System Of Discrete Equations Research Articles

Impact of wavy porous layer on mixed convection flow of a hybrid nanofluid in an enclosure under the effect of partial magnetic field

In this article, the effects of partial magnetic field and wavy porous layer on the characteristics of fluid flow and heat transfer are explored. The system of governing partial differential equations in dimensionless form is computed using the Galerkin based finite element method together for various range of parameters. The obtained discrete system of nonlinear algebraic equations is handled with Newton’s method and related linearized subproblems are solved by geometric multigrid techniques. The impact of involved parameters such as Richardson (Ri) number, Hartmann (Ha) number, porous layer parameters as amplitude (A) of waviness or the number of undulations () or the thickness (Hp ) of the porous layer, inclination angle of partial magnetic field (γ), and Darcy (Da) number on the dynamics of fluid and heat are investigated in detail. Moreover, the computational analysis has been demonstrated using streamlines, isotherms, and plots as well as average Nusselt number () along the hot vertical left wall and average kinetic energy in the entire domain. It is observed that is an increasing function of and Da while it is a decreasing function of also increases at Ha = 100 as 57.3%, which is the highest increase comparing to smaller Ha values, when Ri changes from 0.1 to 10, and significantly decreases at an angle at also rises for as Ri rises.

An efficient data assimilation algorithm using the Allen–Cahn equation

Data assimilation combines observation data and the physical model to provide an estimate of the state which is more accurate than what could be obtained by using just the data or the model alone. We will establish a robust algorithm for the Allen–Cahn (AC) equation by considering the data assimilation term. The proposed approach includes a feedback control scheme for the AC equation, which forces the numerical solutions to preserve the observed data. We utilize the Crank–Nicolson formula and a stable numerical method to solve the data assimilation method using the AC equation. The proposed numerical algorithm has second-order temporal and spatial accuracy. Our computational scheme has a low computational burden because of the linearity of the elliptic equation. The discrete system of equations is unconditionally energy stable. The efficiency of our proposed method has been verified through a number of computational tests. The proposed approach is the first simple algorithm to consider the data assimilation method using the AC equation. The computational results indicate that our method performs well with low spatial resolution observational measurements and can be used for studying the image inpainting and shape transformation processing.

An effective preconditioning strategy for volume penalized incompressible/low Mach multiphase flow solvers

The volume penalization (VP) or the Brinkman penalization (BP) method is a diffuse interface method for simulating multiphase fluid-structure interaction (FSI) problems in ocean engineering and/or phase change problems in thermal sciences and engineering. The method relies on a penalty factor (which is inversely related to body's permeability κ) that must be large to enforce rigid body velocity in the solid domain. When the penalty factor is large, the discrete system of equations becomes stiff and difficult to solve numerically. In this paper, we propose a projection method-based preconditioning strategy for solving volume penalized (VP) incompressible and low-Mach Navier-Stokes equations. The projection preconditioner enables the monolithic solution of the coupled velocity-pressure system in both single phase (uniform density and viscosity) and multiphase (variable density and viscosity) flow settings. In this approach, the penalty force is treated implicitly, which is allowed to take arbitrary large values without affecting the solver's convergence rate or causing numerical stiffness. It is made possible by including the penalty term in the pressure Poisson equation (PPE), which was not included in previous works that solved VP incompressible Navier-Stokes equations using the projection method. We show how and where the Brinkman penalty term enters the PPE by re-deriving the projection algorithm for the VP method. Solver scalability under grid refinement is demonstrated, i.e., convergence is achieved with the same number of iterations regardless of the problem size. A manufactured solution in a single phase setting is used to determine the spatial accuracy of the penalized solution. Various values of body's permeability κ are considered. Second-order pointwise accuracy is achieved for both velocity and pressure solutions for reasonably small values of κ. Error saturation occurs when κ is extremely small, but the convergence rate of the solver does not degrade. The solver converges faster as κ decreases, contrary to prior experience. Two multiphase fluid-structure interaction (FSI) problems from the ocean engineering literature are also simulated to evaluate the solver's robustness and performance (in terms of its number of iterations). The proposed solver also allows us to investigate the effect of κ on the motion of the contact line over the surface of the immersed body. It also allows us to investigate the dynamics of the free surface of a solidifying metal.

