scholarly journals On the Optimal Auxiliary Linear Operator for the Spectral Homotopy Analysis Method Solution of Nonlinear Ordinary Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
S. S. Motsa
Keyword(s):  
Linear Operator ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Homotopy Analysis Method ◽  
Linear Operators ◽  
Nonlinear Ordinary Differential Equations ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Auxiliary Linear Operator ◽  
Spectral Homotopy Analysis Method

The purpose of this study is to identify the auxiliary linear operator that gives the best convergence and accuracy in the implementation of the spectral homotopy analysis method (SHAM) in the solution of nonlinear ordinary differential equations. The auxiliary linear operator is an essential element of the homotopy analysis method (HAM) algorithm that strongly influences the convergence of the method. In this work we introduce new procedures of defining the auxiliary linear operators and compare solutions generated using the new linear operators with solutions obtained using well-known linear operators. The applicability and validity of the proposed linear operators is tested on four highly nonlinear ordinary differential equations with fluid mechanics applications that have recently been reported in the literature. The results from the study reveal that the new linear operators give better results than the previously used linear operators. The identification of the optimal linear operator will direct future research on further applications of HAM-based methods in solving complicated nonlinear differential equations.

scholarly journals Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Z. Pashazadeh Atabakan ◽  
A. Kazemi Nasab ◽  
A. Kılıçman ◽  
Zainidin K. Eshkuvatov
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Solution ◽  
Homotopy Analysis Method ◽  
Existence And Uniqueness ◽  
Lagrange Interpolation ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Uniqueness Of The Solution ◽  
Spectral Homotopy Analysis Method

Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions.

scholarly journals The Consequences of Thermal Radiation and Chemical Reactions on Magneto-hydrodynamics in Two Dimensions Over a Stretching Sheet with Jeffrey Fluid.

2021 ◽  
Author(s):  
Vijay Patel ◽  
Jigisha Pandya
Keyword(s):  
Differential Equations ◽  
Thermal Radiation ◽  
Ordinary Differential Equations ◽  
Homotopy Analysis Method ◽  
Stretching Sheet ◽  
Fluid Velocity ◽  
Jeffrey Fluid ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
The Impact

In this research paper, the Homotopy Analysis Method is used to investigate the twodimensional electrical conduction of a magneto-hydrodynamic (MHD) Jeffrey Fluid across a stretching sheet under various conditions, such as when electrical current and temperature are both present, and when heat is added in the presence of a chemical reaction or thermal radiation. Applying similarity transformation, the governing partial differential equation is transformed into terms of nonlinear coupled ordinary differential equations. The Homotopy Analysis Method is used to solve a system of ordinary differential equations. The impact of different numerical values on velocity, concentration, and temperature is examined and presented in tables and graphs. The fluid velocity reduces as the retardation time parameter(2) grows, while the fluid velocity inside the boundary layer increases as the Deborah number () increases. The velocity profiles decrease when the magnetic parameter M is increased. The results of this study are entirely compatible with those of a viscous fluid. The Homotopy Analysis Method calculations have been carried out on the PARAM Shavak high-performance computing (HPC) machine using the BVPh2.0 Mathematica tool.

scholarly journals On the Bivariate Spectral Homotopy Analysis Method Approach for Solving Nonlinear Evolution Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. S. Motsa
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Analysis Method ◽  
Partial Differential ◽  
New Approach ◽  
Homotopy Analysis ◽  
Spectral Homotopy Analysis Method

This paper presents a new application of the homotopy analysis method (HAM) for solving evolution equations described in terms of nonlinear partial differential equations (PDEs). The new approach, termed bivariate spectral homotopy analysis method (BISHAM), is based on the use of bivariate Lagrange interpolation in the so-called rule of solution expression of the HAM algorithm. The applicability of the new approach has been demonstrated by application on several examples of nonlinear evolution PDEs, namely, Fisher’s, Burgers-Fisher’s, Burger-Huxley’s, and Fitzhugh-Nagumo’s equations. Comparison with known exact results from literature has been used to confirm accuracy and effectiveness of the proposed method.

scholarly journals Implementation of Homotopy Analysis Method on circular permeable slider containing of incompressible Newtonian fluid

2016 ◽  
Vol 34 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Javad Rahimi ◽  
Mazaher Rahimi Esboee ◽  
Davod Domairy Ganji ◽  
Iman Rahimi-Petrodi ◽  
Reza Mohammadyari
Keyword(s):  
Differential Equations ◽  
Newtonian Fluid ◽  
Homotopy Analysis Method ◽  
Classical Problem ◽  
Momentum Equation ◽  
Velocity Fields ◽  
Nonlinear Ordinary Differential Equations ◽  
Analysis Method ◽  
Incompressible Newtonian Fluid ◽  
Homotopy Analysis

The aim of this paper is to examine the classical problem of an incompressible Newtonian fluid through the porous of a circular slider which is moving laterally on a horizontal plan. Employing the similarity variables, the governing differential equations have been reduced to ordinary ones and solved via Homotopy Analysis Method (HAM). The analytical solution for the coupled Nonlinear Ordinary Differential Equations resulting from the momentum equation is obtained and Velocity fields have been computed and discussed for different values of the Reynolds number. Also the fourth-order Runge-Kutta numerical method (NUM) is used for the validity of these analytical methods and excellent agreement are observed between the solutions obtained from HAM and numerical results.

scholarly journals The Spectral Homotopy Analysis Method Extended to Systems of Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
S. S. Motsa ◽  
F. G. Awad ◽  
Z. G. Makukula ◽  
P. Sibanda
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Stretching Sheet ◽  
Nonlinear Partial Differential Equations ◽  
Analysis Method ◽  
Partial Differential ◽  
Homotopy Analysis ◽  
Spectral Homotopy Analysis Method ◽  
Good Agreement

The spectral homotopy analysis method is extended to solutions of systems of nonlinear partial differential equations. The SHAM has previously been successfully used to find solutions of nonlinear ordinary differential equations. We solve the nonlinear system of partial differential equations that model the unsteady nonlinear convective flow caused by an impulsively stretching sheet. The numerical results generated using the spectral homotopy analysis method were compared with those found using the spectral quasilinearisation method (SQLM) and the two results were in good agreement.

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