System Of Nonlinear Equations Research Articles

Nonlinear stochastic model for epidemic disease prediction by optimal filtering perspective

Understanding and predicting novel diseases have become very important owing to the huge global health burden. The organization and study of mathematical models are critical in predicting disease behavior of the disease. In this paper, a new stochastic Susceptible‐Infected‐Recovered‐Death (SIRD) model for spreading epidemic disease is investigated. First, the deterministic SIRD model is considered, and then, by allowing randomness in the recovery and death rates that are not deterministic, the system of nonlinear stochastic differential equations is derived. For the suggested model, the existence and uniqueness of a positive global solution are demonstrated. The parameter estimation is done with the conditional least square estimator for deterministic models and the maximum likelihood estimator for stochastic ones. After that, we investigate a nonadditive state‐space model for spreading epidemic disease by considering infected as the hidden process variable. The problem of the hidden process variable from noisy observations is filtered, predicted, and smoothed using a recursive Bayesian technique. For estimating the hidden number of infected variables, closed‐form solutions are obtained. Finally, numerical simulations with both simulated and real data are performed to demonstrate the efficiency and accuracy of the current work.

Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

A nonlinear system of hybrid fractional differential equations with application to fixed time sliding mode control for Leukemia therapy

In this article, a coupled nonlinear problem of hybrid fractional differential equations (HFDEs) is presented for the qualitative work and numerical results. Two types of operators are involved in the research problem. One of them is DβrR which represent Riemann–Liouville's (RL) fractional derivatives while the operator L(DϱrR) is a series operator and DϱrR's are RL operators such that βr,ϱr∈(0,1]. These operators are joined by Φp operator. As a result, we have a nonlinear coupled system of FDEs. The newly established nonlinear system is studied for the existence, uniqueness criteria, stability of the solutions, and numerical computations. For the theoretical results, we take help from the available literature about the fixed point (FP) techniques. Then a computational scheme is developed with the help of Lagrange's interpolation technique. An application of the problem as a particular case is presented in the sense of the Leukemia mathematical model. The model presents the infection propagation. Leukemia can be managed by providing a chemotherapeutic treatment generally accepted to be safe, and a fractional-order fixed-time terminal sliding mode control has been developed to achieve this goal of removing Leukemic cells while keeping a sufficient number of normal cells. In order to evaluate the proposed controller stability, the fixed-time Lyapunov stability theory is employed. To better illustrate the study, comparison simulations are shown, demonstrating that the suggested control approach has higher tracking and convergence performance.

A Class of Efficient Sixth-Order Iterative Methods for Solving the Nonlinear Shear Model of a Reinforced Concrete Beam

In this paper, we present a three-step sixth-order class of iterative schemes to estimate the solutions of a nonlinear system of equations. This procedure is designed by means of a weight function technique. We apply this procedure for predicting the shear strength of a reinforced concrete beam. The values for the parameters of the nonlinear system describing this problem were randomly selected inside the prescribed ranges by technical standards for structural concrete. Moreover, some of these parameters were fixed taking into consideration the solvability region of the adopted steel constitutive model. The effectiveness of the new class is also compared with other current schemes in terms of the computational efficiency and numerical performance, with very good results. The advantages of this new class come from the low computational cost, due to the existence of an only inverse operator.

Picard and Adomian solutions of nonlinear fractional differential equations system containing Atangana – Baleanu derivative

In this paper, we apply two methods for solving nonlinear system of fractional differential equations (FDEs); these two methods are Picard and Adomian decomposition methods (ADM). The type of fractional derivative in this system will be the Atangana–Baleanu derivative. The existence and uniqueness of the solution will be proved. In addition, the convergence of ADM series solution and the maximum expected error will be discussed. Some numerical examples will be solved by using these two method and a comparison between their solutions will be done. There exist an important application to these types of systems, this application is the fractional-order rabies model and it will be solved here. From the obtained results, it is noticed that the obtained results from using these two methods are coincide with each other, and also these results are coincide with the obtained results from the classical fractional derivatives such as Caputo sense.

Evolutionary equations for the disturbed flame stabilised at the flat burner

The dynamics of the burner-stabilised flame is investigated theoretically in the frame of diffusive-thermal model. The nonlinear system of equations describing dynamics of curved flame front and the flame front temperature is derived in the quasi-stationary approximation from the basic diffusive-thermal model with one-step chemical reaction of Arrhenius type. The reduced model successfully describes nonlinear flame oscillation that is confirmed by comparison of oscillation characteristics obtained in the reduced and full models with the volumetric chemical reaction. The comparison of the instability increments depending on wave numbers agrees with the increments following from the linear stability analysis. The proposed model makes it possible to significantly reduce calculations by reducing the dimension of the system, as well as to obtain estimates of the critical parameters at which oscillations occur. Specifying the critical parameters by simple model will facilitate the performance of complex calculations in the models with detailed kinetics of chemical combustion reactions. A reduced model that describes the flame front spatial instability is proposed too.

