System Of Equations Research Articles

Alternating Projection Method for Intersection of Convex Sets, Multi-Agent Consensus Algorithms and Averaging Inequalities

The history of the alternating projection method for finding a common point of several convex sets in Euclidean space goes back to the well-known Kaczmarz algorithm for solving systems of linear equations, which was devised in the 1930s and later found wide applications in image processing and computed tomography. An important role in the study of this method was played by I.I. Eremin’s, L.M. Bregman’s, and B.T. Polyak’s works, which appeared nearly simultaneously and contained general results concerning the convergence of alternating projections to a point in the intersection of sets, assuming that this intersection is nonempty. In this paper, we consider a modification of the convex set intersection problem that is related to the theory of multi-agent systems and is called the constrained consensus problem. Each convex set in this problem is associated with a certain agent and, generally speaking, is inaccessible to the other agents. A group of agents is interested in finding a common point of these sets, that is, a point satisfying all the constraints. Distributed analogues of the alternating projection method proposed for solving this problem lead to a rather complicated nonlinear system of equations, the convergence of which is usually proved using special Lyapunov functions. A brief survey of these methods is given, and their relation to the theorem ensuring consensus in a system of averaging inequalities recently proved by the first author is shown (this theorem develops convergence results for the usual method of iterative averaging as applied to the consensus problem).

On Sawada-Kotera and Kaup-Kuperschmidt integrable systems

To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods that are based on integrable scalar nonlinear partial differential equations. We show that some systems of integrable equations published recently are the M2 -extension of integrable such scalar equations. For illustration, we give Korteweg–de Vries, Kaup-Kupershmidt, and Sawada-Kotera equations as examples. By the use of such an extension of integrable scalar equations, we obtain some new integrable systems with recursion operators. We also give the soliton solutions of the systems and integrable standard nonlocal and shifted nonlocal reductions of these systems.

Evolution equations for open systems and collective variables: Which equation would you like to solve by molecular dynamics simulation?

In molecular dynamics simulations, the Langevin equation is frequently used to model the dynamics of collective variables and of systems coupled to baths. Often, external forces are added to the Langevin equation (e.g., when using targeted or steered molecular dynamics in biomolecular simulation). It is also popular to add derivatives of thermodynamic potentials to the Langevin equation as effective forces (e.g., when using a potential of mean force in a coarse-grained polymer model). These practices can be adventurous. In this article, we recall derivations of different versions of the Langevin equation and we discuss why care is needed if one would like to make changes to the structure of the equation.

Closed-Form Solutions of Injection-Driven Flow and Heat Transfer Inside an Inclined Horizontal Filter Chamber

Abstract The closed-form solutions of models provide operators and designers with a better understanding of how the systems perform practically, thus improving critical industrial production operations. Due to this importance, the case study at hand seeks to find closed-form solutions of the internal momentum and temperature variation during the filtration process to advance fluid purification. Lie group analysis is used to transform a system of equations representing the flow and heat transfer into a solvable system without changing the dynamics of the case study. The transformed solvable system is then integrated to find closed-form solutions of the internal velocity (momentum) and temperature variation. The obtained closed-form solutions are then used to analyze the effects of physical parameters arising from the process dynamics to find combinations of parameters that yield maximum permeates outflow. The analysis conveys that the internal fluid velocity increases when enhancing the permeation parameter and minimizing Reynolds number, wave speed parameter, and chamber height.

Analytical simulation of Darcy–Forchheimer nanofluid flow over a curved expanding permeable surface

Abstract This research paper presents an analytical simulation of Darcy–Forchheimer flow over a porous curve stretching surface. In fluid dynamics, the Darcy–Forchheimer model combines Forchheimer adjustment and high-velocity effects with Darcy’s formula for porous media flow: two nanofluid particles, molybdenum disulphide, and graphene oxide, form nanofluid with the base fluid blood. The governing partial differential equations for momentum and energy are converted into a nonlinear ordinary differential equations system by applying the appropriate similarity transformations. The homotopy analysis method is used to solve the transform equations analytically. The impact of essential factors includes the Forchheimer parameter, porosity parameter, slip parameter, Eckert number, nanoparticle volume friction, magnetic field parameter, and curvature parameter. The results have applications in the design of sophisticated cooling systems, where effective thermal control is essential.

