HOMOTOPY ANALYSIS METHOD TO SOLVE BOUSSINESQ EQUATIONS

2015 ◽  
Vol 10 (3) ◽  
pp. 2825-2833
Author(s):  
Achala Nargund ◽  
R Madhusudhan ◽  
S B Sathyanarayana
Keyword(s):  
Homotopy Analysis Method ◽  
Degrees Of Freedom ◽  
Initial Approximation ◽  
Boussinesq Equations ◽  
Convergent Series ◽  
Boussinesq System ◽  
Analysis Method ◽  
Analytical Technique ◽  
Homotopy Analysis ◽  
Region Of Convergence

In this paper, Homotopy analysis method is applied to the nonlinear coupleddifferential equations of classical Boussinesq system. We have applied Homotopy analysis method (HAM) for the application problems in [1, 2, 3, 4]. We have also plotted Domb-Sykes plot for the region of convergence. We have applied Pade for the HAM series to identify the singularity and reflect it in the graph. The HAM is a analytical technique which is used to solve non-linear problems to generate a convergent series. HAM gives complete freedom to choose the initial approximation of the solution, it is the auxiliary parameter h which gives us a convenient way to guarantee the convergence of homotopy series solution. It seems that moreartificial degrees of freedom implies larger possibility to gain better approximations by HAM.

Predictor homotopy analysis method for nanofluid flow through expanding or contracting gaps with permeable walls

2015 ◽  
Vol 08 (04) ◽  
pp. 1550050 ◽  
Author(s):  
Navid Freidoonimehr ◽  
Behnam Rostami ◽  
Mohammad Mehdi Rashidi
Keyword(s):  
Homotopy Analysis Method ◽  
Volume Fraction ◽  
Expansion Ratio ◽  
Analysis Method ◽  
Nanofluid Flow ◽  
Analytical Technique ◽  
Study Results ◽  
Permeable Walls ◽  
Homotopy Analysis ◽  
Flow Through

In this paper a definitely new analytical technique, predictor homotopy analysis method (PHAM), is employed to solve the problem of two-dimensional nanofluid flow through expanding or contracting gaps with permeable walls. Moreover, comparison of the PHAM results with numerical results obtained by the shooting method coupled with a Runge–Kutta integration method as well as previously published study results demonstrates high accuracy for this technique. The fluid in the channel is water containing different nanoparticles: silver, copper, copper oxide, titanium oxide, and aluminum oxide. The effects of the nanoparticle volume fraction, Reynolds number, wall expansion ratio, and different types of nanoparticles on the flow are discussed.

scholarly journals Numerical Solution of Higher Order Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Shahid S. Siddiqi ◽  
Muzammal Iftikhar
Keyword(s):  
Boundary Value Problems ◽  
Homotopy Analysis Method ◽  
Present Method ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Higher Order ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Linear And Nonlinear

The aim of this paper is to use the homotopy analysis method (HAM), an approximating technique for solving linear and nonlinear higher order boundary value problems. Using HAM, approximate solutions of seventh-, eighth-, and tenth-order boundary value problems are developed. This approach provides the solution in terms of a convergent series. Approximate results are given for several examples to illustrate the implementation and accuracy of the method. The results obtained from this method are compared with the exact solutions and other methods (Akram and Rehman (2013), Farajeyan and Maleki (2012), Geng and Li (2009), Golbabai and Javidi (2007), He (2007), Inc and Evans (2004), Lamnii et al. (2008), Siddiqi and Akram (2007), Siddiqi et al. (2012), Siddiqi et al. (2009), Siddiqi and Iftikhar (2013), Siddiqi and Twizell (1996), Siddiqi and Twizell (1998), Torvattanabun and Koonprasert (2010), and Kasi Viswanadham and Raju (2012)) revealing that the present method is more accurate.

scholarly journals Homotopy Analysis Method for Solving Foam Drainage Equation with Space- and Time-Fractional Derivatives

