scholarly journals Series Solution of the Multispecies Lotka-Volterra Equations by Means of the Homotopy Analysis Method

2008 ◽  
Vol 2008 ◽  
pp. 1-14 ◽  
Author(s):  
A. Sami Bataineh ◽  
M. S. M. Noorani ◽  
I. Hashim
Keyword(s):  
Homotopy Analysis Method ◽  
Continuous Solution ◽  
Three Dimensional ◽  
Volterra Equations ◽  
Analysis Method ◽  
Runge Kutta Method ◽  
Homotopy Analysis ◽  
Base Functions ◽  
Comparable Accuracy ◽  
Straightforward Approach

The time evolution of the multispecies Lotka-Volterra system is investigated by the homotopy analysis method (HAM). The continuous solution for the nonlinear system is given, which provides a convenient and straightforward approach to calculate the dynamics of the system. The HAM continuous solution generated by polynomial base functions is of comparable accuracy to the purely numerical fourth-order Runge-Kutta method. The convergence theorem for the three-dimensional case is also given.

scholarly journals Direct Solution ofnth-Order IVPs by Homotopy Analysis Method

2009 ◽  
Vol 2009 ◽  
pp. 1-15 ◽  
Author(s):  
A. Sami Bataineh ◽  
M. S. M. Noorani ◽  
I. Hashim
Keyword(s):  
Homotopy Analysis Method ◽  
Direct Solution ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Initial Value ◽  
Convergence Region ◽  
Runge Kutta Method ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
Comparable Accuracy

Direct solution of a class ofnth-order initial value problems (IVPs) is considered based on the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge-Kutta method (RK78).

Analytic Solution of Three-Dimensional Viscous Flow and Heat Transfer Over a Stretching Flat Surface by Homotopy Analysis Method

2008 ◽  
Vol 130 (12) ◽  
Author(s):  
Ahmer Mehmood ◽  
Asif Ali
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Analytic Solution ◽  
Porous Plate ◽  
Three Dimensional ◽  
Parallel Plates ◽  
Analysis Method ◽  
Electrically Conducting ◽  
Homotopy Analysis

In this paper heat transfer in an electrically conducting fluid bonded by two parallel plates is studied in the presence of viscous dissipation. The plates and the fluid rotate with constant angular velocity about a same axis of rotation where the lower plate is a stretching sheet and the upper plate is a porous plate subject to constant injection. The governing partial differential equations are transformed to a system of ordinary differential equations with the help of similarity transformation. Homotopy analysis method is used to get complete analytic solution for velocity and temperature profiles. The effects of different parameters are discussed through graphs.

Study on a Multi-Frequency Homotopy Analysis Method for Period-Doubling Solutions of Nonlinear Systems

2018 ◽  
Vol 28 (04) ◽  
pp. 1850049 ◽  
Author(s):  
H. X. Fu ◽  
Y. H. Qian
Keyword(s):  
Periodic Solutions ◽  
Homotopy Analysis Method ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Algebraic Equations ◽  
Analysis Method ◽  
Duffing System ◽  
Runge Kutta Method ◽  
Homotopy Analysis ◽  
Two Degree Of Freedom

In this paper, a modification of homotopy analysis method (HAM) is applied to study the two-degree-of-freedom coupled Duffing system. Firstly, the process of calculating the two-degree-of-freedom coupled Duffing system is presented. Secondly, the single periodic solutions and double periodic solutions are obtained by solving the constructed nonlinear algebraic equations. Finally, comparing the periodic solutions obtained by the multi-frequency homotopy analysis method (MFHAM) and the fourth-order Runge–Kutta method, it is found that the approximate solution agrees well with the numerical solution.

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.

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