Application of the FEM Method to Modeling and Analysis of the Cold Thread Rolling Process-Part 1: Conditions for Ensuring a Plane State of Deformations.

The article concerns the application of the FEM method for the prediction of stress and deformation states in a workpiece during the thread rolling process (TR). The analysis covered a new kinematic variant of the TR process in which the basket of the head rotates and is torque-driven, while the workpiece is stationary and the head with the rollers moves axially relative to the workpiece. The TR process was considered as a geometrical and physical non-linear initial and boundary problem with non-linear, moving, and variable in time and space boundary conditions. The boundary conditions in the contact areas of the tool with the workpiece were unknown. An updated Lagrange (UL) description was used to describe the physical phenomena at a typical incremental step. The states of strain and strain rate were described by non-linear relationships without linearization. New discrete systems of motion and deformation equations of the object in the TR were introduced, which take into account the change in the stiffness of the system during the TR process. This equation was solved by the central differences method (explicit). The material parameters were estimated during tensile tests to determine the characteristics of the C45 steel, and a new semi-empirical method was used to determine the relationship yield stress, effective true strain, and effective true strain rate in the thread rolling process. A modified Cowper-Symonds material model was also used to model the displacement process of the wedge on an elastic/visco-plastic body reflecting the TR process. A non-linear dependency of material hardening module depending on strain and strain rate was introduced. To confirm the plane state of deformation and spatial state of stress, an experimental investigation was carried out. The computer models were validated, and a good convergence of the results was obtained. Applications in the ANSYS/LS-Dyna program were developed to simulate the TR process. The developed applications enable a comprehensive time analysis of the states of displacement, strain, and stress occurring in an object consisting of a workpiece (shaft) and a tool (roller) for the case of a plane strain state and a spatial stress state. Exemplary results of numerical analyzes are presented to explain the influence of the friction coefficient on the condition of the thread quality, and the state of deformations and stresses were shown.

ПАВУТИНОПОДІБНА МОДЕЛЬ РИНКУ З НАВЧАННЯМ

The article discusses both a simpler dynamic model of a product market (referred to as the «web-like model») and a modified version of this model where sellers set the market price. These dynamic models are described using discrete first-order difference equations and systems of difference equations. The study focuses on the importance of understanding dynamic market models for certain products, particularly the web and web-like market models with training, which are examined in the article. Based on certain assumptions (described further in the article), a Valsar interpretation of these models is provided: the market is regulated by an auctioneer who initially announces the product price, and then buyers and sellers execute agreements and communicate the results to the auctioneer in terms of supply and demand volumes. If the agreements are found to be imbalanced, the auctioneer adjusts the price in an attempt to restore market equilibrium. Final agreements are made once equilibrium is reached. The article considers cases where the initial point coincides or does not coincide with equilibrium, and analyzes the price and production volume trends in these scenarios. A formula is derived that determines the trajectory of price changes in the «spider model», indicating that the market price will fluctuate around the equilibrium price. Additionally, a modification of the simpler spider-like model is discussed, where sellers determine the market price by focusing on a weighted average value between demand and supply from the previous period. Similar to the web model, the equilibrium price is found. The article investigates the case when the initial point does not coincide with equilibrium and examines the trends of prices and production volumes in this scenario. Different types of models are presented, and the properties of general solutions of difference equations and systems of difference equations that describe the web-like model and its modification are analyzed.

A superconvergent meshfree collocation formulation for laminated composite plates with particular focus on convergence analysis

A superconvergent meshfree collocation formulation is presented for the accurate analysis of laminated composite plates, where the convergence characteristic is particularly investigated. The shear locking is resolved through the employment of a mixed formulation that takes the in-plane displacements, deflection, rotations and shear forces as primary variables. The resulting meshfree collocation equations are attained by introducing the recursive gradients constructed from the first order gradients of meshfree shape functions. Subsequently, the convergence of both the standard meshfree collocation approach and the proposed recursive gradient meshfree collocation formulation is theoretically analyzed for the system of mixed discrete equations corresponding to laminated composite plates. These theoretical results clearly identify the basis degree discrepancy issue for meshfree collocation analysis of laminated composite plates. More importantly, this accuracy deficiency is successfully removed by the recursive gradient meshfree collocation formulation, which essentially achieves a desirable superconvergence for the meshfree laminated plate solutions. The superior accuracy and locking free performance of the present meshfree collocation formulation is thoroughly demonstrated by numerical results.