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.

Rotating nonlinear states in trapped binary Bose-Einstein condensates under the action of the spin-orbit coupling

Abstract We report results of systematic analysis of confined steadily rotating patterns in the two-component BEC including the spin-orbit coupling (SOC) of the Rashba type, which acts in the interplay with the attractive or repulsive intra-component and inter-component nonlinear interactions and confining potential. The analysis is based on the system of the Gross-Pitaevskii equations (GPEs) written in the rotating coordinates. The resulting GPE system includes effective Zeeman splitting. In the case of the attractive nonlinearity, the analysis, performed by means of the imaginary-time simulations, produces deformation of the known two-dimensional SOC solitons (semi-vortices and mixed-modes). Essentially novel findings are reported in the case of the repulsive nonlinearity. They demonstrate patterns arranged as chains of unitary vortices which, at smaller values of the rotation velocity $\Omega $, assume the straight (single-string) form. At larger $\Omega $, the straight chains become unstable, being spontaneously replaced by a trilete star-shaped array of
vortices. At still large values of $\Omega $, the trilete pattern rebuilds itself into a star-shaped one formed of five and, then, seven strings. The transitions between the different patterns are accounted for by comparison of their energy. It is shown that the straight chains of vortices, which form the star-shaped structures, are aligned with boundaries between domains populated by plane waves with different wave vectors. A transition from an axisymmetric higher-order (multiple) vortex state to the trilete pattern is investigated too.

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.

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.

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.

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.

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.

Evaluating the mixed convection flow of Casson fluid from the semi-infinite vertical plate with radiation absorption effect

This study examines a steady laminar Casson fluid flow induced by a semi-infinite vertical plate under the impact of the Darcy-Forchheimer relation and thermal radiation. The features of mixed convection, cross-diffusion, radiation absorption, heat generation, chemical reactions and viscous dissipation are also considered to explain the transport phenomenon. The resultant system of equations, concerned with the problem under consideration, is transformed into a group of non-linear ordinary differential equations (ODEs) by means of similarity variables. The bvp4c method, an instrument popular for its numerical accomplishments, is utilized to solve this problem. The effect of flow parameters on heat transfer, concentration and velocity is evaluated via diagrams. To validate our code, we have compared the present outcomes to the prevenient literature, and stable consent has been detected. Moreover, the friction coefficient Cfx, Nusselt number Nux, and Sherwood number Shx are also computed to assess velocity gradient, efficiency of heat transfer and mass transfer process, respec-tively.

Paired piecewise linear regression model with fixed nodes

In this paper we develop a method for constructing a paired regression model in the class of piecewise linear continuous functions with fixed nodes. The concept of linear division of the correlation field, its partial sections and its nodes is introduced into consideration. Using the linear division of the correlation field, a class of piecewise linear functions with fixed nodes is determined. The linear division is revealed during the visual analysis of the correlation field. Using the least squares method, a system of linear equations is compiled to find point estimates of the parameters of the approximating function. With the exception of two unknown angular coefficients (unknown) this system is reduced to a tridiagonal system of equations, the unknowns of which are the nodal values of the approximating function. The tridiagonal system is solved by the run-through method. An algorithm was constructed to demonstrate the operation of the developed method. Its initial data is arrays of the corresponding values of the explanatory and dependent variables, as well as an array of numbers of the right ends of the intervals defining the nodes. An array of fixed nodes is constructed from an array of values of the explanatory variable and an array of numbers of the right ends of the intervals. Next, an array of point estimates of the parameters of the approximating function is constructed. This algorithm is implemented in Python in the form of user-defined functions.