System Of Equations Research Articles

Italian regional econometric model

This paper shifts studies on macroeconomic models from a national to a regional scope. Regions, indeed, often follow growth trajectories that dif- fer greatly from others within the same country. In this study we propose a macroeconometric model for 20 economies corresponding to the Italian administrative Regions. We called it: IREM (Italian Regional Econometric Model). We illustrate the general structure and properties of IREM, especially with regard to the economy’s response to changes in regional fiscal policy, technological advances and other dimensions of the economic environment. One of the model’s key features is the joint representation of the economy on both the demand and the supply side with a multiple estimating equations system calibrated at regional level. Public finance is designed in great detail, with multiple specific equations for local government expenses and revenues, using the Italian database CPT (Conti Pubblici Territoriali). After documenting the model structure and the estimation results with an in-sample simulation, we turn to illustrate the model properties through the study of its response functions to multiple shocks.

A study for analysing predator–prey ecological system using Legendre polynomial

This paper presents a mathematical model to examine the effects of the coexistence of predators on single prey. Based on Caputo operators, we present a newly developed system of differential equations for the predator–prey system using wavelet method. It is well known that a system of nonlinear singular models cannot operate smoothly since they are singular and nonlinear. Therefore, with the help of this numerical approach, we have converted the system into a nonlinear system of algebraic equations by extending it through operational matrix of Legendre wavelets. Using the wavelet collocation scheme, we have calculated these unknown coefficients. It has been demonstrated in tables and graphs that the developed approach is consistent and proficient. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyse its behaviour. In addition, a nonlinear functional analysis have used to establish uniformly boundedness for the proposed model. Also we have discussed residual error analysis and Lyapunov exponent. The applicability and efficacy of this methodology have been demonstrated through this nonlinear system. Additionally, a comparison with existing results highlights the advantages of our numerical approach. All calculations have been done using MATLAB.

An Alternative Finite Element Formulation to Predict Ductile Fracture in Highly Deformable Materials

Abstract An alternative finite element formulation to predict ductile damage and fracture in highly deformable materials is presented. For this purpose, a finite-strain elastoplastic model based on the Gurson–Tvergaard–Needleman (GTN) formulation is employed, in which the level of damage is described by the void volume fraction (or porosity). The model accounts for large strains, associative plasticity, and isotropic hardening, as well as void nucleation, coalescence, and material failure. To avoid severe damage localization, a nonlocal enrichment is adopted, resulting in a mixed finite element whose degrees-of-freedom are the current positions and nonlocal porosity at the nodes. In this work, 2D triangular elements of linear-order and plane-stress conditions are used. Two systems of equations have to be solved: the global variables system, involving the degrees-of-freedom; and the internal variables system, including the damage and plastic variables. To this end, a new numerical strategy has been developed, in which the change in material stiffness due to the evolution of internal variables is embedded in the consistent tangent operator regarding the global system. The performance of the proposed formulation is assessed by three numerical examples involving large elastoplastic strains and ductile fracture. Results confirm that the present formulation is capable of reproducing fracture initiation and evolution, as well as necking instability. Convergence analysis is also performed to evaluate the effect of mesh refinement on the mechanical response. In addition, it is demonstrated that the nonlocal parameter alleviates damage localization, providing smoother porosity fields.

Convergence results for controlled contractions and applications to a system of Volterra integral equations

This work presents the concept of controlled dislocated metric spaces, and the Banach contraction principle in these spaces is examined. Next, we provide extensive examples of numerical convergence of fixed points to demonstrate the practical implications of such spaces. Along with specific instances, it presents the idea of a rational contraction of the ϝ − type to determine common fixed points in such spaces. Moreover, numerical analysis and graphical comparisons are used to further assess the existence of common fixed points based on the provided notions. Lastly, using these ideas to solve a Volterra integral equation shows how useful they are for finding common solutions, improving our comprehension of fixed point theory in these spaces, and creating new avenues for applications.

Mechanochemical topological defects in an active nematic.

