of equations (for systems of linear equations) and the Jacobian of right-hand sides (for nonlinear equations) have the following advantages: (1) in investigating stability in different metrics from the same standpoints, we have obtained a set of sufficient conditions; (2) sufficient conditions are easily expressed; (3) robustness areas of systems are easily determined with respect to the variation of their parameters; (4) in case of impulse action, information on moments of impact distribution is not required; (5) a method to obtain sufficient conditions of stability is extended to other definitions of stability (in particular, to p-moment stability). The obtained sufficient conditions are used to study Hopfield neural networks with discontinuous synapses and discontinuous activation functions. Full article

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

17 pages, 3276 KiB

Open AccessArticle

A Dynamical Model for Financial Market: Among Common Market Strategies Who and How Moves the Price to Fluctuate, Inflate, and Burst?

by Annalisa Fabretti

Mathematics 2022, 10(5), 679; https://doi.org/10.3390/math10050679 - 22 Feb 2022

Cited by 4 | Viewed by 2150

Abstract

A piecewise linear dynamical model is proposed for a stock price. The model considers the price is driven by three rather standard demand components: chartist, fundamental and market makers. The chartist demand component is related to the study of differences between moving averages. [...] Read more.

A piecewise linear dynamical model is proposed for a stock price. The model considers the price is driven by three rather standard demand components: chartist, fundamental and market makers. The chartist demand component is related to the study of differences between moving averages. This generates a high order system characterized by a piecewise linear map not trivial to study. The model has been studied analytically in its fixed points and dynamics and then numerically. Results are in line with the related literature: the fundamental demand component helps the stability of the system and keeps prices bounded; market makers satisfy their role of restoring stability, while the chartist demand component produces irregularity and chaos. However, in some cases, the chartist demand component assumes the role to compensate the fundamental demand component, felt in an autogenerated loop, and pushes the dynamics to equilibrium. This fact suggests that the instability must not be searched into the nature of the different investment styles rather in the relative proportion of the contribution of market actors. Full article

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

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19 pages, 31368 KiB

Open AccessArticle

A Mathematical Model of the Production Inventory Problem for Mixing Liquid Considering Preservation Facility

by Md Sadikur Rahman, Subhajit Das, Amalesh Kumar Manna, Ali Akbar Shaikh, Asoke Kumar Bhunia, Leopoldo Eduardo Cárdenas-Barrón, Gerardo Treviño-Garza and Armando Céspedes-Mota

Mathematics 2021, 9(24), 3166; https://doi.org/10.3390/math9243166 - 09 Dec 2021

Cited by 4 | Viewed by 3193

Abstract

The mixing process of liquid products is a crucial activity in the industry of essential commodities like, medicine, pesticide, detergent, and so on. So, the mathematical study of the mixing problem is very much important to formulate a production inventory model of such [...] Read more.

The mixing process of liquid products is a crucial activity in the industry of essential commodities like, medicine, pesticide, detergent, and so on. So, the mathematical study of the mixing problem is very much important to formulate a production inventory model of such type of items. In this work, the concept of the mixing problem is studied in the branch of production inventory. Here, a production model of mixed liquids with price-dependent demand and a stock-dependent production rate is formulated under preservation technology. In the formulation, first of all, the mixing process is presented mathematically with the help of simultaneous differential equations. Then, the mixed liquid produced in the mixing process is taken as a raw material of a manufacturing system. Then, all the cost components and average profit of the system are calculated. Now, the objective is to maximize the corresponding profit maximization problem along with the highly nonlinear objective function. Because of this, the mentioned maximization problem is solved numerically using MATHEMATICA software. In order to justify the validity of the model, two numerical examples are worked out. Finally, to show the impact of inventory parameters on the optimal policy, sensitivity analyses are performed and the obtained results are presented graphically. Full article

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

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14 pages, 302 KiB

Open AccessArticle

Accidental Degeneracy of an Elliptic Differential Operator: A Clarification in Terms of Ladder Operators

by Roberto De Marchis, Arsen Palestini and Stefano Patrì

Mathematics 2021, 9(23), 3005; https://doi.org/10.3390/math9233005 - 23 Nov 2021

Cited by 1 | Viewed by 1074

Abstract

We consider the linear, second-order elliptic, Schrödinger-type differential operator

L : = - \frac{1}{2} \nabla^{2} + \frac{r^{2}}{2} .

