A new modification of He’s homotopy perturbation method for rapid convergence of nonlinear undamped oscillators

In this paper we present a new efficient modification of the homotopy perturbation method with x ³ force nonlinear undamped oscillators for the first time that will accurate and facilitate the calculations. The He’s homotopy perturbation method is modified by adding a term to linear operator depends on the equation and boundary conditions. We find that this modified homotopy perturbation method works very well for the wide range of time and boundary conditions for nonlinear oscillator. Only two or three iteration leads to high accuracy of the solutions. We then conduct a comparative study between the new modification and the homotopy perturbation method for strongly nonlinear oscillators. Numerical illustrations are investigated to show the accurate of the techniques. The new modified method accelerates the rapid convergence of the solution, reduces the error solution and increases the validity range. The new modification introduces a promising tool for many nonlinear problems.     

He’s homotopy perturbation method for systems of integro-differential equations

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.     

Comparison of two reliable analytical methods based on the solutions of fractional coupled Klein–Gordon–Zakharov equations in plasma physics

In this paper, homotopy perturbation transform method and modified homotopy analysis method have been applied to obtain the approximate solutions of the time fractional coupled Klein–Gordon–Zakharov equations. We consider fractional coupled Klein–Gordon–Zakharov equation with appropriate initial values using homotopy perturbation transform method and modified homotopy analysis method. Here we obtain the solution of fractional coupled Klein–Gordon–Zakharov equation, which is obtained by replacing the time derivatives with a fractional derivatives of order α ∈ (1, 2], β ∈ (1, 2]. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present methods homotopy perturbation transform method and modified homotopy analysis method. The fractional derivatives here are described in Caputo sense.     

Homotopy perturbation method for numerical solutions of KdV‐Burgers' and Lax's seventh‐order KdV equations

In this article, we applied homotopy perturbation method to obtain the solution of the Korteweg‐de Vries Burgers (for short, KdVB) and Lax's seventh‐order KdV (for short, LsKdV) equations. The numerical results show that homotopy perturbation method can be readily implemented to this type of nonlinear equations and excellent accuracy. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Analytical solutions for the fractional nonlinear cable equation using a modified homotopy perturbation and separation of variables methods

In this paper, we first introduce a new homotopy perturbation method for solving a fractional order nonlinear cable equation. By applying proposed method the nonlinear equation it is changed to linear equation for per iteration of homotopy perturbation method. Then, we solve obtained problems with separation method. In examples, we illustrate that the exact solution is obtained in one iteration by convenience separating of source term in given equation.     

Application of He’s homotopy perturbation method to functional integral equations

In this paper, an application of He’s homotopy perturbation method is applied to solve functional integral equations. Comparisons are made between expansion method based on Lagrange interpolation formulae and the homotopy perturbation method. The results reveal that the He’s homotopy perturbation method is very effective and simple and gives the exact solution.     

Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems

A combination of the hybrid spectral collocation technique and the homotopy analysis method is used to construct an iteration algorithm for solving a class of nonlinear optimal control problems (NOCPs). In fact, the nonlinear two-point boundary value problem (TPBVP), derived from the Pontryagin’s Maximum Principle (PMP), is solved by spectral homotopy analysis method (SHAM). For the first time, we present here a convergence proof for SHAM. We treat in detail Legendre collocation and Chebyshev collocation. It is indicated that Legendre collocation gives the same numerical results with Chebyshev collocation. Comparisons are made between SHAM, Matlab bvp4c generated results and results from literature such as homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM) and differential transformations.     

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.     

An efficient algorithm for solving fifth-order boundary value problems

In this paper, we apply the homotopy perturbation method for solving the fifth-order boundary value problems. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the homotopy perturbation method. Comparisons are made to confirm the reliability of the method.     