Variable RPIM and HOCM coupling for non-linear buckling and post-buckling analysis of transverse FG sandwich beams

This paper presents an analytical-based approach to analyze non-linear buckling and post-buckling of transverse functionally graded sandwich beams (TFGSB). The proposed approach combines a high order continuation method (HOCM) and a variable radial point interpolation method (RPIM). First of all, the strong form of the beam buckling governing equations are established in the framework of first order shear deformation theory (FSDT) by taking into account the non-linear tangential term. Then, the spatial approximation of the unknown variables and their partial derivatives using the RPIM with variable shape parameters is the first step of this approach. The RPIM shape parameter is considered to change according to Franke formula, modified Franke formula or reciprocal Franke formula. The non-linear resulting system of discrete equations is linearized by using the Taylor series expansion, whilst the continuation method is employed to obtain the whole solution. To demonstrate the effectiveness of the proposed approach, numerical examples on non-linear buckling and post-buckling analysis of TFGSB are presented. These numerical tests are carried out to define the optimal RPIM with variable shape parameters and to investigate the sensitivity of the accuracy regarding boundary conditions, effects of power-law index, Young’s modulus, skin-core-skin thickness and the span-depth ratios. The results obtained through the proposed approach are compared with those computed by finite element method (FEM) and the available data in the literature.

Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations

Chains of coupled van der Pol equations are considered. The main assumption that motivates the use of special asymptotic methods is that the number of elements in the chain is sufficiently large. This allows moving from a discrete system of equations to the use of a continuity argument and obtaining an integro-differential boundary value problem as the initial model. In the study of the behaviour of all its solutions in a neighbourhood of the equilibrium state, infinite-dimensional critical cases arise in the problem of the stability of solutions. The main results include the construction of special families of quasi-normal forms, namely non-linear boundary value problems of either Schrödinger or Ginzburg–Landau type. Their solutions make it possible to determine the main terms of the asymptotic expansion of both regular and irregular solutions to the original system. The main goal is the study of chains with diffusion- and advective-type couplings, as well as fully connected chains.

Modeling diffusion and types I-V sorption of water vapor in heterogeneous systems

A sorption and diffusion model is developed using a mobile-immobile decomposition of the vapor concentration. Henry’s mode represents the mobile species which diffuses through the solid, while Langmuir and pooling modes represent possible immobilization processes that cause nonlinearities in sorption capacity. The model can simulate the five classical sorption types and can incorporate additional dynamics compared to models based on thermodynamic equilibrium. Additionally, the framework allows for discontinuities in material properties encountered in multi-material systems. The resulting non-linear coupled equations are solved by employing a finite element method to discretize in space, and a backward Euler method to discretize in time. The discretized system of equations is solved via a Newton-type iteration scheme at each time-step. Four different materials are parameterized for the model, which are then used in examples to demonstrate the model’s ability to capture complex sorption processes in multi-material systems.

A computational technique for the Caputo fractal-fractional diabetes mellitus model without genetic factors.

The concept of a Caputo fractal-fractional derivative is a new class of non-integer order derivative with a power-law kernel that has many applications in real-life scenarios. This new derivative is applied newly to model the dynamics of diabetes mellitus disease because the operator can be applied to formulate some models which describe the dynamics with memory effects. Diabetes mellitus as one of the leading diseases of our century is a type of disease that is widely observed worldwide and takes the first place in the evolution of many fatal diseases. Diabetes is tagged as a chronic, metabolic disease signalized by elevated levels of blood glucose (or blood sugar), which results over time in serious damage to the heart, blood vessels, eyes, kidneys, and nerves in the body. The present study is devoted to mathematical modeling and analysis of the diabetes mellitus model without genetic factors in the sense of fractional-fractal derivative. At first, the critical points of the diabetes mellitus model are investigated; then Picard's theorem idea is applied to investigate the existence and uniqueness of the solutions of the model under the fractional-fractal operator. The resulting discretized system of fractal-fractional differential equations is integrated in time with the MATLAB inbuilt Ode45 and Ode15s packages. A step-by-step and easy-to-adapt MATLAB algorithm is also provided for scholars to reproduce. Simulation experiments that revealed the dynamic behavior of the model for different instances of fractal-fractional parameters in the sense of the Caputo operator are displayed in the table and figures. It was observed in the numerical experiments that a decrease in both fractal dimensions and leads to an increase in the number of people living with diabetes mellitus.