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Hadi Hosseini Fadravi ◽  
Hassan Saberi Nik ◽  
Reza Buzhabadi
Keyword(s):  
Analytical Solution ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Convergent Series ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Foam Drainage Equation ◽  
Foam Drainage

The analytical solution of the foam drainage equation with time- and space-fractional derivatives was derived by means of the homotopy analysis method (HAM). The fractional derivatives are described in the Caputo sense. Some examples are given and comparisons are made; the comparisons show that the homotopy analysis method is very effective and convenient. By choosing different values of the parameters in general formal numerical solutions, as a result, a very rapidly convergent series solution is obtained.

scholarly journals Homotopy Analysis Method for Second-Order Boundary Value Problems of Integrodifferential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Ahmad El-Ajou ◽  
Omar Abu Arqub ◽  
Shaher Momani
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Convergent Series ◽  
Second Order ◽  
Integrodifferential Equations ◽  
Analysis Method ◽  
Convergence Region ◽  
New Approach ◽  
Homotopy Analysis ◽  
And Control

In this paper, series solution of second-order integrodifferential equations with boundary conditions of the Fredholm and Volterra types by means of the homotopy analysis method is considered. The new approach provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. The homotopy analysis method provides us with a simple way to adjust and control the convergence region of the infinite series solution by introducing an auxiliary parameter. The proposed technique is applied to a few test examples to illustrate the accuracy, efficiency, and applicability of the method. The results reveal that the method is very effective, straightforward, and simple.

Application of Homotopy Analysis Method in Nonlinear Oscillations

1998 ◽  
Vol 65 (4) ◽  
pp. 914-922 ◽  
Author(s):  
Shi-Jun Liao ◽  
A. T. Chwang
Keyword(s):  
Homotopy Analysis Method ◽  
Nonlinear Oscillations ◽  
Nonlinear Problems ◽  
Analysis Method ◽  
General Validity ◽  
Single Degree Of Freedom ◽  
Analytical Technique ◽  
Single Degree ◽  
Homotopy Analysis ◽  
Odd Nonlinearity

In this paper, we apply a new analytical technique for nonlinear problems, namely the Homotopy Analysis Method (Liao 1992a), to give two-period formulas for oscillations of conservative single-degree-of-freedom systems with odd nonlinearity. These two formulas are uniformly valid for any possible amplitudes of oscillation. Four examples are given to illustrate the validity of the two formulas. This paper also demonstrates the general validity and the great potential of the Homotopy Analysis Method.

scholarly journals Solving Famous Nonlinear Coupled Equations with Parameters Derivative by Homotopy Analysis Method

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Sohrab Effati ◽  
Hassan Saberi Nik ◽  
Reza Buzhabadi
Keyword(s):  
Homotopy Analysis Method ◽  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Reaction Diffusion Equations ◽  
Analysis Method ◽  
Coupled Equations ◽  
Homotopy Analysis ◽  
Special Case

The homotopy analysis method (HAM) is employed to obtain symbolic approximate solutions for nonlinear coupled equations with parameters derivative. These nonlinear coupled equations with parameters derivative contain many important mathematical physics equations and reaction diffusion equations. By choosing different values of the parameters in general formal numerical solutions, as a result, a very rapidly convergent series solution is obtained. The efficiency and accuracy of the method are verified by using two famous examples: coupled Burgers and mKdV equations. The obtained results show that the homotopy perturbation method is a special case of homotopy analysis method.

scholarly journals Analytic Solution of Multipantograph Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Fadi Awawdeh ◽  
Ahmad Adawi ◽  
Safwan Al-Shara'
Keyword(s):  
Homotopy Analysis Method ◽  
Analytic Solution ◽  
Convergent Series ◽  
Analysis Method ◽  
Analytical Results ◽  
Homotopy Analysis

We apply the homotopy analysis method (HAM) for solving the multipantograph equation. The analytical results have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the homotopy analysis method. Comparisons are made to confirm the reliability of the homotopy analysis method.

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