Successive Linearization Method Research Articles

A New Numerical Solution of Maxwell Fluid over a Shrinking Sheet in the Region of a Stagnation Point

The mathematical model for the incompressible two‐dimensional stagnation flow of a Maxwell fluid towards a shrinking sheet is proposed. The developed equations are used to discuss the problem of being two dimensional in the region of stagnation point over a shrinking sheet. The nonlinear partial differential equations are transformed to ordinary differential equations by first‐taking boundary‐layer approximations into account and then using the similarity transformations. The obtained equations are then solved by using a successive linearisation method. The influence of the pertinent fluid parameters on the velocity is discussed through the help of graphs.

A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method

We present a novel application of the successive linearisation method to the classical Van der Pol and Duffing oscillator equations. By recasting the governing equations as nonlinear eigenvalue problems we obtain accurate values of the frequency and amplitude. We demonstrate that the proposed method can be used to obtain the limit cycle and bifurcation diagrams of the governing equations. Comparison with exact and other results in the literature shows that the method is accurate and effective in finding solutions of nonlinear equations with oscillatory solutions, nonlinear eigenvalue problems, and other nonlinear problems with bifurcations.

On the Solution of Double‐Diffusive Convective Flow due to a Cone by a Linearization Method

The paper details the use of a nonperturbation successive linearization method to solve the coupled nonlinear boundary value problem due to double‐diffusive convection from an inverted cone. Diffusion‐thermo and thermal‐diffusion effects have been taken into account. The governing partial differential equations are transformed into ordinary differential equations using a suitable similarity transformation. The SLM is based on successively linearizing the governing nonlinear boundary layer equations and solving the resulting higher‐order deformation equations using spectral methods. The results are compared with the limited cases from previous studies and results obtained using the Matlab inbuilt bvp4c numerical algorithm and a shooting technique that uses Runge‐Kutta‐Fehlberg (RKF45) and Newton‐Raphson schemes. These comparisons reveal the robustness and validate the usage of the linearisation method technique. The results show that the nonperturbation technique in combination with the Chebyshev spectral collocation method is an efficient numerical algorithm with assured convergence that serves as an alternative to numerical methods for solving nonlinear boundary value problems.

Successive Linearization Analysis of the Effects of Partial Slip, Thermal Diffusion, and Diffusion‐Thermo on Steady MHD Convective Flow due to a Rotating Disk

We proposed a general formulation of the successive linearization method for solving highly nonlinear boundary value problem arising in rotating disk flow. The problem was studied under the effects of partial slip, thermal diffusion, and diffusion‐thermo. The governing fundamental conservation equations of mass, momentum, angular momentum, energy, and concentration are transformed into a system of ordinary differential equations by means of similarity transformations. A parametric study illustrating the influence of the magnetic field, slip factor, Eckert number, Dufour and Soret numbers was carried out.

Application of Piecewise Successive Linearization Method for the Solutions of the Chen Chaotic System

This paper centres on the application of the new piecewise successive linearization method (PSLM) in solving the chaotic and nonchaotic Chen system. Numerical simulations are presented graphically and comparison is made between the PSLM and Runge‐Kutta‐based methods. The work shows that the proposed method provides good accuracy and can be easily extended to other dynamical systems including those that are chaotic in nature.

An improved spectral homotopy analysis method for MHD flow in a semi-porous channel

In this paper we report on a novel method for solving systems of highly nonlinear differential equations by blending two recent semi-numerical techniques; the spectral homotopy analysis method and the successive linearisation method. The hybrid method converges rapidly and is an enhancement of the utility of the original spectral homotopy analysis method (Motsa et al., Commun Nonlinear Sci Numer Simul 15:2293---2302, 2010; Computer & Fluids 39:1219---1225, 2010) and an improvement on other recent semi-analytical techniques. We illustrate the application of the method by solving a system of nonlinear differential equations that govern the problem of laminar viscous flow in a semi-porous channel subject to a transverse magnetic field. A comparison with the numerical solution confirms the validity and accuracy of the technique and shows that the method converges rapidly and gives very accurate results.

BOUNDARY LAYER FLOW AND DOUBLE DIFFUSION OVER AN UNSTEADY STRETCHING SURFACE WITH HALL EFFECT

The present investigation is concerned with the effect of Hall currents on boundary layer flow, and heat and mass transfer of an electrically conducting fluid over an unsteady stretching sheet in the presence of a strong magnetic field. The electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist, while the induced magnetic field is neglected. The governing time-dependent boundary layer equations for momentum, thermal energy, and concentration are reduced using a similarity transformation to a set of coupled ordinary differential equations. The similarity ordinary differential equations are then solved numerically by the successive linearization method together with the Chebyshev pseudo-spectral collocation method. Effects of the Prandtl number, Pr, Schmidt number, Sc, magnetic field, M, Hall parameter, m, and the unsteadiness parameter, A, on the velocity, temperature, and concentration profiles as well as the local skin friction coefficient and the heat and mass transfer rates are depicted graphically and/or in tabular form. Favorable comparisons with previously published work on various special cases of the problem are also obtained.

