APPROXIMATE SOLUTIONS OF NONLINEAR DIFFERENTIAL DIFFERENCE EQUATIONS

2009 ◽  
Vol 06 (04) ◽  
pp. 569-583 ◽  
Author(s):  
M. A. ABDOU
Keyword(s):  
Exact Solutions ◽  
Difference Equations ◽  
Homotopy Perturbation Method ◽  
Toda Lattice ◽  
Approximate Solutions ◽  
Mathematical Physics ◽  
Nonlinear Difference Equations ◽  
Science And Engineering ◽  
Homotopy Perturbation ◽  
Relativistic Toda Lattice

An extend of He's homotopy perturbation method (HPM) is used for finding a new approximate and exact solutions of nonlinear difference differential equations arising in mathematical physics. To illustrate the effectiveness and the advantage of the proposed method, two models of nonlinear difference equations of special interest in physics are chosen, namely, Ablowitz–Ladik lattice equations and Relativistic Toda lattice difference equations. Comparisons are made between the results of the proposed method and exact solutions. The results show that the HPM is a attracted method in solving the differential difference equations (DDEs). The proposed method will become a much more interesting method for solving nonlinear DDEs in science and engineering.

scholarly journals Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Khaled A. Gepreel ◽  
Taher A. Nofal ◽  
Fawziah M. Alotaibi
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Difference Equations ◽  
Toda Lattice ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Nonlinear Schrödinger ◽  
Saturable Nonlinearity ◽  
Relativistic Toda Lattice

We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.

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.

Application of homotopy perturbation method to solve two models of delay differential equation systems

2017 ◽  
Vol 10 (06) ◽  
pp. 1750080 ◽  
Author(s):  
Şuayip Yüzbaşi ◽  
Murat Karaçayir
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Lorenz System ◽  
Approximate Solutions ◽  
Mathematical Physics ◽  
Differential Systems ◽  
Homotopy Perturbation ◽  
Delay Differential Systems ◽  
Delay Differential

In this paper, two delay differential systems are considered, namely, a famous model from mathematical biology about the spread of HIV viruses in blood and the advanced Lorenz system from mathematical physics. We then apply the homotopy perturbation method (HPM) to find their approximate solutions. It turns out that the method gives rise to easily obtainable solutions. In addition, residual error functions of the solutions are graphed and it is shown that increasing the parameter [Formula: see text] in the method improves the results in both cases.

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 An efficient approach for solution of fractional-order Helmholtz equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nehad Ali Shah ◽  
Essam R. El-Zahar ◽  
Mona D. Aljoufi ◽  
Jae Dong Chung
Keyword(s):  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Hybrid Technique ◽  
Science And Engineering ◽  
Helmholtz Equations ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Suggested Technique ◽  
Present Technique

AbstractIn this article, a hybrid technique called the homotopy perturbation Elzaki transform method has been implemented to solve fractional-order Helmholtz equations. In the hybrid technique, the Elzaki transform method and the homotopy perturbation method are amalgamated. Three problems are solved to validate and demonstrate the efficacy of the present technique. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the results by other techniques. It is shown that the proposed method is efficient, reliable and easy to implement for various related problems of science and engineering.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

scholarly journals Homotopy Perturbation Method

2017 ◽  
pp. 24-61
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Homotopy Perturbation ◽  
Wide Range ◽  
Transfer Problems

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.

Application of He’s Homotopy Perturbation Method for Solving Fractional Fokker-Planck Equations

2009 ◽  
Vol 64 (12) ◽  
pp. 788-794 ◽  
Author(s):  
Mohamed M. Mousa ◽  
Aidarkhan Kaltayev
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Planck Equation ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Homotopy Perturbation ◽  
External Force Field ◽  
Promising Tool ◽  
Transport Problems ◽  
Fokker Planck

Abstract The fractional Fokker-Planck equation (FFPE) has been used in many physical transport problems which take place under the influence of an external force field and other important applications in various areas of engineering and physics. In this paper, by means of the homotopy perturbation method (HPM), exact and approximate solutions are obtained for two classes of the FFPE initial value problems. The method gives an analytic solution in the form of a convergent series with easily computed components. The obtained results show that the HPM is easy to implement, accurate and reliable for solving FFPEs. The method introduces a promising tool for solving other types of differential equation with fractional order derivatives

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