We propose a reaction-diffusion system that converts topological information of an active nematic into chemical signals. We show that a curvature-activated reaction dipole is sufficient for creating a system that dynamically senses topology by producing a concentration field possessing local extrema coinciding with ±1/2 defects. The enabling term is analogous to polarization charge density seen in dielectric materials. We demonstrate the ability of this system to identify defects in both passive and active nematics. Our results illustrate that a relatively simple feedback scheme, expressed as a system of partial differential equations, is capable of producing chemical signals in response to inherently nonlocal structures in anisotropic media. We posit that such coarse-grained systems can help generate testable hypotheses for regulated processes in biological systems, such as morphogenesis, and motivate the creation of bio-inspired materials that utilize dynamic coupling between nematic structure and biochemistry.

A mathematical model to study the role of dystrophin protein in tumor micro-environment

In this research work, the authors have developed a mathematical model to examine the interaction between dystrophin protein and tumor. The authors formulated a system of ordinary differential equations to describe the dynamics of the dystrophin-tumor interaction system. Jacobian matrix and Routh–Hurwitz stability techniques were used to determine equilibrium points, perform stability and bifurcation analysis, and establish the conditions required for the stability of the proposed model. Numerical simulations are performed using Euler’s method to investigate the temporal evolution of the proposed model under different parameter values, such as tumor growth rate and feedback strength of dystrophin protein. The numerical results are presented in tables, and corresponding to each table, a graphical analysis is done. The graphical analysis includes creating phase portraits to visually represent stability regions around the equilibrium points, bifurcation diagrams to identify critical points, and time series analysis to highlight the behavior of the proposed model. The authors explore how variations in dystrophin expression impact tumor progression, identifying potential therapeutic implications of maintaining higher dystrophin levels. This comprehensive analysis enhances our understanding of the dystrophin-tumor interaction, providing a basis for further experimental validation and potential therapeutic strategies.

Simulation of Nonlinear and Nonisothermal Reactive Liquid Chromatography considering Finite Mass-Transfer Rates and Various Adsorption Scenarios.

A multicomponent nonisothermal lumped kinetic model (LKM) of fixed-bed liquid chromatographic reactor is introduced and numerically approximated. The model incorporates nonlinear Bi-Langmuir and Toth adsorption isotherms, reaction kinetics, axial and temporal variations of concentrations, and enthalpies of reaction and adsorption. The resulting model equations constitute a nonlinear coupled system of convection-diffusion-reaction partial differential equations (CDR-PDEs) for mass and energy balances in both the mobile and stationary phases, augmented by differential mass balances in the stationary phase and algebraic expressions for reaction rates and adsorption isotherms. Due to the nonlinearity of adsorption isotherms and reaction kinetics, which is impeding the derivation of closed-form solutions, a local-projection discontinuous Galerkin finite element (DG-FE) method is applied for space discretization. The emerging semidiscrete system of ordinary differential equations in time is solved numerically by employing a total variation bounded (TVB) Runge-Kutta method. To analyze the process performance, parametric studies are conducted using numerical simulations. A series of rigorous consistency tests are elucidating the interplay between thermal and concentration gradients, showcasing the impact of temperature on reactor performance, and highlighting the aspects of reactant conversion into products. Subsequently, a thorough comparison is performed among the considered adsorption isotherms to scrutinize the numerical findings and to augment the originality of research. The results obtained verify the validity of model equations and precision of the numerical technique used. Moreover, the outcomes of this study can be utilized to understand mass and energy distributions in nonequilibrium and nonisothermal liquid chromatographic reactors and to provide profound insights into the complexities of this separation-conversion process.

Long wavelength limit of the KP-type equation for the ion dynamics system

In this paper, we consider the modified Kadomtsev–Petviashvili equation (KP-type) limit for ion dynamics system in R 2 + 1 . We derive the two-dimensional homogeneous and inhomogeneous KP-type equation from the non-dimensional ion dynamics system by Gardner–Morikawa transforms. The different Gardner–Morikawa transforms in temporal–spatial directions bring the anisotropic problem for energy estimate. By the energy method, we prove that the solution of Euler–Poisson system converges to KP-type equation globally when ϵ → 0 in R 2 + 1 .