An optimization method for solving a general class of the inverse system of nonlinear fractional order PDEs

In this paper, we introduce a general class of the inverse system of nonlinear fractional order partial differential equations (GCISNF-PDEs) with initial-boundary and two overdetermination conditions. An optimization method is considered based on the generalized shifted Legendre polynomials (GSLPs) for solving GCISNV-FPDEs. The concept of the fractional order derivatives (F-Ds) is utilized in the Caputo type. Operational matrices (OMs) of classical derivatives and F-Ds of GSLPs are extracted. Making use of GSLPs, OMs, and Lagrange multipliers method, we reduce the given GCISNF-PDEs into an algebraic system of equations. The proposed approach achieves satisfactory results simply for a small number of the novel GSLPs. In this work, two mathematical examples are illustrated to analyse the introduced method convergence and test its validity as well as applicability.

Алгоритм оптимального управления замкнутой системой рынка труда на заданном временном интервале

The ratio of supply and demand in the labor market is an important criteria for the effectiveness of the functioning of the economic system at the macro level. Improving the effectiveness of employers' personnel policy requires the development of new control tools and forecasting of dynamic labor market processes at the macro and micro levels. The problem of optimal control of the previously proposed closed labor market system is considered. The problem of optimal control of a system described by a system of nonlinear differential equations is formulated in general form. Aim. Finding piecewise constant control values at which the price-quality ratio in the labor market will be maximum over the next three years. Materials and methods. According to the concept of the model, the subjects of the labor market are divided into three categories depending on the amount of demand for their labor: subjects of high, low and medium qualifications. A hypothesis has been adopted according to which vacant places in the labor market are considered to be occupied by low-skilled subjects. Due to the previously proven uniqueness of solving the problem of optimal control of a closed labor market, a greedy algorithm for finding the maximum value of the price-quality ratio in the labor market is justified. Using Wolfram Mathematica, according to the method found earlier, the formulated optimal control problem was solved at each time half-interval. Results. Control values have been obtained at which the price-quality ratio in the labor market of the village of Sylva in the Perm Territory will be maximum over the next three years. The results substantiate the recommendation to introduce stricter requirements for applicants for vacant positions at the city-forming enterprise of the village of Sylva in the Perm Territory.

Solvability Analysis of the Nonlinear Integral Equations System Arising in the Logistic Dynamics Model in the Case of Piecewise Constant Kernels

Solvability Analysis of the Nonlinear Integral Equations System Arising in the Logistic Dynamics Model in the Case of Piecewise Constant Kernels

Knowledge-informed neuro-integrators for aggregation kinetics

We report a novel approach for the efficient computation of solutions of a broad class of large-scale systems of non-linear ordinary differential equations, describing aggregation kinetics. The method is based on a new take on the dimensionality reduction for this class of equations which can be naturally implemented by a cascade of small feed-forward artificial neural networks. We show that this cascade, of otherwise static models, is capable of predicting solutions of the original large-scale system over large intervals of time, using the information about the solution computed over much smaller intervals. The computational cost of the method depends very mildly on the temporal horizon, which is a major improvement over the current state-of-the-art methods, whose complexity increases super-linearly with the system’s size and proportionally to the simulation time. In cases when prior information about the values of solutions over a relatively small interval of time is already available, the method’s computational complexity does not depend explicitly on the system’s size. The successful application of the new method is illustrated for spatially-homogeneous systems, with a source of monomers, for a number of the most representative reaction rates kernels.

Investigation of MHD oxytactic microorganisms with NEPCMs in rectotrapezoidal enclosure with FEM: Applications to energy storage technologies

Nanoparticle-enhanced phase change materials (NEPCMs) have gained significant attention in recent years due to their unique ability to enhance the thermal properties and overall performance of phase change materials (PCMs). The present work investigates the bioconvection flow of nano-encapsulated phase change material within a rectotrapezoidal cavity. The left wall of the is fixed at the constant high temperature whereas the right vertical wall is kept at cold temperature. The higher order Galerkin finite element method (GFEM) has been employed to simulate the proposed problem. The discrete systems of nonlinear equations are linearized by implementing the adaptive Newtons method. Influence of various pertinent parameters such as inclined magnetic field, oxytactic bacteria, radiation and Lewis number has been analyzed. It is observed that the average Sherwood and Nusselt number exhibit a positive correlation with the Hartmann and radiation parameter. Moreover, the average heat transfer exhibits a decreasing trend as the Hartmann, bioconvection Rayleigh and Peclet increase.

