Numerical Solution Of Mathematical Model Research Articles

Экологиялық жүйедегі екі биологиялық түрлердің популяцияларының өзара әрекеттесуін математикалық модельдеу

The interaction of populations of two biological species in an isolated predator-prey ecosystem has been studied. Mathematical models of the interaction of these biological species are considered. The mathematical model is presented in the form of a system of differential equations. For the research, a program was created in the Mathcad environment. With the help of this program, graphical and numerical solutions of mathematical models were obtained. The dynamics of changes in the parameters of two populations has been studied. Experimental results concerning populations of two predator-prey biological species were considered and investigated. The statistical data of the hunting of the hare and lynx of the Canadian fur company are analyzed. The results of experiments on the study of argali and wolf populations in the Kokentau Nature Reserve of the East Kazakhstan region are presented. The correspondence of the theoretical results of graphical data and mathematical models of populations characterizing the interaction of animals in the argali-wolf ecosystem is proved.

Shear coefficient of spin viscosity estimation through velocity profiles of a ferrofluid under the effect of an external rotating magnetic field of low amplitude and frequency

The present article describes a numerical strategy for the estimation of the shear coefficient of spin viscosity for a ferrofluid sample confined to a cylindrical container and exposed to the effect of an external rotating magnetic field with a low amplitude and frequency. As far as we know, there are no experimental measurements of such coefficient. Furthermore, the few analytical values reported differ in several orders of magnitude. First, we describe briefly the mathematical model of the system and its numerical solution. Then, the definition of the direct and inverse problems is given as a part of the methodology for estimating such coefficient. Finally, we solve the inverse problem using simulated measurements and two global optimization algorithms. We generate this type of measurements by adding white Gaussian noise signals to the numerical solution of the ferrohydrodynamic mathematical model. Several noise levels in the range of 10 to 40 dB were used to increase the number of scenarios for validation purpose. Results showed an excellent agreement between the estimated values and those used in the numerical solution of the mathematical model. A statistical analysis revealed a normal distribution that was dependent on the noise level. This variation did not affect the results, but showed instead the validity of the proposed method. Additionally, this strategy stands as a computational tool for validating experimental results of the future in situ measurements. Tables 7, Figs 11, Refs 17.

Discrete Gradient Methods for Solving SIRI Epidemic Model Numerically While Preserving First Integrals

In this paper, we use discrete gradient methods for numerical solutions of mathematical models of epidemiology: a susceptible infected recovered infected (SIRI) model. The numerical results mimic the true and qualitatively correct behavior of the SIRI model. The first integrals (also known as the constant of motion) of the SIRI model obtained in literature using partial Lagrangian approach are well preserved by the discrete gradient method. The graphical results verify the advantages of the proposed schemes.

Chaste: Cancer, Heart and Soft Tissue Environment.

Chaste (Cancer, Heart And Soft Tissue Environment) is an open source simulation package for the numerical solution of mathematical models arising in physiology and biology. To date, Chaste development has been driven primarily by applications that include continuum modelling of cardiac electrophysiology ('Cardiac Chaste'), discrete cell-based modelling of soft tissues ('Cell-based Chaste'), and modelling of ventilation in lungs ('Lung Chaste'). Cardiac Chaste addresses the need for a high-performance, generic, and verified simulation framework for cardiac electrophysiology that is freely available to the scientific community. Cardiac chaste provides a software package capable of realistic heart simulations that is efficient, rigorously tested, and runs on HPC platforms. Cell-based Chaste addresses the need for efficient and verified implementations of cell-based modelling frameworks, providing a set of extensible tools for simulating biological tissues. Computational modelling, along with live imaging techniques, plays an important role in understanding the processes of tissue growth and repair. A wide range of cell-based modelling frameworks have been developed that have each been successfully applied in a range of biological applications. Cell-based Chaste includes implementations of the cellular automaton model, the cellular Potts model, cell-centre models with cell representations as overlapping spheres or Voronoi tessellations, and the vertex model. Lung Chaste addresses the need for a novel, generic and efficient lung modelling software package that is both tested and verified. It aims to couple biophysically-detailed models of airway mechanics with organ-scale ventilation models in a package that is freely available to the scientific community. Chaste is designed to be modular and extensible, providing libraries for common scientific computing infrastructure such as linear algebra operations, finite element meshes, and ordinary and partial differential equation solvers. This infrastructure is used by libraries for specific applications, such as continuum mechanics, cardiac models, and cell-based models. The software engineering techniques used to develop Chaste are intended to ensure code quality, re-usability and reliability. Primary applications of the software include cardiac and respiratory physiology, cancer and developmental biology.