Generating discrete dynamical system equations from input–output data using neural network identification models

This research presents a novel framework for generating equations describing discrete dynamical systems from only input–output data. The framework operates in two steps, creating a system identification model from input–output data using neural networks and then performing sensitivity analysis on the model. The sensitivity analysis is driven by a uniquely constrained functional decomposition of the identification model that breaks a complex identification problem into a group of small curve fitting problems. The resultant system equation represents the neural network identification model and by proxy the original system from which the input–output data belongs. The analysis allows for system equations to be generated from both black box systems and identification models, which can then be used for transparent and interpretable replacement of opaque system models. Transparent models can be better understood, leading to increased trustworthiness, safety, and reliability. An open source code implementation of the framework is created and made publicly available.

A Generic and Strictly Banded Spectral Petrov–Galerkin Method for Differential Equations with Polynomial Coefficients

In this paper we describe a generic spectral Petrov–Galerkin method that is sparse and strictly banded for any linear ordinary differential equation with polynomial coefficients. The method applies to all subdivisions of Jacobi polynomials (e.g., Chebyshev and Legendre), utilizes well-known recurrence relations of orthogonal polynomials, and leads to almost exactly the same discretized system of equations as the integration preconditioners [E. A. Coutsias, T. Hagstrom, and D. Torres, Math. Comp., 65 (1996), pp. 611–635] if this method was redesigned to make use of trial functions that satisfy a given problem’s boundary conditions. A link between the new Petrov–Galerkin method and IP is revealed through a new recursion relation for Jacobi polynomials. Because of the strictly banded nature of all coefficient matrices, the new method extends easily and efficiently to multiple dimensions though the use of tensor product methods.

Projector Approach to Constructing the Asymptotics of Solution of Initial Value Problems for Weakly Nonlinear Discrete Systems with Small Step in the Critical Case

An algorithm for constructing an asymptotic solution containing boundary functions for an initial value problem for a weakly nonlinear system of discrete equations with small step in the critical case under certain conditions is given in the article by V.F. Butuzov and A.B. Vasil’eva in Differ. Uravn., 1970, vol. 6, no. 4, pp. 650–664. In the present paper, orthogonal projectors are used to construct the asymptotics of the solution of this problem. This projector approach greatly simplifies the understanding of the algorithm for constructing the asymptotics and permits explicitly writing the problems from which one can find the terms of any order in the asymptotics of the solution.

Analysis of a Modified System of Infectious Disease in a Closed and Convex Subset of a Function Space with Numerical Study

In this article, the transmission dynamical model of the deadly infectious disease named Ebola is investigated. This disease identified in the Democratic Republic of Congo (DRC) and Sudan (now South Sudan) and was identified in 1976. The novelty of the model under discussion is the inclusion of advection and diffusion in each compartmental equation. The addition of these two terms makes the model more general. Similar to a simple population dynamic system, the prescribed model also has two equilibrium points and an important threshold, known as the basic reproductive number. The current work comprises the existence and uniqueness of the solution, the numerical analysis of the model, and finally, the graphical simulations. In the section on the existence and uniqueness of the solutions, the optimal existence is assessed in a closed and convex subset of function space. For the numerical study, a nonstandard finite difference (NSFD) scheme is adopted to approximate the solution of the continuous mathematical model. The main reason for the adoption of this technique is delineated in the form of the positivity of the state variables, which is necessary for any population model. The positivity of the applied scheme is verified by the concept of M-matrices. Since the numerical method gives a discrete system of difference equations corresponding to a continuous system, some other relevant properties are also needed to describe it. In this respect, the consistency and stability of the designed technique are corroborated by using Taylor’s series expansion and Von Neumann’s stability criteria, respectively. To authenticate the proposed NSFD method, two other illustrious techniques are applied for the sake of comparison. In the end, numerical simulations are also performed that show the efficiency of the prescribed technique, while the existing techniques fail to do so.

