A new HPM for ordinary differential equations

In this paper, we introduce a new version of the homotopy perturbation method (NHPM) that efficiently solves linear and non‐linear ordinary differential equations. Several examples, including Euler‐Lagrange, Bernoulli and Ricatti differential equations, are given to demonstrate the efficiency of the new method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Solving systems of ODEs by homotopy analysis method

This paper applies the homotopy analysis method (HAM) to systems of ordinary differential equations (ODEs). The systems investigated include stiff systems, the chaotic Genesio system and the matrix Riccati-type differential equation. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge–Kutta method (RK78).     

Solution of Davey-Stewartson equations by homotopy perturbation method

In this paper, we extend the homotopy perturbation method to solve the Davey-Stewartson equations. The homotopy perturbation method is employed to compute an approximation to the solution of the equations. Computation the absolute errors between the exact solutions of the Davey-Stewartson equations and the HPM solutions are presented. Some plots are given to show the simplicity the method. The article is published in the original.     

A new analytical modelling for fractional telegraph equation via Laplace transform

The main aim of the present work is to propose a new and simple algorithm for space-fractional telegraph equation, namely new fractional homotopy analysis transform method (HATM). The fractional homotopy analysis transform method is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method (HPTM). The beauty of the paper is error analysis which shows that our solution obtained by proposed method converges very rapidly to the known exact solution. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method.     

He's homotopy perturbation method for solving the shock wave equation

In this article, we discuss the analytic solution of the fully developed shock waves. The homotopy perturbation method is used to solve the shock wave equation, which describes the flow of gases. Unlike the various numerical techniques, which are usually valid for short period of time, the solution of the presented equation is analytic for 0 < t < ∞. The results presented converge very rapidly, indicating that the method is reliable and accurate.     

The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations

In this work, a combined form of the Laplace transform method with the Adomian decomposition method is developed for analytic treatment of the nonlinear Volterra integro-differential equations. The combined method is capable of handling both equations of the first and second kind. Illustrative examples will be examined to support the proposed analysis.     

Comparison between variational iteration method and homotopy perturbation method for linear and nonlinear partial differential equations with the nonhomogeneous initial conditions

In this study, linear and nonlinear partial differential equations with the nonhomogeneous initial conditions are considered. We used Variational iteration method (VIM) and Homotopy perturbation method (HPM) for solving these equations. Both methods are used to obtain analytic solutions for different types of differential equations. Four examples are presented to show the application of the present techniques. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Beyond sumudu transform and natural transform: J-transform properties and applications

In this paper, we introduce an efficient integral transform called the ${\mathbb J}$-transform which is a modification of the well-known Sumudu transform and the Natural transform for solving differential equations with real applications in applied physical sciences and engineering. The ${\mathbb J}$-transform is more advantageous than both the Sumudu transform and the Natural transform. Interestingly, our proposed ${\mathbb J}$-transform can be applied successfully to solve complex problems that are ordinarily beyond the scope of either Sumudu transform or Natural transform. As a proof of concept, we consider some classic examples and highlight the limitations of the previously reported integral transforms and lastly demonstrate the superiority of the proposed ${\mathbb J}$-transform in addressing those limitations.     

An analytical approximation for coupled viscous Burgers’ equation

In this paper, the combined Laplace transform and new homotopy perturbation methods (LTNHPM) is employed to obtain the closed form solutions of the coupled Burgers equation. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply.     

Application of adapted homotopy perturbation method for approximate solution of Henon‐Heiles system

We performed adapted homotopy perturbation method on the Henon‐Heiles system with the help of the symbolic computation of package Maple 10 (User Manual by Maplesoft. www.maplesoft.com ). We obtained a new approximate solution of the Henon‐Heiles system. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Exact solutions for nonlinear Burgers' equation by homotopy perturbation method

The aim of this article is to construct a new efficient recurrent relation to solve nonlinear Burgers' equation. The homotopy perturbation method is used to solve this equation. Because Burgers' equation arises in many applications, it is worth trying new solution methods. Comparison of the results with those of Adomian's decomposition method leads to significant consequences. Four standard problems are used to illustrate the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The homotopy analysis method for explicit analytical solutions of Jaulent–Miodek equations

In this work, the homotopy analysis method (HAM) is applied to obtain the explicit analytical solutions for system of the Jaulent–Miodek equations. The validity of the method is verified by comparing the approximation series solutions with the exact solutions. Unlike perturbation methods, the HAM does not depend on any small physical parameters at all. Thus, it is valid for both weakly and strongly nonlinear problems. Besides, different from all other analytic techniques, the HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter ?. Briefly speaking, this work verifies the validity and the potential of the HAM for the study of nonlinear systems. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

伴有边界摄动的Volterra型积分微分方程组的奇摄动

讨论了伴有边界摄动的二阶非线性Volterra型积分微分方程组的奇摄动.在适当的条件下,利用对角化技巧证明了解的存在性,构造出解的渐近展开式并给出余项的一致有效估计.     

A new technique of using homotopy analysis method for solving high‐order nonlinear differential equations

In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high‐order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth‐order nonlinear differential equation to a system of n first‐order equations. Second‐ and third‐ order problems are solved as illustration examples of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

An efficient numerical method for solving coupled Burgers' equation by combining homotopy perturbation and pade techniques

In this study, we combined homotopy perturbation and Pade techniques for solving homogeneous and inhomogeneous two‐dimensional parabolic equation. Also, we apply our combined method for coupled Burgers' equations. The numerical results demonstrate that our combined method gives the approximate solution with faster convergence rate and higher accuracy than using the classic homotopy perturbation method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 982–995, 2011     

The travelling wave solutions for non-linear initial-value problems using the homotopy perturbation method

In this article, we have used the homotopy perturbation method (HPM) to find the travelling wave solutions for some non-linear initial-value problems in the mathematical physics. These problems consist of the Burgers–Fisher equation, the Kuramoto–Sivashinsky equation, the coupled Schordinger KdV equations and the long–short wave resonance equations together with initial conditions. The results of these problems reveal that the HPM is very powerful, effective, convenient and quite accurate to the systems of non-linear equations. It is predicted that this method can be found widely applicable in engineering and physics.     

Application of He’s homotopy perturbation method for nth-order integro-differential equations

In this paper, He’s homotopy perturbation method is proposed to solve nth-order integro-differential equations. The results reveal that the method is very effective and simple.     

Rational Runge-Kutta methods are not suitable for stiff systems of ODEs

The present paper shows that rational RK-methods are not very appropriate to solve stiff differential equations. The CA₀-stability (i.e. componentwise contractivity) is defined and the non-existence of CA₀-stable rational RK-methods is demonstrated. Furthermore it is shown that the stepsizes which can be expected when solving a stiff differential system with a rational or with an explicit linear RK-method are of the same order of magnitude.