MHD Free Convection of Localized Heat Source/Sink in Hybrid Nanofluid-Filled Square Cavity

In the field of heat transfer enhancement, there is a very new growing strategy to use hybrid nanofluid consisting of two or more nanoparticles dispersed in a base fluid. The current perusal introduces a numerical analysis to investigate the importance of hybrid nanofluid in the free convection inside a partially heated square cavity, and subjected to inclined magnetic field with heat generation/absorption. The horizontal walls are kept adiabatic. Numerical solution of mathematical model which describes the problem is achieved by finite difference method. The Conventional models of Brinkman and Maxwell were applied to assess the viscosity and thermal conductivity of the Cu-Al2O3-Water hybrid nanofluid. The considered problem was studied by varying some parameters such as the size and position of the heat source/sink, heat generation parameter, Hartmann number, and three combinations of nanoparticles volume fractions. The results show that the Nusselt number of hybrid nanofluid is higher than the Alumina-water nanofluid and less than the Copper-water nanofluid.

Creating molecular weight distribution of butadiene rubber on neodymium-based catalytic system

The method of calculation of the molecular weight distribution (MWD) of rubber SDRN synthesized in the cascade of reactors with a continuous mixer in the presence of chain transfer to the polymer (by direct numerical solution system of equations arising from the kinetic scheme of polymerization with special selection of boundary conditions, maximum length of macromolecules and calculation step value). Numerical solution of mathematical model that is systems of differential equations for concentrations of monomer, co-catalyst and macromolecules of polymer together with thermal balance is obtained. The influence of regime parameters on MWD and molecular mass characteristics of the final product obtained using molecular weight distribution are analyzed.

Experimental and numerical simulation on the influence of anisotropic fracture network deformation to shale gas percolation

Shale gas reservoirs usually have extremely low porosity and permeability. By massive fracturing, shale gas industry development has well-developed fracture network, which generally has the characteristics of anisotropy and pressure sensitivity. Present studies on fracture network fail to consider both anisotropy and pressure sensitivity. The percolation experiments on pressure sensitivity, gas slippage effect, and water damage mechanism are carried out by a new physical experiment method with similar materials. It is not only more convenient but also more secure to use similar materials to observe fracture deformation and the flow law of fluid under low pressure. Then a new percolation mathematical model considering both anisotropy and pressure sensitivity is established. And a numerical solution of mathematical model with 19-point fully implicit finite difference is conducted. Finally, a discussion is presented of the results of numerical simulation and experiment test. The most significant finding is that (a) the gas slip effect has a greater influence on gas apparent permeability of fracture network media than pressure-sensitivity effect does; (b) the gas apparent permeability of fracture network media was affected greatly by fracturing fluid due to the formation damage and relative permeability; (c) the permeability of fracture network increases with initial formation pressure increases, decreases with well bottom pressure decreases; and (d) the anisotropic fracture network deformation will significantly decrease permeability, and the cumulative gas production is positively correlated with angle between main facture and horizontal well. The study will can play a significant role in the simulation of shale gas production.

Analysis of inlet and gap effect in hydrodynamics and mass transport performance of a multipurpose electrochemical reactor: CFD simulation and experimental validation

This work deals with the reaction environment analysis (hydrodynamic and mass transport characterization) inside the flow channel of a parallel plate electrolyzer. Different electrode gap and inlet configurations were tested using computational and experimental techniques. Computational methods are based on numerical solution of mathematical models that describe fluid dynamics inside reactor meanwhile, experimental characterization was carried out by means of residence time distribution (RTD) and limiting current tests.In order to evidence the phenomenological influence of hydrodynamic pattern generated by different design parameters, a 3-D diffusion-convection model was implemented taking into account the local velocity vectors obtained by solution of Navier-Stokes equations to obtain theoretical RTD and by diffusion-convection equation in order to describe hydrodynamics and overall mass transport performance. Average Re numbers considered in this work were between 102 ≤ Re ≤ 702, which are characteristic of a laminar flow regime.Velocity fields obtained by CFD analysis evidenced the existence of prominent jet flow zones, particularly when 45° inlet angle is used. These effects diminished with 60° inlet angle configuration. RTD analysis also showed important tailing, indicating presence of internal flow by-passing channeling effects. Mass flux distribution (associated with global mass transport coefficient and limiting current) obtained by CFD analysis evidenced the existence of non-homogeneous distribution provoked by the flow pattern inside of our reactor. From phenomenological point of view, the analysis carried out in this work, demonstrates that mass flux distribution is strongly influenced by geometric design parameters of a multipurpose parallel plate reactor. The higher mass transport coefficient was obtained with 7 mm of interelectrode gap and 60° inlet angle configuration.