Successive linearisation analysis of unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and thermal radiation effects

AbstractIn this study, we investigate the application of the new successive linearisation method (SLM) to the problem of unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and thermal radiation effects. The governing nonlinear momentum, energy and mass transfer equations are successfully solved numerically using the SLM approach coupled with the spectral collocation method for iteratively solving the governing linearised equations. Comparison of the SLM results for various flow parameters against numerical results and other published results, obtained using the homotopy analysis method (HAM) and Runge–Kutta methods, for related problems indicates that the SLM is a very powerful tool which is much more accurate and efficient than other methods. The SLM converges much faster than the traditional methods like the HAM and is very easy to implement. © 2011 Canadian Society for Chemical Engineering

Hydromagnetic non‐Darcy flow, heat and mass transfer over a stretching sheet in the presence of thermal radiation and Ohmic dissipation

AbstractA numerical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristics on steady two‐dimensional flow of an electrically conducting fluid over a stretching sheet embedded in a non‐Darcy porous medium in the presence of thermal radiation and viscous dissipation. The governing partial differential equations are convected into a system of nonlinear ordinary differential equations by similarity transformation and are solved numerically by using the Successive linearisation method, together with the Chebyshev pseudo‐spectral collocation method. The effects of various parameters on the velocity, temperature, and concentration fields as well as on the skin‐friction coefficient are presented graphically and in tabular forms.

A Novel Numerical Technique for Two-Dimensional Laminar Flow between Two Moving Porous Walls

We investigate the steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain that is bounded by two permeable surfaces. The governing fourth-order nonlinear differential equation is solved by applying the spectral-homotopy analysis method and a novel successive linearisation method. Semianalytical results are obtained and the convergence rate of the solution series was compared with numerical approximations and with earlier results where the homotopy analysis and homotopy perturbation methods were used. We show that both the spectral-homotopy analysis method and successive linearisation method are computationally efficient and accurate in finding solutions of nonlinear boundary value problems.

A Note on the Solution of the Von Kármán Equations Using Series and Chebyshev Spectral Methods

The classical von Karman equations governing the boundary layer flow induced by a rotating disk are solved using the spectral homotopy analysis method and a novel successive linearisation method. The methods combine nonperturbation techniques with the Chebyshev spectral collocation method, and this study seeks to show the accuracy and reliability of the two methods in finding solutions of nonlinear systems of equations. The rapid convergence of the methods is determined by comparing the current results with numerical results and previous results in the literature.

Variable Viscosity on Magnetohydrodynamic Fluid Flow and Heat Transfer over an Unsteady Stretching Surface with Hall Effect

The problem of magnetohydrodynamic flow and heat transfer of a viscous, incompressible, and electrically conducting fluid past a semi-infinite unsteady stretching sheet is analyzed numerically. The problem was studied under the effects of Hall currents, variable viscosity, and variable thermal diffusivity. Using a similarity transformation, the governing fundamental equations are approximated by a system of nonlinear ordinary differential equations. The resultant system of ordinary differential equations is then solved numerically by the successive linearization method together with the Chebyshev pseudospectral method. Details of the velocity and temperature fields as well as the local skin friction and the local Nusselt number for various values of the parameters of the problem are presented. It is noted that the axial velocity decreases with increasing the values of the unsteadiness parameter, variable viscosity parameter, or the Hartmann number, while the transverse velocity increases as the Hartmann number increases. Due to increases in thermal diffusivity parameter, temperature is found to increase.

Error calculation for corrective machining with allowance requirements

Corrective machining is fundamental to obtain higher precision than the machine tool based on error inspection and calculation techniques. Generally, the error to be corrected is calculated by minimizing the material removal volume. However, there are often constraints for corrective machining, such as nonnegative error and allowance requirements, making it a constrained minimization problem. The basic algorithm for error calculation combines the alternating optimization method and the successive linearization method. Whereas, the sequence of objective function values is not guaranteed monotonously decreasing because of the local validity of linearization. Therefore, in the improved algorithm, the problem is reformulated as an inequality-constrained least square problem. Bound constraints are imposed on the optimization variables to keep local validity. Then, the line search and Palacios-type adjustment strategy are incorporated in each iteration in order to find a better point reducing the value of objective function. This point is chosen as the start of the next iteration. Finally, better convergence of the improved algorithm is verified by numerical simulations and a case of application in optical machining.

