Adams Predictor-corrector Method Research Articles

OPTIMAL CONTROL ON FRACTIONAL ENZYME KINETICS WITH INHIBITORS

In enzyme kinetic chemical reaction, inhibitors are working as a regulators of the metabolism of the systems and the biochemical activities as well. The main focus of this study is to explore the fractional dynamics of enzyme kinetic biochemical reactions with competitive and uncompetitive inhibitors. Further, the analysis continuous with the optimal controls on the system which accelerate biochemical reactions and maximize product generation. Eventually, the numerical analysis have been done by the Adams predictor–corrector method and the forward–backward sweep method for the dynamics and optimal control study, respectively.

Analysis of models describing thermocline depth-ocean temperature and dissolved oxygen concentration in the ocean-plankton community

In the present work, the two mathematical models describing components of the climate are examined using the numerical method. The first model depicts the relation between the change in thermocline depth and ocean temperature. The second model is simplified by eliminating the natural depletion of the Phytoplankton population. The stability and convergences analysis is conducted for the modified system. The numerical simulation is conducted with the help of the most proven numerical method, called the Adams Predictor-Corrector method. The effect of the oxygen production rate by Phytoplankton (τ) is illustrated with 2D plots. The graphs of the solutions of the models demonstrate the influence of different parameter values on the solutions. The present investigation helps to work for more complex models, and the model can be examined with novel operators.

Adams predictor–corrector method for solving uncertain differential equation

Uncertain differential equation (UDE) is a type of differential equation driven by Liu process, which is used to describe uncertain dynamic systems. Since the analytic solution of UDE is difficult to obtain or the representation is complex, this paper proposes a new Adams predictor–corrector method to solve UDE. Compared with Euler Method, Runge–Kutta method, Adams method, Milne method, Adam–Simpson method and Hamming method of solving UDE, the proposed algorithm has higher accuracy. The extreme value and time integral of solution to UDE are obtained in virtue of the proposed method. Meanwhile, the convergence properties of the method are discussed. Finally, some illustrated examples are provided to demonstrate the efficiency and accuracy of the proposed technique.

Numerical approximations of 2-point Adams predictor-corrector block method for neutral delay using Newton’s divided difference formulation

Numerical approximations of 2-point Adams predictor-corrector block method for neutral delay using Newton’s divided difference formulation

A Modified Fractional Differential Love Model

Abstract A modified system of nonlinear fractional-order differential equations become used to classify humans of various personalities and different Impact Factors of Memory (IFM); with unique set of model parameters. The model was used to interpret and predict the functions of the union of various people with external circumstance(s) and adapted to neighborhood environment in which the statistics collections were achieved to analyze numerous measures affecting marriages, unique challenges in marriage and associated occasions were investigated through the use of questionnaire. Data had been analyzed and the outcomes have been carried out as parameters to validate the model. Adams Predictor-Corrector Method was used to test the chaoticity of the system and it was confirmed via numerical simulations. Numerical simulation outcomes had been presented to reveal the effectiveness of the model and the accuracy of the statements established. The mathematical information implied by the model unveil an underlying mechanism which can give an explanation for couple disruption in relationships that were initially deliberated to remain all the time. Despite the terrible aspects of relationships, some human beings were still satisfied in their intimate members of the family. The study was addressed on a field survey (use of questionnaires) and with the aid of interrogating the members one on one. A feasible path for future work is the choice to attain balance through modelling and to validate the results with the aid of numerical simulations.

Time-varying nonlinear modeling and analysis of algal bloom dynamics

The ecological dynamic models of algal bloom in existence are of poor environmental adaptability and hard to reflect nonlinear dynamic change of algal bloom formation mechanism. To solve this problem, time variable is applied to mechanism-based model according to algal bloom nonlinear dynamic in this paper, and mechanism feature of algal bloom is represented by time function model and effecting function model. Data-driven modeling approaches including tabu search and genetic algorithm are also adopted to optimize structure and parameter for the time-varying nonlinear modeling to improve environmental adaptability and accuracy of the model. High-precision numerical solutions of the optimized time-varying nonlinear model is obtained by fourth-order Adams predictor–corrector method. This method finally realizes effective prediction of algal bloom.

