scholarly journals On Extending the Quasilinearization Method to Higher Order Convergent Hybrid Schemes Using the Spectral Homotopy Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Sandile S. Motsa ◽  
Precious Sibanda
Keyword(s):  
Higher Order ◽  
Nonlinear Boundary Value Problems ◽  
Hybrid Schemes ◽  
Iterative Schemes ◽  
Flow Problems ◽  
Highly Nonlinear ◽  
Homotopy Analysis ◽  
Layer Flow ◽  
Type Boundary ◽  
Significant Figures

We propose a sequence of highly accurate higher order convergent iterative schemes by embedding the quasilinearization algorithm within a spectral collocation method. The iterative schemes are simple to use and significantly reduce the time and number of iterations required to find solutions of highly nonlinear boundary value problems to any arbitrary level of accuracy. The accuracy and convergence properties of the proposed algorithms are tested numerically by solving three Falkner-Skan type boundary layer flow problems and comparing the results to the most accurate results currently available in the literature. We show, for instance, that precision of up to 29 significant figures can be attained with no more than 5 iterations of each algorithm.

scholarly journals A New Spectral Local Linearization Method for Nonlinear Boundary Layer Flow Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
S. S. Motsa
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Nonlinear Boundary ◽  
Linearization Method ◽  
Variable Boundary ◽  
Flow Problems ◽  
Local Linearization ◽  
Highly Nonlinear ◽  
Layer Flow ◽  
Local Linearization Method

We propose a simple and efficient method for solving highly nonlinear systems of boundary layer flow problems with exponentially decaying profiles. The algorithm of the proposed method is based on an innovative idea of linearizing and decoupling the governing systems of equations and reducing them into a sequence of subsystems of differential equations which are solved using spectral collocation methods. The applicability of the proposed method, hereinafter referred to as the spectral local linearization method (SLLM), is tested on some well-known boundary layer flow equations. The numerical results presented in this investigation indicate that the proposed method, despite being easy to develop and numerically implement, is very robust in that it converges rapidly to yield accurate results and is more efficient in solving very large systems of nonlinear boundary value problems of the similarity variable boundary layer type. The accuracy and numerical stability of the SLLM can further be improved by using successive overrelaxation techniques.

scholarly journals A Spectral Relaxation Approach for Unsteady Boundary-Layer Flow and Heat Transfer of a Nanofluid over a Permeable Stretching/Shrinking Sheet

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
S. S. Motsa ◽  
P. Sibanda ◽  
J. M. Ngnotchouye ◽  
G. T. Marewo
Keyword(s):  
Numerical Algorithms ◽  
Relaxation Method ◽  
Coupled Systems ◽  
Shrinking Sheet ◽  
Nonlinear Boundary Value Problems ◽  
Flow And Heat Transfer ◽  
Highly Nonlinear ◽  
Spectral Relaxation Method ◽  
Layer Flow ◽  
Spectral Relaxation

This paper introduces two novel numerical algorithms for the efficient solution of coupled systems of nonlinear boundary value problems. The methods are benchmarked against existing methods by finding dual solutions of the highly nonlinear system of equations that model the flow of a viscoelastic liquid of Oldroyd-B type in a channel of infinite extent. The methods discussed here are the spectral relaxation method and spectral quasi-linearisation method. To verify the accuracy and efficiency of the proposed methods a comparative evaluation of the performance of the methods against established numerical techniques is given.

scholarly journals Mass Transpiration in Nonlinear MHD Flow Due to Porous Stretching Sheet

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Jitender Singh ◽  
U. S. Mahabaleshwar ◽  
Gabriella Bognár
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Boundary Layer Flow ◽  
Stretching Sheet ◽  
Flow Characteristics ◽  
Nonlinear Flow ◽  
Numerical Work ◽  
Highly Nonlinear ◽  
Homotopy Analysis ◽  
Layer Flow

AbstractMotivated from numerous practical applications, the present theoretical and numerical work investigates the nonlinear magnetohydrodynamic (MHD) laminar boundary layer flow of an incompressible, viscous fluid over a porous stretching sheet in the presence of suction/injection (mass transpiration). The flow characteristics are obtained by solving the underlying highly nonlinear ordinary differential equation using homotopy analysis method. The effect of parameters corresponding to suction/injection (mass transpiration), applied magnetic field, and porous stretching sheet parameters on the nonlinear flow is investigated. The asymptotic limits of the parameters regarding the flow characteristics are obtained mathematically, which compare very well with those obtained using the homotopy analysis technique. A detailed numerical study of the laminar boundary layer flow in the vicinity of the porous stretching sheet in MHD and offers a particular choice of the parametric values to be taken in order to practically model a particular type of the event among suction and injection at the sheet surface.