Regulation of Photovoltaic Voltage

In photovoltaic power systems, both photovoltaic modules and switching-mode converters present nonlinear and time-variant characteristics, which result in a difficult control problem. This paper presents an in-depth analysis and modeling to discover the inherent features of a photovoltaic power system. The method of successive linearization simplifies the nonlinear problem back to the linear case. This paper also presents the use of Youla parameterization to design a stable control system for regulating the photovoltaic voltage. The experimental and simulation results demonstrate the effectiveness of the presented analysis, design, and implementation.

SOLVABILITY OF A QUASI-STEADY ROLLING PROBLEM WITH SLIGHTLY COMPRESSIBLE MATERIAL MODEL

A quasi-steady rolling problem with slightly compressible, rigid-viscoplastic and isotropic hardening material model and nonlinear Coulomb friction law are studied. The problem is stated in the form of a nonlinear variational inequality coupled with an equivalent strain evolution equation. Under restrictions on the material characteristics, existence and uniqueness results are obtained and the convergence of a successive linearization method of solution is proved. An algorithm, combining this method with the finite element method, is proposed and applied to solve numerically an example problem.

Theory Analysis of Nonlinear Data Reconciliation and Application to a Coking Plant

The estimation methodology of process variables usually consists of three parts: classification of process variables, gross error detection, and data reconciliation. In this paper, we proposed a modified M-estimator method for the covariance estimator which depends on the results from robust statistics to reduce the effect of the gross errors. We consider the Lagrange multipliers method and successive linearization method for nonlinear data reconciliation. Finally, the example of a coking plant is presented to illustrate the effectiveness of the revised M-estimator method and nonlinear data reconciliation methods. In this paper, the classifying, estimating, and adjusting of the process variables are based on a components balance and total flow rates balance. The comparative results of the introduced methods are given and demonstrate the successful application of the proposed method to reconcile actual plant data from a complex chemical process.

Iterative algorithm for subaperture stitching interferometry for general surfaces

A novel iterative algorithm for subaperture stitching interferometry for general surfaces is presented. It is based on the alternating optimization technique and the successive linearization method. The computer-aided-design model of the tested surface is used to determine the overlapping region precisely. Subapertures are simultaneously stitched by minimizing deviations among them as well as deviations from the nominal surface. Precise prior knowledge of the six degrees-of-freedom nulling and alignment motion is no longer required.

Successive Linearization Methods for Nonlinear Semidefinite Programs

We present a successive linearization method with a trust region-type globalization for the solution of nonlinear semidefinite programs. At each iteration, the method solves a quadratic semidefinite program, which can be converted to a linear semidefinite program with a second order cone constraint. A subproblem of this kind can be solved quite efficiently by using some recent software for semidefinite and second-order cone programs. The method is shown to be globally convergent under certain assumptions. Numerical results on some nonlinear semidefinite programs including optimization problems with bilinear matrix inequalities are reported to illustrate the behaviour of the proposed method.

Seismic Attenuation Tomography Using the Successive Linearization Method

AbstractBased on the attenuation tomographical method in time domain, we propose a successive linearation method which is similar to the current travel‐time tomographical methods. We deduce a series of successive linearization formulae and obtain the iterative arithmetic and the executable technique of the seismic attenuation tomography using the successive linearization method. The method feasibility is validated by numerical experiments. As a supplement, multi‐parameters tomography is realized. We have applied the successive linearization method to engineering reconnaissance practice and gained preferable results.

Control of Copolymer Properties in a Semibatch Methyl Methacrylate/Methyl Acrylate Copolymerization Reactor by Using a Learning-Based Nonlinear Model Predictive Controller

This paper is concerned with the production of copolymers with uniform copolymer composition and desired weight-average molecular weight by using a learning-based nonlinear model predictive control (NLMPC). For an effective control in a semibatch copolymerization system, the successive linearization method used in NLMPC is modified. The nonlinear model for the copolymerization system is linearized by using the previous batch data within the prediction horizon in such a way that the linear time-varying model is obtained as a function of the increment of the inputs between the two consecutive batches. The learning algorithm is incorporated into the controller by virtue of the model structure, and the effectiveness of the proposed controller is shown by comparison with a conventional nonlinear model predictive controller as well as a linear model-based learning controller. Through the implementation of the controller to a semibatch methyl methacrylate/methyl acrylate copolymerization reactor, it is proven that copolymers with desired properties are produced effectively using the algorithm proposed in this study.