On the large time behaviour of the solutions of an evolutionary-epidemic system with spatial dispersal.

This paper investigates some properties of the large time behaviour of the solutions of a spatially distributed system of equations modelling the evolutionary epidemiology of a plant-pathogen system. The model takes into account the phenotypic trait and the mutation of the pathogen, which is described by a non-local operator. We roughly speaking prove that the solutions separate the phenotype trait from the spatio-temporal evolution in the large time asymptotic. This feature is obtained by investigating the positive and bounded entire solutions of the problem, that are shown to exhibit such a separation of the variables property, by reformulating them as the positive solutions of suitable integral equations in some ordered Banach space. In addition, some numerical simulations are performed to support our theoretical results.

Performance of latent heat storage exchangers: Evaluation framework and fast prediction model

Given the lack of consensus on the selection and design of appropriate latent heat storage exchangers (LHSEs) for practical applications, this study presents a framework for evaluating the performance of various LHSEs and a novel prediction model without involving complex differential equation systems is proposed to quickly predict the performance of LHSEs. The prediction accuracy is guaranteed by comparing against the validated numerical simulations under different geometries, inlet heat transfer fluid (HTF) parameters, and PCM properties. The proposed model performs well in predicting the LHSE performance under different geometries, inlet HTF parameters, and PCM properties. The maximum prediction errors for the effective operating time and air outlet temperature are 0.9 h and 1.9 °C, respectively. It implies that the proposed model has the potential to predict the performance of the LHSE under various conditions. Due to ignoring the temperature gradient within the PCM containers and the sensible thermal energy storage of the PCM, the predicted average PCM temperature is slightly overestimated during the first half and underestimated during the second half of the melting process. This study is anticipated to provide a new solution for performance evaluation and fast prediction of LHSEs.

SOLUTION OF THE INVERSE PROBLEM OF CHEMICAL KINETICS BASED ON FUZZY LOGIC MODELS

The possibility of using “soft” calculation methods to solve the inverse problem of chemical kinetics to identify fuzzy kinetic laws based on the data of stationary experiments without a priori specifying the type of the kinetic law has been investigated. “Soft” computing combines artificial intelligence methods such as fuzzy logic, neural networks, genetic algorithms, etc. (F. Rosenblatt, L. Zadeh, J. Holland, etc.). Combinations of these methods will allow the creation of various hybrid systems. Dynamic kinetic models of chemical reactions are described by systems of ordinary differential equations (clear deterministic models), on the basis of which, using various optimization methods (least squares, etc.), the inverse problem of chemical kinetics is solved. B. Kosko proved that any mathematical system can be approximated as accurately as desired by a non-deterministic or weakly deterministic model based on fuzzy logic using simple algorithmic constructions of the “if ... then” type (conditional operators) without direct quantitative or qualitative analysis of systems of differential equations. A fuzzy kinetic model is an algorithmic design based on fuzzy algebra, which allows one to calculate the reaction rate for any values of reagent concentrations of interest to the experimenter. This model allows solving the inverse problem without a priori specifying the type of kinetic law, does not have an unambiguous formulaic representation and can be described by regression (clear) equations. In this article, based on fuzzy logic methods, regression kinetic models are obtained and studied to solve the inverse problem for a simple model reaction and the acetylene hydrochlorination reaction. It has been shown that clear regression models demonstrate good accuracy and can be used to establish alternative kinetic laws and mechanisms for chemical reactions. For citation: Kol'tsov N.I. Solution of the inverse problem of chemical kinetics based on fuzzy logic models. ChemChemTech [Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol.]. 2024. V. 67. N 12. P. 54-63. DOI: 10.6060/ivkkt.20246712.7067.