Physical and mathematical modeling of gas production in shale matrix

Shale gas mainly stores in shale matrix, and gas production in shale matrix is very important during exploration. In order to clarify gas production and transport mechanism in shale matrix, an experimental modeling of gas production in shale matrix was designed and conducted with Longmaxi shale samples collected from South of Sichuan. The experimental results show that gas production decline curve displays a “L” pattern which indicates initial production is high and declines rapidly, while late production is low and declines moderately; meanwhile, pressure propagation in shale matrix is quite slow due to ultralow permeability. Based on the results, a mathematical model was derived to describe gas production in shale matrix. The comparison between numerical solution of mathematical model and experimental results shows that the mathematical model can well describe gas transport in shale matrix. In addition, factors affecting gas production were investigated on the basis of the mathematical model. Adsorbed gas can replenish gas pressure in pores by desorption and delay pressure propagation, and gas production decreases very quickly when there is no adsorbed gas. Other parameters (diffusion coefficient, permeability and porosity) also need to be considered in shale gas development.

Ultrasonic-Assisted Osmotic Dehydration of Carrot Followed by Convective Drying with Continuous and Intermittent Heating

The aim of the studies was to analyze the effect of ultrasound action on osmotic pretreatment and then on drying kinetics in continuous and intermittent drying conditions, and on the final product quality. This article presents the results of intermittent-convective drying of carrot preceded by ultrasonic assisted osmotic dehydration in fructose aqueous solutions. The theoretical drying kinetics developed from the numerical solution of mathematical model is validated using the experimental data. It has been shown that combination of ultrasonic assisted osmosis with intermittent-convective drying accelerates the drying process and improves the quality of dried biomaterial. A good adherence of the numerically determined kinetic curves confirms the usefulness of the presented model and its possible application to construction of controlled and optimized drying processes.

The G-Scheme: A framework for multi-scale adaptive model reduction

The numerical solution of mathematical models for reaction systems in general, and reacting flows in particular, is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics. However, the dynamics can develop very-slow and very-fast time scales separated by a range of active time scales. An opportunity to reduce the complexity of the problem arises when the fast/active and slow/active time scales gaps becomes large. We propose a numerical technique consisting of an algorithmic framework, named the G-Scheme, to achieve multi-scale adaptive model reduction along-with the integration of the differential equations (DEs). The method is directly applicable to initial-value ODEs and (by using the method of lines) PDEs. We assume that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. The G-Scheme introduces locally a curvilinear frame of reference, defined by a set of orthonormal basis vectors with corresponding coordinates, attached to this decomposition. The evolution of the curvilinear coordinates associated with the active subspace is described by non-stiff DEs, whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem. Adjusting the active DEs dynamically during the time integration is the most significant feature of the G-Scheme, since the numerical integration is accomplished by solving a number of DEs typically much smaller than the dimension of the original problem, with corresponding saving in computational work. To demonstrate the effectiveness of the G-Scheme, we present results from illustrative as well as from relevant problems.

Transient and steady-state models for simulated moving bed processes: numerical solutions

The objective of this work is to compare different strategies for the numerical solution of mathematical models of simulated moving bed (SMB) processes to predict the transient and steady-state behaviour. The mathematical model is based on the true moving bed (TMB) description of SMB processes. The model assumes axial dispersion flow for the liquid phase and plug flow for the solid phase. Intraparticle mass transfer was described in terms of a simple linear driving force (LDF) approximation. The mathematical models for transient situations (TM_1 and TM_2) can be written in terms of system of partial differential-algebraic equations (PDAEs) or partial differential equations (PDEs), respectively, when the algebraic equation representing the adsorption equilibrium isotherm is inserted into the mass conservation/mass transfer kinetics equations or written as separate equation. Similarly, for steady-state the mathematical models (SSM_3 and SSM_4) are represented by a system of differential-algebraic equations (DAEs) or ordinary differential equations (ODEs). Different public domain solvers were used to numerically solve the four mathematical models described above. For the transient model TM_1 the DASSL solver was used to solve the system of PDAEs and for TM_2 the PDECOL package was used to solve the system of PDEs. For the steady-state model, the solvers COLDAE and COLNEW were used to solve the DAEs and ODEs systems, corresponding to the models SSM_3 and SSM_4, respectively. The glucose/fructose separation was used as a test case in a SMB with a 12 column configuration considering both linear and non-linear isotherms. The study of the effect of the operating conditions on the TMB performance and separation regions was also analysed. For the calculation of steady-state concentration profiles and process performance COLDAE solver is recommended because it requires less computing time and allows working with the adsorption equilibrium isotherm separately permitting easy changes on the algebraic relationship.

