Theory Of Differential Equations Research Articles

Dynamical analysis of a Lotka–Volterra competition model with both Allee and fear effects

Population ecology theory is replete with density-dependent processes. However, trait-mediated or behavioral indirect interactions can both reinforce or oppose density-dependent effects. This paper presents the first two species competitive ODE and PDE systems, where the non-consumptive behavioral fear effect and the Allee effect, a density-dependent process, are both present. The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations. It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points, but they do not affect the stability of the positive equilibria. We also observe standard co-dimension one bifurcation in the system by varying the Allee or fear parameter. Interestingly, we find that the Allee effect working in conjunction with the fear effect can bring about several dynamical changes to the system with only fear. There are three parametric regimes of interest in the fear parameter. For small and intermediate amounts of fear, the Allee + fear effect opposes dynamics driven by the fear effect. However, for large amounts of fear the Allee + fear effect reinforces the dynamics driven by the fear effect. The analysis of the corresponding spatially explicit model is also presented. To this end, the comparison principle for parabolic PDE is used. The conclusions of this paper have strong implications for conservation biology, biological control as well as the preservation of biodiversity.

Modelling the Impact of Decomposed Disease-Induced Dead Cashew Plants on Fusarium Wilt Dynamics in South-Eastern Tanzania

Cashew Fusarium wilt disease transmitted by Fusarium oxysporum is dangerous and destroys many cashew plants in Tanzania. The effect of this disease leads households and the government to experience a decrease in overall cashew production and income due to this disease’s capacity to harm cashew plants. This study aimed to ascertain the role of Fusarium wilt-decayed disease-induced dead plants in spreading the disease among cashew plants. A mathematical model was created based on the dynamics of the disease and a stability analysis was conducted using theories of ordinary differential equations. Data from two regions in Tanzania, Lindi, and Mtwara, were used in model fitting and parameter estimation. Additionally, the parameters were estimated using maximum likelihood estimation (MLE). The outcome suggests that the disease’s prevalence and spread increase during an outbreak as decomposed diseased-induced dead plants contribute to the saturation of chlamydospores in the soil.

EN General thermodynamic relations for thermal variables formed by first-order partial derivatives and not mixed second-order derivatives

The theory of differential equations is a powerful tool necessary for constructing mathematical models of various applied problems and their solution. The study and analysis of the general differential relations between the derivatives of the thermodynamic functions of substances is a new direction in the theory of their thermodynamic properties. This research is the development of thermodynamics associated with the establishment and use of differential equations. This paper presents two new thermodynamic relationships between thermal variables pressure, absolute temperature, and specific volume. They relate first-order partial derivatives with unmixed second-order derivatives for these variables. Revealing new general relations of thermodynamics made it possible to formulate and solve a number of problems, among which we can note: creation of algorithms for calculating second-order mixed derivatives for a given thermal equation of state; analysis of the features of the second mixed derivatives at the critical point of pure substance; investigations of the general thermodynamic equations with entropy. The procedure for establishing new equations is based on the use of a rigorous mathematical apparatus and Maxwell's differential equations. Therefore, they are of a general nature, that is, they are valid for any pure substance or any mixture of fixed composition. They are also applicable to any model of the equation of state of these substances. With a formal approach to the concept of the term "equation of state", the new relations themselves can be considered as the equation of state of substance. The obtained new relations can be used to develop the theory of the thermodynamic properties of substances. In particular, in this work it is shown that the ratios of two partial derivatives of pressure over the volume, which are zero critical conditions, have no singularities at the critical point itself

PROPAGATION OF MECHANIC WAVE IN TWO-DIMENSION STRATIFIED MEDIUM

The principle of emergence encourages us to look for new properties in future materials as a result of structuring of constituent elements, and mostly we are talking about non-obvious structuring. For example, understanding and simulating isotropy is on the way to robustness against internal vibrations. Structurality itself is a characteristic feature of the world around a person (protein molecules, crystal lattices, etc.). However, the variety of natural structures is so numerous that directly studying them to identify new properties appears to be quite a difficult task. Therefore, to a certain extent, the principle of emergence supposedly contrasts the need to copy or imitate natural structures with the ability to carry out inventions. Just as a wide variety of patterns are formed in an optical kaleidoscope thanks to 2 or 3 optical elements, the principle of emergence gives reason to expect that as a result of structuring only a few components of the future environment, new properties can be obtained. The development of technology, in particular, 3D printing, opens up opportunities to look at the methodology of structuring the environment in a new way. If, in addition, this process is relatively economic, then empirically, by carrying out structuring, it is possible to obtain experimental laboratory samples. Today they are talking about the possibility of achieving such effects as lensing of mechanical waves precisely thanks to emergent properties. The corresponding devices promise to find application, for example, in ultrasound diagnostics in medicine (Assoc. Prof. J. Memoli, University of Sussex, UK). The paper considers the problem of the propagation of mechanical waves in a folding periodic two-layer medium - a flat model. For such a medium, a wave equation is written and solved by the method of separation of variables. Such an equation turns out to be a linear differential equation with periodic piecewise-stable coefficients. From the basic theory of differential equations with periodic coefficients (Floquet theory), the method of transfer matrices (a method for calculating the passage of waves through multilayer media) is well known, which makes it possible to obtain a composite condition for the solvability of the wave equation. The work develops an approach to obtaining solvability conditions (constructing a dispersion equation) along with the classical method of transfer matrices. The proposed approach is in a certain sense equivalent to the transfer matrix method, however, it has some remarkable differences, in particular, it provides a rigorous mathematical basis for the transition to media with a finite number of layers.