The mixed displacement method to avoid shear locking in problems in elasticity

AbstractThe mixed displacement (MD) method was initially developed to mitigate geometrical locking effects in beams, plates, and shells with the intention of having intrinsically locking‐free characteristics while using equal‐order interpolation for all degrees of freedom. In other words, it is an unlocking scheme that works independent of the element shape, polynomial order, and discretization scheme. It includes additional degrees of freedom that adhere to a carefully designed differential relation that can be interpreted as a kinematic law, incorporated in a mixed sense. Certain constraints are to be enforced on these additional degrees of freedom to obtain a well‐posed system of equations. In this work, the MD method is extended for problems in solid mechanics. We present the underlying variational formulation, followed by its application to 2D solid elements. Additionally, we showcase an idea to enforce the additional constraints in a general sense. Various numerical examples, within the framework of the finite element method and isogeometric analysis, are outlined to demonstrate the performance of the MD method in the geometrically linear and geometrically nonlinear cases.

On modeling flow between adjacent surfaces where the fluid is governed by implicit algebraic constitutive relations

We consider pressure-driven flow between adjacent surfaces, where the fluid is assumed to have constant density. The main novelty lies in using implicit algebraic constitutive relations to describe the fluid’s response to external stimuli, enabling the modeling of fluids whose responses cannot be accurately captured by conventional methods. When the implicit algebraic constitutive relations cannot be solved for the Cauchy stress in terms of the symmetric part of the velocity gradient, the traditional approach of inserting the expression for the Cauchy stress into the equation for the balance of linear momentum to derive the governing equation for the velocity becomes inapplicable. Instead, a non-standard system of first-order equations governs the flow. This system is highly complex, making it important to develop simplified models. Our primary contribution is the development of a framework for achieving this. Additionally, we apply our findings to a fluid that exhibits an S-shaped curve in the shear stress versus shear rate plot, as observed in some colloidal solutions.

Lie symmetry method to discuss the optimal system of Lie subalgebras and similarity solutions for shock propagation in a perfectly conducting dusty gas with magnetic field in rotating medium

In this paper, the similarity solutions for the shock wave propagation in a perfectly conducting dusty gas under the effect of axial or azimuthal magnetic field in the case of rotating medium by using Lie symmetry method have been discussed. The optimal system of one-dimensional Lie subalgebras related to the hyperbolic system of nonlinear partial differential equations had been constructed by using general invariant function and general adjoint transformation matrix. The optimal system is the collection of inequivalent optimal classes, which provides crucial information regarding different types of the similarity solutions. On the basis of the optimal system, we have discussed all possible similarity solutions for inequivalent optimal classes individually. Also, we have found that the similarity solutions exist in two cases for power law shock path and in one case with exponential law shock path. The shock wave decay is due to an increment in the physical parameter such as shock Cowling number or non-idealness parameter or adiabatic exponent or rotational parameter. For smaller value of the ratio of density of solid particles to initial species density of the gas i.e. [Formula: see text], the shock strength decreases with an increment in the mass fraction of solid particles in the mixture [Formula: see text], but the shock strength decreases as [Formula: see text] increases for [Formula: see text]. Also, the shock strength increases with an increment in [Formula: see text] at constant [Formula: see text]. It is observed that the shock wave is stronger in the presence of axial magnetic field as compared to azimuthal magnetic field.

Vector-borne disease outbreak control via instant releases

This paper is devoted to the study of optimal release strategies to control vector-borne diseases, such as dengue, Zika, chikungunya and malaria. Two techniques are considered: the sterile insect one (SIT), which consists in releasing sterilized males among wild vectors in order to perturb their reproduction, and the Wolbachia one (presently used mainly for mosquitoes), which consists in releasing vectors, that are infected with a bacterium limiting their vectorial capacity, in order to replace the wild population by one with reduced vectorial capacity. In each case, the time dynamics of the vector population is modeled by a system of ordinary differential equations in which the releases are represented by linear combinations of Dirac measures with positive coefficients determining their intensity. We introduce optimal control problems that we solve numerically using ad-hoc algorithms, based on writing first-order optimality conditions characterizing the best combination of Dirac measures. We then discuss the results obtained, focusing in particular on the complexity and efficiency of optimal controls and comparing the strategies obtained. Mathematical modeling can help testing a great number of scenarios that are potentially interesting in future interventions (even those that are orthogonal to the present strategies) but that would be hard, costly or even impossible to test in the field in present conditions.