Because of its rotational invariance, that is it does not change under

S O (3)

transformations, the eigenvalue problem [...] Read more.

We consider the linear, second-order elliptic, Schrödinger-type differential operator

L : = - \frac{1}{2} \nabla^{2} + \frac{r^{2}}{2} .

Because of its rotational invariance, that is it does not change under

S O (3)

transformations, the eigenvalue problem

[- \frac{1}{2} \nabla^{2} + \frac{r^{2}}{2}] f (x, y, z) = λ f (x, y, z)

can be studied more conveniently in spherical polar coordinates. It is already known that the eigenfunctions of the problem depend on three parameters. The so-called accidental degeneracy of

L

occurs when the eigenvalues of the problem depend on one of such parameters only. We exploited ladder operators to reformulate accidental degeneracy, so as to provide a new way to describe degeneracy in elliptic PDE problems. Full article

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

16 pages, 2696 KiB

Open AccessArticle

SEIR Mathematical Model of Convalescent Plasma Transfusion to Reduce COVID-19 Disease Transmission

by Hennie Husniah, Ruhanda Ruhanda, Asep K. Supriatna and Md. H. A. Biswas

Mathematics 2021, 9(22), 2857; https://doi.org/10.3390/math9222857 - 10 Nov 2021

Cited by 5 | Viewed by 2792

Abstract

In some diseases, due to the restrictive availability of vaccines on the market (e.g., during the early emergence of a new disease that may cause a pandemic such as COVID-19), the use of plasma transfusion is among the available options for handling such [...] Read more.

In some diseases, due to the restrictive availability of vaccines on the market (e.g., during the early emergence of a new disease that may cause a pandemic such as COVID-19), the use of plasma transfusion is among the available options for handling such a disease. In this study, we developed an SEIR mathematical model of disease transmission dynamics, considering the use of convalescent plasma transfusion (CPT). In this model, we assumed that the effect of CPT increases patient survival or, equivalently, leads to a reduction in the length of stay during an infectious period. We attempted to answer the question of what the effects are of different rates of CPT applications in decreasing the number of infectives at the population level. Herein, we analyzed the model using standard procedures in mathematical epidemiology, i.e., finding the trivial and non-trivial equilibrium points of the system including their stability and their relation to basic and effective reproduction numbers. We showed that, in general, the effects of the application of CPT resulted in a lower peak of infection cases and other epidemiological measures. As a consequence, in the presence of CPT, lowering the height of an infective peak can be regarded as an increase in the number of remaining healthy individuals; thus, the use of CPT may decrease the burden of COVID-19 transmission. Full article

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

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11 pages, 286 KiB

Open AccessArticle

Evolutionary Derivation of Runge–Kutta Pairs of Orders 5(4) Specially Tuned for Problems with Periodic Solutions

by Vladislav N. Kovalnogov, Ruslan V. Fedorov, Andrey V. Chukalin, Theodore E. Simos and Charalampos Tsitouras

Mathematics 2021, 9(18), 2306; https://doi.org/10.3390/math9182306 - 18 Sep 2021

Cited by 12 | Viewed by 1409

Abstract

The purpose of the present work is to construct a new Runge–Kutta pair of orders five and four to outperform the state-of-the-art in these kind of methods when addressing problems with periodic solutions. We consider the family of such pairs that the celebrated [...] Read more.

The purpose of the present work is to construct a new Runge–Kutta pair of orders five and four to outperform the state-of-the-art in these kind of methods when addressing problems with periodic solutions. We consider the family of such pairs that the celebrated Dormand–Prince pair also belongs. The chosen family comes with coefficients that all depend on five free parameters. These latter parameters are tuned in a way to furnish a new method that performs best on a couple of oscillators. Then, we observe that this trained pair outperforms other well known methods in the relevant literature in a standard set of problems with periodic solutions. This is remarkable since no special property holds such as high phase-lag order or an extended interval of periodicity. Full article

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

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9 pages, 221 KiB

Open AccessArticle

Human Networks and Toxic Relationships

by Nazaria Solferino and Maria Elisabetta Tessitore

Mathematics 2021, 9(18), 2258; https://doi.org/10.3390/math9182258 - 14 Sep 2021

Cited by 2 | Viewed by 9924

Abstract

We devise a theoretical model to shed light on the dynamics leading to toxic relationships. We investigate what intervention policy people could advocate to protect themselves and to reduce suffocating addiction in order to escape from physical or psychological abuses either inside family [...] Read more.

