Asymptotic expansions for the Sturm-Liouville problem by homotopy perturbation method

The asymptotic formulae for the eigenvalues and eigenfunctions of Sturm-Liouville problem with the Dirichlet boundary conditions when the potential is square integrable on [0, 1] are obtained by using homotopy perturbation method.     

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.     

Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation

In this paper, the homotopy analysis method (HAM) proposed by Liao in 1992 and the homotopy perturbation method (HPM) proposed by He in 1998 are compared through an evolution equation used as the second example in a recent paper by Ganji et al. [D.D. Ganji, H. Tari, M.B. Jooybari, Variational iteration method and homotopy perturbation method for nonlinear evolution equations. Comput. Math. Appl. 54 (2007) 1018–1027]. It is found that the HPM is a special case of the HAM when =-1. However, the HPM solution is divergent for all x and t except t=0. It is also found that the solution given by the variational iteration method (VIM) is divergent too. On the other hand, using the HAM, one obtains convergent series solutions which agree well with the exact solution. This example illustrates that it is very important to investigate the convergence of approximation series. Otherwise, one might get useless results.     

Note on the homotopy perturbation method for multivariate vector-value oscillatory integrals

This paper considers a homotopy perturbation method for approximating multivariate vector-value highly oscillatory integrals. The asymptotic formulae of the integrals and the asymptotic order of the asymptotic method are presented. Numerical examples show the efficiency of the approximation method.     

A homotopy perturbation method for solving a neutron transport equation

The purpose of this paper consists in the finding of the solution for a stationary transport equation using the techniques of homotopy perturbation method (HPM). The results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method.     

Solving two-point boundary value problems using combined homotopy perturbation method and Green’s function method

In this paper, an algorithm is proposed for the solution of second-order boundary value problems with two-point boundary conditions. The Green’s function method is applied first to transform the ordinary differential equation into an equivalent integral one, which has already satisfied the boundary conditions. And then, the homotopy perturbation method is used to the resulting equation to construct the numerical solution for such problems. Numerical examples demonstrate the efficiency and reliability of the algorithm developed, it is quite accurate and readily implemented for both linear and nonlinear differential equations with homogeneous and nonhomogeneous boundary conditions. Furthermore, the lower order approximation is of higher accuracy for most cases. Some other extended applications of this algorithm are also exhibited.     

Study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method

This paper deals with the study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method. The paper considers a micro-beam suspended between two conductive micro-plates, subjected to a same actuation voltage. The nonlinear governing differential equation of motion about static equilibrium position using calculus of variation theory and Taylor series expansion has been linearized and implementing a Galerkin based reduced order model a Mathieu type equation has been obtained. By improving variational iteration method combining with method of strained parameters transition curves, separating stable from unstable regions have been obtained. The results of variational iteration method, perturbation and direct numerical integration methods for some cases selected from different regions (stable and unstable regions) have been compared.     

A homotopy perturbation algorithm to solve a system of Fredholm–Volterra type integral equations

In this paper, an application of He’s homotopy perturbation (HPM) method is applied to solve the system of Fredholm and Volterra type integral equations, the results revealing that the HPM is very effective and simple.     

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.     

Homotopy perturbation method for fractional KdV equation

In this paper, the homotopy perturbation method is directly applied to derive approximate solutions of the fractional KdV equation. The results reveal that the proposed method is very effective and simple for solving approximate solutions of fractional differential equations.     

Solution of nonlinear fractional differential equations using homotopy analysis method

In this article, the homotopy analysis method has been applied to solve nonlinear differential equations of fractional order. The validity of this method has successfully been accomplished by applying it to find the solution of two nonlinear fractional equations. The results obtained by homotopy analysis method have been compared with those exact solutions. The results show that the solution of homotopy analysis method is good agreement with the exact solution.     

Piecewise homotopy perturbation method for solving linear and nonlinear weakly singular VIE of second kind

The purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular Volterra integral equation of the second kind, especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem. For this purpose, we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like. Through modifying homotopy perturbation method, an algorithm is successfully established to solve the linear and nonlinear weakly singular Volterra integral equation of the second kind. And the convergence of the algorithm is proved strictly. Comparisons are made between our method and other methods, and the results reveal that the modified homotopy perturbation is effective.     

Analytical approximations to nonlinear vibration of an electrostatically actuated microbeam

This paper employs the homotopy analysis method (HAM) to derive analytical approximate solutions for the nonlinear problem with high-order nonlinearity. Such a problem corresponds to the large-amplitude vibration of electrostatically actuated microbeams. The HAM is also optimized to accelerate the convergence of approximate solutions. To verify the accuracy of the present approach, illustrative examples are provided and compared with other analytical and exact solutions.     

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.     

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.     

Homotopy perturbation method for linear programming problems

In this paper, He’s homotopy perturbation method (HPM) is applied for solving linear programming (LP) problems. This paper shows that some recent findings about this topic cannot be applied for all cases. Furthermore, we provide the correct application of HPM for LP problems. The proposed method has a simple and graceful structure. Finally, a numerical example is displayed to illustrate the proposed method.     

On exact solutions to partial differential equations by the modified homotopy perturbation method

Based on the modified homotopy perturbation method (MHPM), exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a series in p as exact solutions. Under suitable initial conditions, the PDE is transformed into an ODE. Some illustrative examples reveal the effciency of the proposed method.     

Comparison of the Adomian decomposition method with homotopy perturbation method for the solutions of seventh order boundary value problems

The aim of this paper is to compare the Adomian decomposition method and the homotopy perturbation method for solving the linear and nonlinear seventh order boundary value problems. The approximate solutions of the problems obtained with a small amount of computation in both methods. Two numerical examples have been considered to illustrate the accuracy and implementation of the methods.