MHD and Radiation Effects on Mixed Convection Unsteady Flow of Micropolar Fluid Over A Stretching Sheet

This present paper is concerned with the study of the magnetohydrodyamics (MHD) effects on mixed convection flow of an incompressible micropolar fluid over a stretching sheet in case of unsteady flow. Energy equation takes into account of thermal radiation. The stretching velocity is assumed to vary linearly with the distance along the sheet. Two equal and opposite forces are assumed to be impulsively applied along axial direction. The governing non-linear equations and their associated boundary conditions are first cast into a dimensionless form using local non-similarity transformations. The resulting equations are solved numerically using the Adams-Predictor Corrector method. A representative set of numerical results is displaced graphically to illustrate the influence of various physical parameters on velocity, microrotation profiles as well as the skin friction coefficient. It is found that there is a smooth transition from small-time solution to the large-time solution.

Adams Predictor-Corrector Systems for Solving Fuzzy Differential Equations

A predictor-corrector algorithm and an improved predictor-corrector (IPC) algorithm based on Adams method are proposed to solve first-order differential equations with fuzzy initial condition. These algorithms are generated by updating the Adams predictor-corrector method and their convergence is also analyzed. Finally, the proposed methods are illustrated by solving an example.

Development of a thermal–hydraulic safety analysis code RETAC for AP1000

In the present study, a thermal–hydraulic safety analysis code for AP1000 named RETAC (REactor Transient Analysis Code) has been developed using FORTRAN 90 language. A point reactor neutron kinetics model with six groups of delayed neutrons was adopted to describe the core thermal power transient. A distributed parameter model with two-phase drift flux model was used in the U-tube steam generator simulation. In the pressurizer simulation, the RETAC code was equipped with three-region non-equilibrium model and multi-region non-equilibrium model respectively. Similar to current large commercial codes such as RELAP5 and RETRAN series, the four-quadrant analogy curves were adopted for the solution of the transient behaviors of the main coolant pumps. In this paper, a new and reasonable model for the passive residual heat removal heat exchanger (PRHR HX) was proposed based on basic equations of mass, momentum and energy. Gear method and Adams predictor-corrector method were adopted alternately for a better solution to ill-condition differential equations corresponding to detail processes. The PRHR HX inadvertent operation accident and the ADS (automatic depressurization system) inadvertent operation accident were chosen in the transient accident analysis. Furthermore, the simulation results obtained by RETAC were compared with that by Westinghouse-developed LOFTRAN code. The comparison results showed a good agreement and thus proved the accuracy and reliability of the RETAC code. With the adoption of modular programming techniques, the RETAC code can be easily modified and applied to higher power passive safety reactors in the future.

Turbulent flow in converging nozzles, part one: boundary layer solution

The boundary layer integral method is used to investigate the development of the turbulent swirling flow at the entrance region of a conical nozzle. The governing equations in the spherical coordinate system are simplified with the boundary layer assumptions and integrated through the boundary layer. The resulting sets of differential equations are then solved by the fourth-order Adams predictor-corrector method. The free vortex and uniform velocity profiles are applied for the tangential and axial velocities at the inlet region, respectively. Due to the lack of experimental data for swirling flows in converging nozzles, the developed model is validated against the numerical simulations. The results of numerical simulations demonstrate the capability of the analytical model in predicting boundary layer parameters such as the boundary layer growth, the shear rate, the boundary layer thickness, and the swirl intensity decay rate for different cone angles. The proposed method introduces a simple and robust procedure to investigate the boundary layer parameters inside the converging geometries.