Analytical solutions and instability analysis of truncated M-fractional coupled dispersionless equations

Abstract This paper comprehensively investigates the truncated M-fractional coupled dispersionless equations, a nonlinear system of partial differential equations characterized by its M-fractional derivative. The Jacobi elliptic function expansion method is employed to derive analytical solutions for the coupled system. In addition, the modulation instability of the solutions is thoroughly explored, providing a detailed exposition of the mathematical framework governing the system. The analytical solutions are graphically illustrated and analyzed to highlight their physical significance. These findings have significant applications in nonlinear optics, offering new insights into wave propagation and stability within such systems.

Non-axisymmetric coupled non-stationary problem of thermoelectroelasticity for a long piezoceramic cylinder

A new closed solution to the non-axisymmetric coupled non-stationary problem of thermoelectroelasticity was constructed for a long piezoceramic cylinder for the case of satisfaction of the first and the third kind boundary conditions. Cylindrical surfaces were made as electrodes and connected to a measurement device with large input resistance. Limitation of a temperature change “load” rate made it possible to include equations of statics, electrostatics and thermal conductivity in the initial formula. The finite biorthogonal transforms are applying to explore a non-selfadjoint system of differential equations and to develop a closed solution. The obtained relations made it possible to determine the temperature and electric fields, and the stress-strain state in the piezoceramic cylinder, as well as the potential difference between cylindrical surfaces (electrodes) under non-stationary non-axisymmetric temperature impact.

Compatibility of strains and the three-fold differentiability of the displacement field

The problem of the necessary class of smoothness of solutions to quasi-static problems of deformable solid mechanics in terms of displacements was discussed. It is shown that in order for the equations of compatibility of deformations to become identities when displacements are substituted in them, the existence of some third mixed derivatives of displacements is required. A counterexample for a linearly elastic compressible isotropic elastic medium was given. In this counterexample, the displacement field, being a doubly differentiable solution to the boundary value problem for the system of Lame equations in the entire domain, is not a solution to the displacement problem at all points in this domain.

Classical-quantum simulation of non-equilibrium Marshak waves

In the radiation hydrodynamic simulations used to design inertial confinement fusion (ICF) and pulsed power experiments, nonlinear radiation diffusion tends to dominate CPU time. This raises the interesting question of whether a quantum algorithm can be found for nonlinear radiation diffusion which provides a quantum speedup. Recently, such a quantum algorithm was introduced based on a quantum algorithm for solving systems of nonlinear partial differential equations (PDEs) which provides a quadratic quantum speedup. Here, we apply this quantum PDE (QPDE) algorithm to the problem of a non-equilibrium Marshak wave propagating through a cold, semi-infinite, optically thick target, where the radiation and matter fields are not assumed to be in local thermodynamic equilibrium. The dynamics is governed by a coupled pair of nonlinear PDEs which are solved using the QPDE algorithm, as well as two standard PDE solvers: (i) Python's py-pde solver; and (ii) the KULL ICF simulation code developed at Lawrence-Livermore National Laboratory. We compare the simulation results obtained using the QPDE algorithm and the standard PDE solvers and find excellent agreement.

Non-fragile sampled-data observers resisting measured noise for a class of nonlinear systems

In this paper, we study non-fragile observers for a class of nonlinear systems with sampled outputs and unknown measurement noise. After an output filtering transformation, an extended nonlinear system is constructed. The sampled outputs with unknown measurement noise are transformed into the state equation of the extended system. Then, an extended non-fragile observer is designed to estimate the states of the extended system, and the unknown measurement noise is estimated by a first-order estimator. On the basis of a time-varying matrix inequality (TMI), it is proven that the observer system can track the extended system well even if there are observer gain perturbations and measured noise. Finally, a nonlinear dynamic model is used to verify the feasibility and validity of the proposed methods.