We devise a theoretical model to shed light on the dynamics leading to toxic relationships. We investigate what intervention policy people could advocate to protect themselves and to reduce suffocating addiction in order to escape from physical or psychological abuses either inside family or at work. Assuming that the toxic partner’s behavior is exogenous and that the main source of addiction is income or wealth we find that an asymptotically stable equilibrium with positive love is always possible. The existence of a third unconditionally reciprocating part as a benchmark, i.e., presence of another partner, support from family, friends, private organizations in helping victims, plays an important role in reducing the toxic partner’s appeal. Analyzing our model, we outline the conditions for the best policy to heal from a toxic relationship. Full article

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

9 pages, 324 KiB

Open AccessArticle

Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits

by Yu-Cheng Shen, Chia-Liang Lin, Theodore E. Simos and Charalampos Tsitouras

Mathematics 2021, 9(12), 1342; https://doi.org/10.3390/math9121342 - 10 Jun 2021

Cited by 7 | Viewed by 2376

Abstract

We consider a family of explicit Runge–Kutta pairs of orders six and five without any additional property (reduced truncation errors, Hamiltonian preservation, symplecticness, etc.). This family offers five parameters that someone chooses freely. Then, we train them in order for the presented method [...] Read more.

We consider a family of explicit Runge–Kutta pairs of orders six and five without any additional property (reduced truncation errors, Hamiltonian preservation, symplecticness, etc.). This family offers five parameters that someone chooses freely. Then, we train them in order for the presented method to furnish the best results on a couple of Kepler orbits, a certain interval and tolerance. Consequently, we observe an efficient performance on a wide range of orbital problems (i.e., Kepler for a variety of eccentricities, perturbed Kepler with various disturbances, Arenstorf and Pleiades). About

1.8

digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order. Full article

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

10 pages, 936 KiB

Open AccessArticle

A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie Symmetries

by Daniele Ritelli

Mathematics 2021, 9(11), 1312; https://doi.org/10.3390/math9111312 - 07 Jun 2021

Cited by 2 | Viewed by 1870

Abstract

In this paper we present a two parameter family of differential equations treated by Jacopo Riccati, which does not appear in any modern repertoires and we extend the original solution method to a four parameter family of equations, translating the Riccati approach in [...] Read more.

In this paper we present a two parameter family of differential equations treated by Jacopo Riccati, which does not appear in any modern repertoires and we extend the original solution method to a four parameter family of equations, translating the Riccati approach in terms of Lie symmetries. To get the complete solution, hypergeometric functions come into play, which, of course, were unknown in Riccati’s time. Re-discovering the method introduced by Riccati, called by himself dimidiata separazione (splitted separation), we arrive at the closed form integration of a differential equation, more general to the one treated in Riccati’s contribution, and which also does not appear in the known repertoires. Full article

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

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12 pages, 783 KiB

Open AccessArticle

Some Properties Involving q-Hermite Polynomials Arising from Differential Equations and Location of Their Zeros

by Cheon-Seoung Ryoo and Jungyoog Kang

Mathematics 2021, 9(11), 1168; https://doi.org/10.3390/math9111168 - 22 May 2021

Cited by 3 | Viewed by 1566

Abstract

Hermite polynomials are one of the Apell polynomials and various results were found by the researchers. Using Hermit polynomials combined with q-numbers, we derive different types of differential equations and study these equations. From these equations, we investigate some identities and properties [...] Read more.

Hermite polynomials are one of the Apell polynomials and various results were found by the researchers. Using Hermit polynomials combined with q-numbers, we derive different types of differential equations and study these equations. From these equations, we investigate some identities and properties of q-Hermite polynomials. We also find the position of the roots of these polynomials under certain conditions and their stacked structures. Furthermore, we locate the roots of various forms of q-Hermite polynomials according to the conditions of q-numbers, and look for values which have approximate roots that are real numbers. Full article

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

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