Mathematical modeling and optimal control of a deterministic SHATR model of HIV/AIDS with possibility of rehabilitation: a dynamic analysis

In the present work, we developed a deterministic SHATR (Susceptible - HIV infected -AIDS infected - Antiretroviral Treatment - Recovered) compartment model for HIV/AIDS. This model considers the disease outbreak due to a lack of awareness and treatment. The steady states of the proposed model system are obtained and analyzed by using the nonlinear stability theory of differential equations. The basic reproduction number is derived and explored to determine the stability and sensitivity index of some important relative parameters. Further, to know the global behavior of the model one parameter bifurcation study is discussed. Moreover, the optimal control theory has been applied to identify the optimal strategy by taking treatment and awareness for safe intercourse as control parameters. The control problem is solved analytically by using Pontryagin’s maximum principle. Finally, the model is simulated to describe the optimality under various assumptions and the stability of equilibrium points.

A Mathematical Model of the Dynamics of Coffee Berry Disease

Coffee berry disease (CBD) is a fungal disease caused by Colletotrichum kahawae. CBD is a major constraint to coffee production to Kenya and Africa at large. In this research paper, we formulate a mathematical model of the dynamics of the coffee berry disease. The model consists of coffee plant population in a plantation and Colletotrichum kahawae pathogen population. We derived the basic reproduction number R k 0 , and analyzed the dynamical behaviors of both disease-free equilibrium and endemic equilibrium by the theory of ordinary differential equations. Using the MATLAB ode45 solver, we carried out numerical simulation, and the findings are consistent with the theoretical results.

Analytic in the unit polydisc functions of bounded L-index in direction

The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.

Formulation and mathematical analysis of a three-step model of anaerobic digestion with inhibition by ammonia

In this work, we are interested in the construction and mathematical analysis of a three-step anaerobic digestion model (Hydrolysis–Acidogenesis–Methanogenesis). Our aim is to investigate the qualitative properties of the system. In addition to the hypotheses used for formulating the well-known baseline two-step anaerobic model, we take into account the inhibition by ammonia together with the decay rate of acidogenic and methanogenic microorganisms, and consider the hydrolysis step to determine its effect on the growth of the methanogens. By so doing, we obtain a system of six ordinary differential equations describing the whole dynamics of the aforementioned phenomenon. As far as the mathematical analysis of this model is concerned, we first prove the existence of a unique positive and bounded global solution. Then we derive local and global stability results. The proofs of the main results rely essentially on some tools from the theory of ordinary differential equations. Finally, by performing some numerical simulations, we illustrate the theoretical results obtained and highlight the key parameters acting on the model’s behavior.

Parameter estimation in multifactor uncertain population model

This paper extends the traditional multifactor population model from a probabilistic background to an uncertain background, allowing us to better account for the fact that the birth and death rates are sometimes affected by different, independent uncertain noises. The theory of multifactor uncertain differential equation is applied to analyze the novel model. First, a multifactor uncertain population model is developed and the solution of the model is obtained. Secondly, the stability in measure and stability in distribution of the multifactor uncertain population model are studied respectively, and the relationship between the two kinds of stability is further discussed. Then, a method of moment estimation based on residuals is proposed for the unknown parameters in the multifactor uncertain population model. Finally, an example is given to illustrate the validity and rationality of the parameter estimation method.

Self-Consistent Modeling of Gravitational Theories beyond General Relativity.

The majority of extensions to general relativity (GR) display mathematical pathologies-higher derivatives, character change in equations that can be classified within partial differential equation theory, and even unclassifiable ones-that cause severe difficulties to study them, especially in dynamical regimes. We present here an approach that enables their consistent treatment and extraction of physical consequences. We illustrate this method in the context of single and merging black holes in a highly challenging beyond GR theory.

