Extension of the Homotopy Perturbation Method for Solving Nonlinear Differential-Difference Equations

2010 ◽  
Vol 65 (12) ◽  
pp. 1060-1064 ◽  
Author(s):  
Mohamed Medhat Mousa ◽  
Aidarkan Kaltayev ◽  
Hasan Bulut
Keyword(s):  
Perturbation Method ◽  
Exact Solutions ◽  
Difference Equations ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Lattice Equation ◽  
Convenient Tool ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
De Vries

In this paper, we have extended the homotopy perturbation method (HPM) to find approximate analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Korteweg-de Vries (mKdV) lattice equation and the discretized nonlinear Schr¨odinger equation are taken as examples to demonstrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons are made between the results of the presented method and exact solutions. The obtained results reveal that the HPM is a very effective and convenient tool for solving such kind of equations.

Homotopy Perturbation Method for Solving Nonlinear Differential- Difference Equations

2010 ◽  
Vol 65 (6-7) ◽  
pp. 511-517 ◽  
Author(s):  
Mohamed M. Mousa ◽  
Aidarkhan Kaltayev
Keyword(s):  
Schrödinger Equation ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Difference Equations ◽  
Analytical Solutions ◽  
Schrodinger Equation ◽  
Homotopy Perturbation Method ◽  
Nonlinear Schrodinger Equation ◽  
Lattice Equation ◽  
Homotopy Perturbation

In this paper, the homotopy perturbation method (HPM) is extended to obtain analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Kortewegde Vries (mKdV) lattice equation and the discretized nonlinear Schrodinger equation are taken as examples to illustrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons between the results of the presented method and exact solutions are made. The results reveal that the HPM is very effective and convenient for solving such kind of equations.

scholarly journals Hybridization of homotopy perturbation method and Laplace transformation for the partial differential equations

2017 ◽  
Vol 21 (4) ◽  
pp. 1843-1846 ◽  
Author(s):  
Zhen-Jiang Liu ◽  
Magaji Adamu ◽  
Enoch Suleiman ◽  
Ji-Huan He
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Analytical Solutions ◽  
Laplace Transformation ◽  
Homotopy Perturbation Method ◽  
Solution Process ◽  
Initial Approximation ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
Linear Differential

Homotopy perturbation method is combined with Laplace transformation to obtain approximate analytical solutions of non-linear differential equations. An example is given to elucidate the solution process and confirm reliability of the method. The result indicates superiority of the method over the conventional homotopy perturbation method due its flexibility in choosing its initial approximation.

scholarly journals Approximate Analytical Solutions of the Regularized Long Wave Equation Using the Optimal Homotopy Perturbation Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Perturbation Method ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equations ◽  
Accelerated Convergence ◽  
Homotopy Perturbation ◽  
Long Wave ◽  
Approximate Analytical Solutions ◽  
Regularized Long Wave Equation ◽  
Regular Homotopy

The paper presents the optimal homotopy perturbation method, which is a new method to find approximate analytical solutions for nonlinear partial differential equations. Based on the well-known homotopy perturbation method, the optimal homotopy perturbation method presents an accelerated convergence compared to the regular homotopy perturbation method. The applications presented emphasize the high accuracy of the method by means of a comparison with previous results.

scholarly journals Application of Homotopy Perturbation Method with an Auxiliary Term for Nonlinear Dropping Equations Arisen in Polymer Packaging System

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Xiang Hong ◽  
Jun Wang ◽  
Li-xin Lu
Keyword(s):  
Numerical Simulation ◽  
Perturbation Method ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Equation Of Motion ◽  
High Accuracy ◽  
Second Order ◽  
Packaging System ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions

The homotopy perturbation method (HPM) with an auxiliary term was applied to obtain approximate analytical solutions of polymer cushioning packaging system. The second-order solution of the equation of motion was obtained and compared with the numerical simulation solution solved by the Runge-Kutta algorithm. The results showed the high accuracy of this modified HPM with convenient calculation.

scholarly journals Application of He’s Homotopy Perturbation Method to Fractional Diffusion Equations

2010 ◽  
Vol 65 (1-2) ◽  
pp. 53-58 ◽  
Author(s):  
Subir Das ◽  
Praveen Kumar Gupta ◽  
Vinod Sankar Pandey ◽  
Kabindra Nath Rai
Keyword(s):  
Perturbation Method ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Time Derivative ◽  
Fractional Diffusion Equations ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
And Performance ◽  
Fractional Time Derivative

AbstractIn this paper, the approximate analytical solutions of a general diffusion equation with fractional time derivative in the presence of a linear external force are obtained with the help of the homotopy perturbation method (HPM). The explicit solutions of the problem for the initial condition as a function of x have been obtained. It reveals that a few iterations are needed to obtain accurate approximate analytical solutions. The numerical calculations are carried out when the initial conditions are like exponential and periodic functions and the results are depicted through graphs. The examples prove that the method is extremely effective due to its simplistic approach and performance.

scholarly journals Homotopy Perturbation Method for Solving Wave-Like Nonlinear Equations with Initial-Boundary Conditions

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Afgan Aslanov
Keyword(s):  
Boundary Conditions ◽  
Perturbation Method ◽  
Nonlinear Equations ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Initial Boundary ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions

The homotopy perturbation method is employed to obtain approximate analytical solutions of the wave-like nonlinear equations with initial-boundary conditions. An efficient way of choosing the auxiliary operator is presented. The results demonstrate reliability and efficiency of the method.

Sign in / Sign up

Export Citation Format

Share Document