Analytical investigation of boundary layer growth and swirl intensity decay rate in a pipe

In this research, the developing turbulent swirling flow in the entrance region of a pipe is investigated analytically by using the boundary layer integral method. The governing equations are integrated through the boundary layer and obtained differential equations are solved with forth-order Adams predictor-corrector method. The general tangential velocity is applied at the inlet region to consider both free and forced vortex velocity profiles. The comparison between present model and available experimental data demonstrates the capability of the model in predicting boundary layer parameters (e.g. boundary layer growth, shear rate and swirl intensity decay rate). Analytical results showed that the free vortex velocity profile can better predict the boundary layer parameters in the entrance region than in the forced one. Also, effects of pressure gradient inside the boundary layer is investigated and showed that if pressure gradient is ignored inside the boundary layer, results deviate greatly from the experimental data.

A staggered-grid numerical algorithm for the extended Boussinesq equations

A second-order accurate numerical scheme is developed to solve Nwogu’s extended Boussinesq equations. A staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. The numerical method is validated against available analytical solutions, other numerical results of Navier–Stokes equations, and experimental data for both 1D and 2D nonlinear wave transformation problems. It is shown that the new algorithm has very good conservative characteristics for mass calculation. As a result, the model can provide accurate and stable results for long-term simulation. The model has proven to be a useful modeling tool for a wide range of water wave problems.

Development of a thermal-hydraulic analysis code for CARR

The China advanced research reactor (CARR) being built in Beijing, China, is a multipurpose research reactor for a variety of fields. Theoretical calculation of thermal hydraulic characteristics of CARR is presented in this paper. The theoretical analysis consists of initial steady and transient accidental analyses. Point reactor neutron kinetics model with six groups of delayed neutron is adopted for the solution of reactor power. All possible flow and heat transfer conditions are considered and the corresponding optional models are supplied in the theoretical calculations. A new simple and convenient model is proposed for the resolution of the transient behaviors of main pump instead of the complicated four-quadrant model. Gear method and Adams predictor–corrector method are adopted alternately for a better solution to such ill-conditioned differential equations corresponding to detail process. The initial multi-channel analysis shows that the effects of geometrical size on flow distribution play dominant role and the effects of core power distribution may be neglected. The temperature fields of fuel elements under asymmetrical cooling condition are also obtained, which are the bases for further study on transient-induced stress analysis, etc. Accidental analyses show that the activity of emergency cooling system apparently reduces the peak temperatures of fuel and coolant, peak quality and other operation parameters. Thus it effectively ensures the safety in operation of CARR. Because of the adoption of modular programming techniques, this code is expected to be applied to accidental analysis of other types of reactors by easily modifying the corresponding function modules. Also, this code is expected to be validated against experimental data.

New Predictor-Corrector Methods Based on Exponential Time Differencing for Systems of Nonlinear Differential Equations**The project supported by National Natural Science Foundation of China under Grant No. 19902002

We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponential time differencing and Adams–Bashforth–Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.

Nonlinear regenerative chatter in turning

Chatter is a topic of immense engineering importance because its occurrence in machining results in poor surface finish, promotes tool wear and hampers productivity. In this paper, a single degree of freedom, delay differential equation model is presented. The central idea of the model is the study of regenerative chatter effect, which qualitatively explains the nonlinear dynamics in machining. Stability charts are derived for the linearized case in a two dimensional phase plane ( μ, Ω) with the bifurcation parameter μ representing the dynamic variation of the chip thickness, and the critical rotational speed Ω. The governing nonlinear equations are integrated in time by Adams Predictor corrector Method. Phase portraits are drawn between the instantaneous tool position x 1 and the relative tangential velocity x 2. It has been shown that in the absence of a cubic nonlinearity, the chatter response appears to be more chaotic.