A discrete model for force-based elasticity and plasticity

The article presents a mathematical model that simulates the elastic and plastic behaviour of discrete systems representing isotropic materials. The systems consist of one lattice of nodes connected by edges and a second lattice with nodes placed at the centres of the existing edges. The derivation is based on the assumption that the kinematics of the second lattice is induced by the kinematics of the first, and uses stored energies in edges of both lattices to derive a edge forces in the first lattice. This leads to a non-linear system of algebraic equations describing elasticity and plasticity in lattices. A numerical solution to the non-linear system is proposed by providing a matrix formulation necessary for software implementation. An illustrative example is given to justify the formulation and demonstrate the system behaviour.

Population dynamic study of two prey one predator system with disease in first prey using fuzzy impulsive control

Abstract Objectives The prey-predator model provides a mathematical framework for understanding the population dynamics of interacting species, highlighting the delicate balance between predator and prey populations in ecological systems. The four-species predator-prey model extends the Lotka-Volterra framework to explore the dynamics of ecosystems with multiple interacting species. It provides a theoretical foundation for understanding how the populations of multiple prey and predator species influence each other over time. Apart from the traditional methods like direct approach for solving the non-linear system of equations, recent Fuzzy method approaches have been developed. The solution of non-linear systems using classical methods is not easy due to its non-linearity, analytical complexity, chaotic behavior, etc. and the T-S method is very much effective to analyze the non-linear models. Methods In this study, we considered an eco-epidemic model with two populations of prey and one population of predator, with the only infectious disease infecting the first prey population. The four-dimensional Lotka-Volterra predator-prey system’s model stability has been examined using the Takagi-Sugeno (T-S) impulsive control model and the Fuzzy impulsive control model. Following the formulation of the model, the global stability and the Fuzzy solution are carried out through numerical simulations and graphical representations with appropriate discussion for a better understanding the dynamics of our proposed model. Results The Takagi-Sugeno method has diverse applications in modeling, control, pattern recognition, and decision-making in systems where uncertainty and non-linearity play a significant role. Its ability to combine fuzzy logic with traditional mathematical models provides a powerful tool for addressing complex real-world problems. Conclusions The impulse control approach, what is considered within the foundation of fuzzy systems established on T-S model, is found to be suitable for extremely complex and non-linear systems with impulse effects.

Convective flow generated in viscous liquid by isothermal and isoconcentration distribution in crown enclosure with novel aspects of inclined magnetic field

ABSTRACT Combined buoyancy shifts from temperature and concentricity gradients create double-diffusive convection. Complex Crown form enclosure with heat and mass source brings originality and practicality by enhancing use in a broad variety of technical, industrial, and geophysical operations. Current artifact is to explore thermosolutal diffusion in buoyantly driven flowing of viscid liquid contained in a crown enclosure with the installation of a uniformly heated and soluted circular cylinder. Novel physical characteristics of inclining magnetic field are introduced, and distributions affected by cylinder radius change are compared. Dimensionless PDEs are solved numerically using finite elements. Grid independence test ensures simulation mesh level. Hybrid meshing with triangular and rectangular components discretizes domain. Linear interpolating polynomials approximate pressure, whereas quadratic polynomials approach velocity, temperature, and concentricity. PARDISO solves a non-linear system of equations efficiently. Average Nusselt and Sherwood numbers are estimated by comparing relevant parameters to cylinder radius change. Heat transference rate rises to 40% and mass transport upsurges to 60% when radius of cylinder is increased from 0.1 to 0.2. Hartmann number tends to affect a decline in Nusselt and Sherwood quantities. Reduction in thermal and solutal boundary layers near surface of the cylinder is observed against the improve in radius of cylinder.

Convergence and Stability Improvement of Quasi-Newton Methods by Full-Rank Update of the Jacobian Approximates

A system of simultaneous multi-variable nonlinear equations can be solved by Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (Secant equation) does not completely specify the Jacobian approximate in multi-dimensional cases, so its full-rank update is not possible with classic variants of the method. The suggested new iteration strategy (“T-Secant”) allows for a full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates; then, the Jacobian approximate can be fully updated. It is shown that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the superlinear convergence of the Secant method φS=1.618… to super-quadratic φT=φS+1=2.618… and the quadratic convergence of the Newton method φN=2 to cubic φT=φN+1=3 in one-dimensional cases. In multi-dimensional cases, the Broyden-type efficiency (mean convergence rate) of the suggested method is an order higher than the efficiency of other classic low-rank-update quasi-Newton methods, as shown by numerical examples on a Rosenbrock-type test function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single-variable cases helps explain the basic operations, and a vector-space description is also given in multi-variable cases.