On Spectral-Homotopy Perturbation Method Solution of Nonlinear Differential Equations in Bounded Domains

2013 ◽  
Vol 1 (1) ◽  
pp. 25-37
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Iteration Algorithm ◽  
Test Problems ◽  
Nonlinear Boundary Value Problems ◽  
Homotopy Perturbation ◽  
Spectral Technique ◽  
Homotopy Analysis

In this study, a combination of the hybrid Chebyshev spectral technique and the homotopy perturbation method is used to construct an iteration algorithm for solving nonlinear boundary value problems. Test problems are solved in order to demonstrate the efficiency, accuracy and reliability of the new technique and comparisons are made between the obtained results and exact solutions. The results demonstrate that the new spectral homotopy perturbation method is more efficient and converges faster than the standard homotopy analysis method. The methodology presented in the work is useful for solving the BVPs consisting of more than one differential equation in bounded domains. 

scholarly journals Modelling Study on Internal Energy Loss Due to Entropy Generation for Non-Darcy Poiseuille Flow of Silver-Water Nanofluid: An Application of Purification

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 851 ◽  
Author(s):  
Nasir Shehzad ◽  
Ahmed Zeeshan ◽  
Rahmat Ellahi ◽  
Saman Rashidi
Keyword(s):  
Porous Media ◽  
Internal Energy ◽  
Entropy Generation ◽  
Poiseuille Flow ◽  
Volume Fraction ◽  
Low Cost ◽  
Theoretical Estimate ◽  
Industrial Process ◽  
Highly Nonlinear ◽  
Homotopy Analysis

In this paper, an analytical study of internal energy losses for the non-Darcy Poiseuille flow of silver-water nanofluid due to entropy generation in porous media is investigated. Spherical-shaped silver (Ag) nanosize particles with volume fraction 0.3%, 0.6%, and 0.9% are utilized. Four illustrative models are considered: (i) heat transfer irreversibility (HTI), (ii) fluid friction irreversibility (FFI), (iii) Joule dissipation irreversibility (JDI), and (iv) non-Darcy porous media irreversibility (NDI). The governing equations of continuity, momentum, energy, and entropy generation are simplified by taking long wavelength approximations on the channel walls. The results represent highly nonlinear coupled ordinary differential equations that are solved analytically with the help of the homotopy analysis method. It is shown that for minimum and maximum averaged entropy generation, 0.3% by vol and 0.9% by vol of nanoparticles, respectively, are observed. Also, a rise in entropy is evident due to an increase in pressure gradient. The current analysis provides an adequate theoretical estimate for low-cost purification of drinking water by silver nanoparticles in an industrial process.

scholarly journals Homotopy Analysis of Carreau Fluid Flow Over a Stretching Cylinder

2021 ◽  
Vol 88 (2) ◽  
pp. 80-92
Author(s):  
Lim Yeou Jiann ◽  
Sharidan Shafie ◽  
Ahmad Qushairi Mohamad ◽  
Noraihan Afiqah Rawi
Keyword(s):  
Nonlinear Differential Equations ◽  
Velocity Profiles ◽  
Weissenberg Number ◽  
Stretching Cylinder ◽  
Series Solutions ◽  
Carreau Fluid ◽  
Keller Box Method ◽  
Highly Nonlinear ◽  
Homotopy Analysis ◽  
Curvature Parameter

Carreau fluid flows past a stretching cylinder is elucidated in the present study. The transformed self-similarity and dimensionless boundary layer equations are solved by using the Homotopy analysis method. A convergence study of the method is illustrated explicitly. Series solutions of the highly nonlinear differential equations are computed and it is very efficient in demonstrating the characteristic of the Carreau fluid. Validation of the series solutions is achieved via comparing with earlier published results. Those results are obtained by using the Keller-Box method. The effects of the Weissenberg number and curvature parameter on the velocity profiles are discussed by graphs and tabular. The velocity curves have shown different behavior in and for an increase of the Weissenberg number. Further, the curvature parameter K does increase the velocity profiles.

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