Computer Simulation of a Variable Fill Hydraulic Dynamometer Part 3: Closed-Loop Performance

An engine and hydraulic dynamometer system is dynamically modelled with two different control systems. The closed-loop system behaviour is studied for a hydromechanical back-pressure water outlet valve and for an electrohydraulic servo-controlled water outlet valve. This is an extension of the open-loop dynamic model of Part 2, which incorporates the mathematical model of the torque absorption processes in a variable fill Froude-type hydraulic dynamometer presented in Part 1. The models are based on a system of first-order differential equations which are solved numerically using an Adams predictor corrector method. Both the analogue hack-pressure valve and digital servo-controlled valve are accurately simulated. In both cases the influence of the dynamic dynamometer model on the transient response is apparent. Variation in digital control system parameters across the operating envelope illustrates the non-linearity of dynamometer response. The computer model and experimental results are compared and implications of the closed-loop behaviour discussed.

A study on the temperature distributions of steel frames in a fire environment

Abstract This study examines the temperature distributions of steel frames exposed to a prescribed fire environment. The effects of radiation, natural convection, and heat conduction are fully considered in formulating the mathematical model. The method of orthogonal collocation and the variable order Adams predictor corrector method are adopted for the numerical solution. The numerical results show that the temperature distributions of steel structures are highly dependent on the fire environment, and the temperature in structural members directly exposed to the fire load increases rapidly. The geometry of structures, and the ratio between the perimeter and the area of cross section are found to have significant influence on the distributions of temperature. In addition, the results indicate that the radiation has a predominant effect on the temperature distributions among the three modes of heat transfer.

Efficient calculation of chemically reacting flow

A semi-implicit finite volume formulation is used to study flows with chemical reactions. In this formulation the source terms resulting from the chemical reactions are treated implicitly and the resulting system of partial differential equations is solved using two time-stepping schemes. The first is based on the Runge-Kutta method while the second is based on an Adams predictor-corrector method. Results show that improvements in computational efficiency depend to a large extent on the manner in which the source term is treated. Further, analysis and computation indicate that the Runge-Kutta method is more efficient than the Adams methods. Finally, an adaptive time stepping scheme is developed to study problems involving shock ignition. Calculations for a hydrogen-air system agree well with other methods.

Invariance theorems concerning reflection at numerical boundaries

Downwind computational boundaries in the numerical approximation of hyperbolic equations are in general not transparent, and they create spurious reflection. A useful measure of this is given by the energy (or usual sum of squares) of the reflected solution in response to an arbitrary solution which originates from within the computing domain. We prove, in that respect, a somewhat unexpected property: namely, that for those full-discretizations which are obtained by applying to a space-discretization of the equations an energy conservative discrete time-marching method, the energy reflected at the boundary is independent of the value of Δt, and is strictly equal to the energy reflected in the semidiscrete case. This is verified in numerical experiments. Optimal boundary equations may be defined in the semidiscrete case of those which maximize the rate of convergence to zero of the reflected energy when h-+0. A corollary of the preceding result is that those boundary equations remain optimal, in the same sense, when used in an energy conservative full discretization. Moreover, this convergence result continues to hold when a nonconservative but stable (i.e., dissipative) time-discretization is used. This is verified numerically with the first-order Adams predictor-corrector method. The results of this paper are derived with the mathematics of the simple three-point central difference discretization of spatial derivatives. Obvious generalizations to other cases are mentioned at the end of the paper.

On the numerical integration of elasto-plastic constitutive model structures for nonproportional cyclic loading

The classical Runge-Kutta method with Gill coefficients, a non-iterative Adams predictorcorrector method, an Euler's method with automatic step-size control, an iterative Adams predictorcorrector method with automatic step-size control and Gear's method are the numerical solution algorithms considered in this study. Their computational accuracy and efficiency are evaluated for two cases of axial-torsional loading with transient, nonproportional, cyclic plasticity. The constitive equations implemented include a modified classical single-surface theory, a two-surface theory and a unified creep-plasticity or state variable theory.