Nonlinear complex generalized zakharov dynamical system inconformal sense utilizing new kudryashov method

We take into account the nonlinear complex generalized Zakharov dynamical system which models the spread of the Langmuir waves in ionized plasma, in the conformal sense in this manuscript. Fractional wave transformation is enforced to convert the nonlinear fractional system to a nonlinear ordinary differential equation system. The new Kudryashov method which was recently introduced and is an efficient method, is implemented to the presented equation to acquire analytical solutions. The required constraint conditions are offered to ensure the validity of the obtained solutions. To analyze the physical interpretations for some of the produced solutions, we illustrate some graphical representations. We derive the bright and singular solitons. Furthermore, 2D views of the behavior of the solitons are represented to investigate the effect of the values of the parameters in the proposed model and fractional parameters. Also, the modulation instability of the model is investigated to ensure the obtained results are stable.

A comparative study of discretization techniques for augmented Urysohn type nonlinear functional Volterra integral equations and their convergence analysis

This article considers the nonlinear functional Volterra integral equation, the Urysohn type class of nonlinear integral equations. In an approach based on the Picard iterative method, the solution's existence and uniqueness are demonstrated. The trapezoidal and Euler discretization methods are used for the approximation of a numerical solution, and a nonlinear algebraic system of equations is obtained. The first and second order of convergence for the Euler and the trapezoidal methods respectively to the solution is manifested by employing the Gronwall inequality and its discrete form. A new Gronwall inequality is devised in order to prove the trapezoidal method's convergence. Finally, some numerical examples are provided which attest to the application, effectiveness, and reliability of the methods.

Boundary Value Problem for a Coupled System of Nonlinear Fractional q-Difference Equations with Caputo Fractional Derivatives

This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic friction. To investigate this system, we introduce an operator that exhibits fixed points corresponding to the solutions of the problem, effectively transforming the system into an equivalent fixed-point problem. We established the necessary conditions for the existence and uniqueness of solutions using the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle, respectively. Stability results in the Ulam sense for the coupled system are also discussed, along with a sensitivity analysis of the range parameters. To support the validity of their findings, we provide illustrative examples. Overall, the paper offers a thorough examination and analysis of the considered coupled system, making important contributions to the understanding of multi-atomic systems and their mathematical modeling.

Numerical Study for MHD Flow of an Oldroyd-B Fluid Over a Stretching Sheet in the Presence of Thermal Radiation with Soret and Dufour Effects

This paper investigates the impact of Soret and Dufour's MHD flow of an Oldroyd-B fluid over a stretching sheet in the presence of thermal radiation. By a similarity transformation, the controlling partial differential equations are transformed into a system of nonlinear ordinary differential equations. Using the successive linearization method (SLM), the linear system is solved. A determination and discussion of the impacts of specific fluid parameters on the temperature, concentration distribution, and velocity are presented. As the magnetic field increases, we observe that the temperature and concentration profiles rise, while the velocity profile falls. In addition, increases in the Dufour and Soret levels will also result in an improvement in the temperature and concentration distribution. The validity of the acquired results is tested by comparing them to previously published works, with particular attention paid to the accuracy and convergence of the solution.

Efficient passive GDQLL scrutinization of an advanced steady EMHD mixed convective nanofluid flow problem via Wakif–Buongiorno approach and generalized transport laws

Owing to the practical importance of nanofluids and their adjustable thermal capabilities, this study intends to develop a robust generalized differential quadrature local linearization (GDQLL) algorithm for examining realistically the heat and mass aspects of electrically conducting nanofluids during their non-Darcian laminar motion nearby a convectively heating vertical surface of an active electromagnetic actuator (i.e., Riga plate). By invoking Wakif–Buongiorno model and Oberbeck–Boussinesq approximations along with other generalized transport laws (i.e., Cattaneo–Christov and non-Fick’s laws) and Grinberg’s concept, a set of gigantic partial differential equations is stated appropriately in the sense of the boundary layer approximations for describing exhaustively the present EMHD mixed convective nonhomogeneous flow under the passive control strategy of nanoparticles within the nanofluidic medium. Operationally, the dimensionless differential forms of the governing boundary equations are derived properly by introducing reasonable mathematical adjustments into the preliminary formulation. In this case, the differential complexity of the leading differential structure is reduced to a nonlinear coupled system of ordinary differential equations, whose discrete numerical solutions are computed perfectly via a well-structured GDQLL algorithm. As foremost outcomes, it is demonstrated that the nanofluid motion and its surface thermal enhancement rate can be reinforced significantly through the thermal strengthening in the convective heating and mixed convective process as well as via the electromagnetic improvement in the driven aspect of Lorentz’s forces.