The solutions of vibration control problems using artificial neural networks

This paper introduces an alternative method artificial neural networks (ANN) used to obtain numerical solutions of mathematical models of dynamic systems, represented by ordinary differential equations (ODEs) and partial differential equations (PDEs). The proposed trial solution of differential equations (DEs) consists of two parts: The initial and boundary conditions (BCs) should be satisfied by the first part. However, the second part is not affected from initial and BCs, but it only tries to satisfy DE. This part involves a feedforward ANN containing adjustable parameters (weight and bias). The proposed solution satisfying boundary and initial condition uses a feedforward ANN with one hidden layer varying the neuron number in the hidden layer according to complexity of the considered problem. The ANN having appropriate architecture has been trained with backpropagation algorithm using an adaptive learning rate to satisfy DE. Moreover, we have, first, developed the general formula for the numerical solutions of nth-order initial-value problems by using ANN. For numerical applications, the ODEs that are the mathematical models of linear and non-linear mass-damper-spring systems and the second- and fourth-order PDEs that are the mathematical models of the control of longitudinal vibrations of rods and lateral vibrations of beams have been considered. Finally, the responses of the controlled and non-controlled systems have been obtained. The obtained results have been graphically presented and some conclusion remarks are given.

Kinetics of Steel Scrap Melting in Molten Fe-C Bath

The melting phenomena of steel scrap in molten steel bath has been investigated by several workers. Their works are reviewed in the present paper. The phenomena are, in general, so complicated that both heat and mass transfer take place simultaneously. Many workers made small scale experiments in such condition that only mass transfer was taken into consideration. The results were summarized in the form of dimensionless correlation among Sh, Gr. Re, Sc and J D . Only one paper reported the experiment in which pure heat transfer was taken into consideration. The overall heat transfer coefficient was estimated by fitting numerical solution of heat transfer equations to the results of immersion experiments. The simultaneous heat and mass transfer were analyzed by mathematical model in relation to BOF converter operation. Some workers made experiments and numerical solution of mathematical model to obtain the heat and mass transfer coefficients in pilot plant converter and commercial converter.

Solving multivariable mathematical models by the quadrature and cubature methods

AbstractThe utilization and generalization of quadrature and cubature approximations for numerical solution of mathematical models of multivariable transport processes involving integral, differential, and integro‐differential operators, and for numerical interpolation and extrapolation, are presented. The methodology for determination of the quadrature and cubature weights for composite operators is developed to accommodate for general functional representations. Application of these methods is demonstrated by solving two‐dimensional steady‐state and one‐dimensional transient‐state problems. The solutions are compared with exact‐analytical solutions to evaluate the performance of these methods. It is demonstrated that the quadrature and cubature approximations are simple and universal; i.e., the same formula is applicable irrespective of the order of accuracy of the numerical approximation, the type of linear operator, and the number of temporal and/or spatial variables. Since the quadrature and cubature methods can produce solutions with sufficient accuracy even when using fewer discrete points, both the programming task and computational effort are reduced considerably. Therefore, the quadrature and cubature methods appear to be very practical in solving the mathematical models of a variety of transport processes. © 1994 John Wiley & Sons, Inc.

Use of temporal moment analysis to study reactive solute transport in aggregated porous media

The paper presents an overview of the use of temporal moment analysis as a useful alternative to more complete analytical or numerical solutions of mathematical models describing adsorbing solute transport in aggregated, saturated porous media. The goal of temporal moment analysis is to derive analytical formulas for the lower-order time moments of the solute breakthrough curve, thereby yielding significant physical insight into the behavior of solute transport models. Only the bi-continuum conceptual model and one-dimensional systems are emphasized in this paper. The transfer function approach for calculating temporal moments is described and illustrated with several examples involving diffusion into spherical aggregates, first-order mass transfer, and kinetically controlled intra-aggregate adsorption. The use of moment analysis for model comparison and sensitivity analysis is discussed; in particular, the paper shows how simplified empirical first-order mass transfer models can be utilized to approximate more complex models involving diffusion into nonuniformly sized spherical aggregates.