scholarly journals A Note on Solutions of the SIR Models of Epidemics Using HAM

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
M. Sajid ◽  
Z. Abbas ◽  
N. Ali ◽  
T. Javed
Keyword(s):  
Excellent Agreement ◽  
Homotopy Analysis Method ◽  
Numerical Results ◽  
Analysis Method ◽  
New Approach ◽  
Entire Domain ◽  
Ham Solution ◽  
Homotopy Analysis

Recently, Awawdeh et al. (2009) discussed the solutions of SIR epidemics model using homotopy analysis method. This comment points out some crucial flaws in (Awawdeh et al. 2009). Particularly, results presented in Figure 1 of the (Awawdeh et al. 2009) do not represent the 20 term solution of the considered problem as stated. The present paper also provides a new approach for solving SIR epidemics model using homotopy analysis method. The new approach is based on dividing the entire domain into subintervals. In each subinterval the three-term HAM solution is sufficient for obtaining accurate and convergent results. The comparison of the obtained solution using new approach is made with the numerical results and found in excellent agreement.

scholarly journals Homotopy Analysis Decomposition Method for the Solution of Viscous Boundary Layer Flow Due to a Moving Sheet

2019 ◽  
pp. 1-7
Author(s):  
S. Alao ◽  
R. A. Oderinu ◽  
F. O. Akinpelu ◽  
E. I. Akinola
Keyword(s):  
Boundary Layer ◽  
Homotopy Analysis Method ◽  
Decomposition Method ◽  
Analysis Method ◽  
Viscous Boundary Layer ◽  
New Approach ◽  
Homotopy Analysis ◽  
Layer Flow ◽  
Viscous Boundary ◽  
Adomian Polynomial

This paper investigates a new approach called Homotopy Analysis Decomposition Method (HADM) for solving nonlinear differential equations, the method was developed by incorporating Adomian polynomial into Homotopy Analysis Method. The Adomian polynomial was used to decompose the nonlinear term in the equation then apply the scheme of homotopy analysis method. The accuracy and efficiency of the proposed method was validated by considering algebraically decaying viscous boundary layer  flow due to a moving sheet. Diagonal Pade approximation was used to get the skin friction. The obtained results were presented along with other methods in the literature in tabular form to show the computational efficiency of the new approach. The results were found to agree with those in literature. Owing to its small size of computation, the method is not aected by discretization error as the results are presented in form of polynomials.

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 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.

A NEW APPROACH IN THE SEARCH FOR PERIODIC ORBITS

2013 ◽  
Vol 23 (11) ◽  
pp. 1350186 ◽  
Author(s):  
ROMINA COBIAGA ◽  
WALTER REARTES
Keyword(s):  
Dynamical Systems ◽  
Linear Operator ◽  
Periodic Orbits ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Initial Conditions ◽  
Formal Series ◽  
Analysis Method ◽  
New Approach ◽  
Homotopy Analysis

In this paper, we present a new approach for finding periodic orbits in dynamical systems modeled by differential equations. It is based on the Homotopy Analysis Method (HAM) but it differs from the usual way it is applied. We apply HAM to construct approximations of a formal series solution of the equation. These approximations exist for any value of the frequency. They can be obtained by choosing a suitable linear operator and allowing the initial conditions to vary freely. Then we study the behavior of the obtained expressions as a function of the frequency. This procedure allows us to find those values of the frequency for which the series converges and therefore to find the periodic orbits. We show several examples of application of the proposed method. Mathematica and MATCONT were applied for all the calculations.

Homotopy analysis method for solving Abel differential equation of fractional order

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Hossein Jafari ◽  
Khosro Sayevand ◽  
Haleh Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Numerical Results ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Abel Differential Equation

AbstractIn this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented.

The Series Solution of Problems in the Calculus of Variations via the Homotopy Analysis Method

2009 ◽  
Vol 64 (1-2) ◽  
pp. 30-36 ◽  
Author(s):  
Saeid Abbasbandy ◽  
Ahmand Shirzadi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Calculus Of Variations ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Numerical Results ◽  
Analysis Method ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is used for solving the ordinary differential equations which arise from problems of the calculus of variations. Some numerical results are given to demonstrate the validity and applicability of the presented technique. The method is very effective and yields very accurate results.

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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