Stability of the Epidemiological Model SIR with Loss of Immunity

This study approaches the analysis of the stability of the epidemiological model SIR with loss of immunity. This is a model given by a system of ordinary differential equations. Initially, we present the model and its interpretation. Then we define the constants and elements that compose the model, so we present the results obtained using the qualitative theory of ordinary differential equations, especially the theory of planar systems related to the dynamics of fixed points. Finally, we show that the system representing the SIR model is globally stable and they have two types of dynamic that {depend on model constants}, and their meaning for epidemiology.

Polynomial differential systems with hyperbolic limit cycles

One of the main problems of the qualitative theory of the planar differential equations is the study their limit cycles. In this paper given an algebraic curve of degree n we characterize the planar polynomial differential systems of degree greater or equal than n which admit the ovals of the algebraic curve as hyperbolic limit cycles.

Технология построения имитационных моделей силовых трансформаторов с произвольной конструкцией активной части

Existing simulation models of power transformers that operate, for example, in the MATLAM Simulink SimPowerSystem environment, are based on an ideal transformer model. It allows using this model as an autonomous link in a complex electrical circuit. The main disadvantage of such models is the difficulty to consider non-standard versions of the magnetic circuit and special schemes for connecting winding elements when modeling. Particularly significant problems arise when modeling special transformers of the certain classes. At the same time, traditionally, when modeling the transient modes of transformers, a different approach is used. It is based on the use of inductance matrices which allows considering all the design features of the active part of the transformer. The disadvantage of this approach is the need to describe an external electrical circuit in addition to the transformer. Therefore, the problem to develop a modern technology for simulation models of transformers with an arbitrary design of the active part is topical. This model meets the requirement for the autonomy of the transformer model from the external circuit model. The authors have used the methods of modeling electrical and magnetic circuits based on the theory of ordinary differential equations, and simulation method using the MatLab Simulink SimPowerSystems package. A technology has been developed for simulation models of transformers with an arbitrary design of the active part based on the use of typical subsystems. An algorithm for matrix of inductances based on the main magnetic field of the transformer with the existing equivalent circuit of the magnetic circuit is given. The authors have presented a diagram of a three-phase transformer model developed using the proposed technology, as well as the results of comparing the current curves in the primary and secondary windings of the transformer when it is turned on at idle and with a resistive load. The results are obtained using the existing and new models. The results of the study can be used to design general industrial and special transformers in design companies and in manufacturing environment. The developed technology can give a special effect when it is used at the R&D stage to study the operating modes of transformers in case the enterprise has no experience to design and manufacture them.

Controllability of discrete-time semilinear Riemann–Liouville-like fractional equations

This article utilizes the Riemann–Liouville-like fractional operator to investigate certain sufficient conditions for the approximate controllability of discrete-time fractional evolution equations. We describe our main results utilizing a sequential approach, the theory of difference equations, and the connection between a class of sequences of operators and C0-semigroups. On the nonlinear term, we impose the Lipschitz-type condition. Finally, a few examples are provided to demonstrate how the outcomes might be used.

Закони керування авіаційним газотурбінним двигуном з турбовентиляторною приставкою

The main aspects of the general task of integrating an aviation engine (AE) with a turbofan additional unit (TAU) of a multi-mode aircraft are the selection of the scheme and design parameters of the AE with the TAU (including parameters of the work process), as well as methods and means of automatic control of the AE with the TAU for better matching its characteristics with certain flight modes. These two tasks are closely interrelated: on the one hand, when determining the appearance of the aircraft and its power plant (PP), it is necessary to consider what means will ensure the adaptation of its characteristics to the variable flight mode, and on the other hand, the purpose of the aircraft, its parameters and parameters of the PP, and the flight modes mostly determine the choice of control laws. The subject of the research is the formation of the laws of control of the aviation gas turbine engine with TAU. The goal is to improve the dynamic characteristics of the aviation gas turbine engine by applying adaptive control of the gas turbine engine with the use of wireless information exchange technologies. Objectives: to generalize the concept of adaptive control at the stage of determining the appearance of an engine with TAU; to determine the methods of regulating the GTE with TAU and their influence on the speed characteristics; to describe the process of formation of the laws governing AE with TAU; to investigate thermal processes in order to find functional dependencies in the optimal control of gas turbines with TAU. Research methods: system analysis, mathematical and computer modeling were used in the formation of control laws; the methods of philosophical knowledge were used to build an approach to the design of adaptive control systems of gas turbines with TAU. The theory of aircraft engines, the theory of differential equations, difference grids, and numerical methods were used to optimize the control of gas turbines with TAU. Results: methods of regulating the GTE with TAU and their influence on the speed characteristics of the aviation engine; formulas for researching the thermal processes of the working blades of the GTE with TAU in order to form the influence of the regulator on the executive mechanisms. Scientific and practical novelty: formation of a paradigm for the development of models of adaptive control of GTE with TAU, considering different flight modes of the aircraft and engine operation modes. The selection of control laws using multi-parameter optimization methods for finding the relationship between structural and component schemes of the gas generator and turbofan additional unit. The character of the engine thrust change depending on the input temperature is shown, which in turn, will allow to increase the efficiency of the fan and obtain a thrust reserve. The research directions of the temperature field and stress field were formed.