A hybrid epidemiological model based on individual dynamics

This article proposes a hybrid susceptible–infected–removed model which takes into consideration the spatiotemporal dynamics of the individuals. The model is based on a system of discrete stochastic diffusion equations. To build these equations, the two-dimensional diffusion equations coming from a balanced method are coupled with the human displacement probability law pattern through a discretization made by the finite volume method for complex geometries. The validation of this model is applied to COVID-19 spread. Since it is actual and the statistics are available elsewhere. The case of a developed country is used for simulation under some assumptions. Firstly to fit the chosen displacement pattern, then the accuracy of the statistics provided helps to analyze the sensitivity of the parameters of the model. The results explain the influence of the population movement on the evolution of the spread.

Global asymptotic behavior of a discrete system of difference equations with delays

In the present paper, we mainly investigate the qualitative behavior of the solutions of a discrete system of difference equations xn+1 = ? + Pmi =1 xn?i yn , yn+1 = ? + Pmi =1 yn?i xn , n ? N where ?, ? ? (0,?), m ? Z+, x?i and y?i are non-negative real numbers for i ? {0, 1, ...,m}. Namely, we discuss the boundedness character and the asymptotic stability properties of steady states of the mentioned system. Finally, for this system, we give a rate of convergence result which has an important place in the discrete dynamical systems. Besides, some numerical simulations with graphs are given to emphasize the efficiency of our theoretical results in the article.

A study on lung cancer using nabla discrete fractional-order model

This study proposes a nabla discrete fractional-order system of differential equations to model lung cancer and its interactions with lung epithelial cells, mutated cells, oncogenes, tumor suppressor genes, immune cells, cytokines, growth factors, angiogenic factors, and extracellular matrix. The proposed model can help predict cancer growth, metastasis, and response to treatment. Analytical results show the system is stable with a unique solution, and the model predicts that the immune system responds to cancer cells but eventually becomes overpowered. The numerical analysis employed the forward and backward Euler method and demonstrated that changes in parameter values have significant effects on the steady-state solution. The findings show that the growth of lung epithelial cells or their interaction with immune cells can cause an increase in the number of lung cancer cells. Conversely, an increase in cell death or a reduction in the interaction between lung epithelial cells and immune cells can decrease the number of lung cancer cells. The study highlights the usefulness of the nabla discrete fractional model in studying lung cancer dynamics.

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting (DS) method into the multiscale interpolating element-free Galerkin (VMIEFG) method, a dimension-splitting multiscale interpolating element-free Galerkin (DS-VMIEFG) method is proposed for three-dimensional (3D) singular perturbed convection-diffusion (SPCD) problems. In the DS-VMIEFG method, the 3D problem is decomposed into a series of 2D problems by the DS method, and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method. The improved interpolation-type moving least squares (IIMLS) method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems. The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems. The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes. For extremely small singular diffusion coefficients, the numerical solution will avoid numerical oscillation and has high computational stability.

A meshless finite point method for the improved Boussinesq equation using stabilized moving least squares approximation and Richardson extrapolation

AbstractA meshless finite point method (FPM) is developed in this paper for the numerical solution of the nonlinear improved Boussinesq equation. A time discrete technique is used to approximate time derivatives, and then a linearized procedure is presented to deal with the nonlinearity. To achieve stable convergence numerical results in space, the stabilized moving least squares approximation is used to obtain the shape function, and then the FPM is adopted to establish the linear system of discrete algebraic equations. To enhance the accuracy and convergence order in time, the Richardson extrapolation is finally incorporated into the FPM. Numerical results show that the FPM is fourth‐order accuracy in both space and time and can obtain highly accurate results in simulating the propagation of a single solitary wave, the interaction of two solitary waves, the solitary wave break‐up and the solution blow‐up phenomena.

A Mathematical Model of Susceptible Diabetes Complication (SDC) Model in Discrete Time Fuzzy and Crisp Environment

In this study, we examined the mathematical model of the discrete-time equation system with susceptible diabetes complication (SDC), which is known to be caused by environmental and genetic factors in a fuzzy environment. From the diabetes complication (DC) model, the susceptible diabetes complication (SDC) model is being developed. It was obtained using definitions of how the behavior of this model changes in a fuzzy environment. A nonlinear differential equation system transforms the sensitive diabetes complication (SDC) model into a discrete time equation system. Stability analysis of the model with jury criterion was examined. In addition, numerical solutions and graphics of the analysis of the discrete model in fuzzy environment are obtained by using the MATLAB package program.