The general analytical and numerical solution for the modified KdV equation with convergence analysis

We investigate the analytical and numerical solutions of the modified Kortweg de Vries equation by applying the idea of commutative hypercomplex mathematics, He's homotopy perturbation method as a simple particular procedure, and the Runge–Kutta discontinuous Galerkin methods. Moreover, we discuss at great length the convergence conditions for this equation by using the Banach fixed point theory, which could provide a good iteration algorithm. Finally, we compare the homotopy perturbation method with some standard ideas same as the Runge–Kutta discontinuous Galerkin method by some numerical illustrations. Copyright © 2012 John Wiley & Sons, Ltd.     

Convergence of the homotopy perturbation method for partial differential equations

In this paper, we introduce a homotopy perturbation method to obtain exact solutions to some linear and nonlinear partial differential equations. This method is a powerful device for solving a wide variety of problems. Using the homotopy perturbation method, it is possible to find the exact solution or an approximate solution of the problem. Convergence of the method is proved. Some examples such as Burgers’, Schrödinger and fourth order parabolic partial differential equations are presented, to verify convergence hypothesis, and illustrating the efficiency and simplicity of the method.     

Homotopy perturbation method for fractional KdV-Burgers equation

In the paper, we extend the homotopy perturbation method to solve nonlinear fractional partial differential equations. The time- and space-fractional KdV-Burgers equations with initial conditions are chosen to illustrate our method. As a result, we successfully obtain some available approximate solutions of them. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.     

Solutions of nonlinear thickness-shear vibrations of an infinite isotropic plate with the homotopy analysis method

As a preliminary attempt for the study on nonlinear vibrations of a finite crystal plate, the thickness-shear mode of an infinite and isotropic plate is investigated. By including nonlinear constitutive relations and strain components, we have established nonlinear equations of thickness-shear vibrations. Through further assuming the mode shape of linear vibrations, we utilized the standard Galerkin approximation to obtain a nonlinear ordinary differential equation depending only on time. We solved this nonlinear equation and obtained its amplitude–frequency relation by the homotopy analysis method (HAM). The accuracy of the present results is shown by comparison between our results and the perturbation method. Numerical results show that the homotopy analysis solutions can be adjusted to improve the accuracy. These equations and results are useful in verifying the available methods and improving our further solution strategy for the coupled nonlinear vibrations of finite piezoelectric plates.     

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.     

Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled KdV equation by an analytical technique

In this work, we implement a relatively analytical technique, the homotopy perturbation method (HPM), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo derivatives. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. He’s homotopy perturbation method (HPM) which does not need small parameter is implemented for solving the differential equations. It is predicted that HPM can be found widely applicable in engineering.     

The homotopy perturbation method applied to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations

The fractional derivatives in the sense of Caputo, and the homotopy perturbation method are used to construct approximate solutions for nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equations with respect to time and space fractional derivatives. Also, we apply complex transformation to convert a time and space fractional nonlinear KPP equation to an ordinary differential equation and use the homotopy perturbation method to calculate the approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.     

Application of homotopy perturbation methods for solving systems of linear equations

In this paper, homotopy perturbation methods (HPMs) are applied to obtain the solution of linear systems, and conditions are deduced to check the convergence of the homotopy series. Moreover, we have adapted the Richardson method, the Jacobi method, and the Gauss-Seidel method to choose the splitting matrix. The numerical results indicate that the homotopy series converges much more rapidly than the direct methods for large sparse linear systems with a small spectrum radius.     

Hypercomplex mathematics and HPM for the time-delayed Burgers equation with convergence analysis

We investigate the analytical and numerical solutions of the time-delayed Burgers equation, by applying the idea of commutative hypercomplex mathematics and the homotopy perturbation method. Moreover, we discuss at great length the convergence conditions of the homotopy perturbation Method (HPM) by using the Banach fixed point theory , which could provide a good iteration algorithm. Finally, we also give some numerical illustrations to the obtained results.     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

Comparison between the homotopy analysis method and homotopy perturbation method to solve coupled Schrodinger-KdV equation

In this paper, we apply the homotopy analysis method (HAM) and the homotopy perturbation method (HPM) to obtain approximate analytical solutions of the coupled Schrodinger-KdV equation. The results show that HAM is a very efficient method and that HPM is a special case of HAM.