Kinetic Behavior and Optimal Control of a Fractional-Order Hepatitis B Model

The fractional-order calculus model is suitable for describing real-world problems that contain non-local effects and have memory genetic effects. Based on the definition of the Caputo derivative, the article proposes a class of fractional hepatitis B epidemic model with a general incidence rate. Firstly, the existence, uniqueness, positivity and boundedness of model solutions, basic reproduction number, equilibrium points, and local stability of equilibrium points are studied employing fractional differential equation theory, stability theory, and infectious disease dynamics theory. Secondly, the fractional necessary optimality conditions for fractional optimal control problems are derived by applying the Pontryagin maximum principle. Finally, the optimization simulation results of fractional optimal control problem are discussed. To control the spread of the hepatitis B virus, three control variables (isolation, treatment, and vaccination) are applied, and the optimal control theory is used to formulate the optimal control strategy. Specifically, by isolating infected and non-infected people, treating patients, and vaccinating susceptible people at the same time, the number of hepatitis B patients can be minimized, the number of recovered people can be increased, and the purpose of ultimately eliminating the transmission of hepatitis B virus can be achieved.

Fuzzy solutions of some variants of the fractional order Korteweg-de-Vries equations via an analytical method

Under uncertainty, the analytical behaviour of fractional partial differential equations is frequently puzzling and challenging to predict. Therefore, in order to address these problems, it is essential to create a proper, extensive, and highly effective theory. The theory of fuzzy fractional partial differential equations is a relatively new idea with applications in applied mathematics and engineering. In this article, we examined the different types of fuzzy fractional-order Korteweg-de-Vries (KdV) equations along with their analytical fuzzy solutions. To construct a series-type solution under a fuzzy notion for the relevant research, we utilise the fractional reduced differential transform method (FRDTM) in the Caputo sense. The solutions of various kinds of fuzzy fractional KdV equations that have been developed are more broadly applicable. The fuzzy concept is used to overrule the uncertainty in physical models based on the parametric form of the fuzzy number. The numerical and graphical presentation demonstrates the symmetry between the lower and upper cut representations of the fuzzy solutions and may be helpful in improving understanding of automatic control models, measure theory, physics, biology, computer science, quantum optics, economics, artificial intelligence, and other domains, as well as non-financial analysis.

A weighted $$L_p$$-regularity theory for parabolic partial differential equations with time-measurable pseudo-differential operators

A weighted $$L_p$$-regularity theory for parabolic partial differential equations with time-measurable pseudo-differential operators

Asymptotic collective behaviors of a delayed multi‐particle system with multiple processes

It is a significant issue to explore the collective behaviors of the delayed complex systems. This paper studies asymptotic collective behaviors of a delayed multi‐particle system with multiple processes while particles interact directionally and locally. Up to now, the studies on dynamic behaviors of multi‐particle systems do not involve limited processing capacity, that is, the information queuing phenomenon is ignored. To this aim, we introduce multiple processes to formulate the evolution of a delayed multi‐particle system. To optimize the delay parameters, we choose the greedy principle to set the processing order. The main idea is to decompose the velocity into each process and use functional differential equation theory and matrix theory to constrain the velocity variation rate while the processing order is fixed. As the new results, in the non‐critical and general situation, when the time delay is less than or equal to the threshold value and the eigenvalue 1 of the average matrix is semi‐simple, we achieve the criterion of flocking and multi‐cluster with exponential convergence rate involving multiple processes. Furthermore, we get the lower bound estimate of the processing capacity.

A refined shear deformation theory of an asymmetric sandwich beam with porous core: Linear bending problem

This paper is devoted to a sandwich beam with an asymmetric structure under uniformly distributed load. The first end of this beam is simply supported, while second end is clamped. The individual nonlinear deformation function of a plane beam cross section – the shear effect with consideration of the classical shear stress formula is analytically developed. Displacements, strains and stresses taking into account the individual nonlinear deformation function for successive layers of this beam are elaborated in detail. The system of two differential equation of equilibrium of this beam, based on the principle of stationary potential energy, is obtained. This system of equations is precisely solved analytically, according to the theory of differential equations. The individual nonlinear deformation functions, shear stresses and deflections of two cases of sample beams are determined in detail: the first with variable core porosity in the direction of the beam depth, the second with constant core porosity – standard sandwich structure. The calculations results specified in Tables and graphically presented in Figures.