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In this paper, we establish the asymptotic expansions for the integration containing the Bessel functions with high oscillations by using the homotopy technique. The proposed method is constructed by approximating the non-oscillatory or low-oscillatory factor in practice. Two cases are considered, that is, the case 0 [image omitted] [a, b] and the case 0∈[a, b], where [a, b] denotes the integration interval. The efficiency of the technique and the correctness of the theory are shown by several examples. [ABSTRACT FROM AUTHOR]

In this paper, an application of homotopy perturbation method (HPM) is applied to solve the kindly of multidimensional partial differential equation such as Helmholtz equation. Comparisons are made between the Adomians decomposition method and HPM. The results reveal that the HPM is very effective and simple and gives the exact solution. [ABSTRACT FROM AUTHOR]

This paper studies the fractional Lotka-Volterra equations for three competitors, since the fractional derivatives possess the properties of good memory and have great biological significance. First of all, the equilibrium points and asymptotic stability for the equations are studied by the stability analysis method. As expected, the fractional-order differential equations are, at least, as stable as their integer-order counterpart. Second, some approximate analytic solutions for this systems are obtained by the stability analysis method and the homotopy perturbation method, which are expressed in the form of the Mittag-Leffler function. The results show that it takes less time for the population to get close to the equilibrium point as the time derivatives increase. Comparing with the classical ones, the fractional Lotka-Volterra equations, no matter whether the fractional derivatives have big or small order, if both take more time, even multiplied time, for the system to reach the equilibrium point, then that may explain the memory properties. Furthermore, some numerical analyses are carried out and verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, the homotopy perturbation method (HPM) is effectively applied to obtain the approximate analytic solutions of the time-fractional Fisher equation (TFFE) with initial conditions. The fractional derivatives are described in the Caputo sense. The initial approximation can be determined by imposing the initial conditions. Some examples are given. Numerical results show that the HPM is easy to implement and is accurate when applied to TFFE. [ABSTRACT FROM AUTHOR]

This paper proposes a fractional model for nonlinear waves in hyperelastic rods, which describes far-field, finite length, finite amplitude radial deformation waves in cylindrical compressible hyperelastic rods. In this model, fractional derivatives are described in the Caputo sense. The error analysis shows that our approximate solution converges very rapidly to the exact solution and the numerical solution is compared with the known analytical solution which is nearly identical with the exact solution. The method introduces a promising tool for solving time the fractional hyperelastic rod equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fractional model of telegraph equation in terms of voltage and current. The fractional derivatives are taken in the Caputo sense. The numerical algorithm based on the homotopy perturbation transform method (HPTM) is applied to obtain analytic and approximate solutions of the space-time fractional telegraph equations. The HPTM is combined in the form of Laplace transform and homotopy perturbation method. The results obtained by the HPTM show that the approach is easy to implement and computationally very attractive. [ABSTRACT FROM AUTHOR]

In this paper, we have obtained approximate solution of the nonlinear (3+1)-dimensional breaking soliton Equation (BSE) by using Homotopy Perturbation Method (HPM) and then formed a chart including numerical conclusion for this equation by drawing graphics of the the obtained solution by means of programming language Mathematica. [ABSTRACT FROM AUTHOR]

A mathematical model of mediated bioelectrocatalysis is restudied in this paper. Here He's Homotopy perturbation method is implemented to give approximate and analytical solutions of steady-state non-linear reaction diffusion equation containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reaction. Approximate analytical expressions for mediator concentration and current have been derived for all values of saturation parameter α and reaction diffusion parameter k for the various types of boundary conditions. The Homotopy perturbation method which produces the solutions in terms of convergent series, requiring no linearization or small perturbation. These analytical results are compared with numerical results (Matlab program) and are found to be in good agreement. [ABSTRACT FROM AUTHOR]

In this paper, Homotopy Perturbation Method (HPM) is applied to find the approximate solutions of ZK equations. It is proved that the HPM gives a powerful tool for solving a large number of nonlinear partial differential equations in mathematical physics. The solutions obtained by HPM are presented graphically. [ABSTRACT FROM AUTHOR]

In this paper, a new tool for the solution of nonlinear differential equations is presented. It is named rational homotopy perturbation method (RHPM). It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. Furthermore, in order to deal with BVP problems, we propose a modification of RHPM method. The obtained results show that RHPM is a powerful tool capable to generate highly accurate rational solutions. [ABSTRACT FROM AUTHOR]

Purpose |!|#8211; The purpose of this paper is to report the effect of radiation on flow of a magneto-micropolar fluid past a continuously moving plate with suction and blowing. Design/methodology/approach |!|#8211; The governing equations are transformed into dimensionless nonlinear ordinary differential equations by similarity transformation. Analytical technique, namely the homotopy perturbation method (HPM) combining with Pad|!|#233; approximants and finite difference method, are used to solve dimensionless non-linear ordinary differential equations. The skin friction coefficient and local Nusselt numbers are also calculated. Beside this, the comparison of the analytical solution with numerical solution is illustrated by the graphs for different values of dimensionless pertinent parameters. Findings |!|#8211; The authors have studied laminar magneto-micropolar flow in the presence of radiation by using HPM-Pad|!|#233; and finite difference methods. Results obtained by HPM-Pad|!|#233; are in excellent agreement with the results of numerical solution. Originality/value |!|#8211; The HPM-Pad|!|#233; is used in a direct way without using linearization, discritization or restrictive assumption. The authors have attempted to show the capabilities and wide-range applications of the HPM-Pad|!|#233; in comparison with the finite difference solution of magneto-micropolar flow in the presence of radiation problem. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a new method to compute approximate analytical solutions for nonlinear differential equations, called the Least Squares Homotopy Perturbation Method. The method is based on the well-known Homotopy Perturbation Method and its main feature is an accelerated convergence compared to the regular Homotopy Perturbation Method. The comparison with previous results emphasizes the high accuracy of the method. [ABSTRACT FROM AUTHOR]

This paper presents a modified Laplace transform homotopy perturbation method with finite boundary conditions (MLT-HPM) designed to improve the accuracy of the approximate solutions obtained by LT-HPM and other methods. To this purpose, a suitable initial approximation will be introduced, in addition, the residual error in several points of the interest interval (RECP) will be canceled. In order to prove the efficiency of the proposed method a couple of nonlinear ordinary differential equations with mixed boundary conditions, indeed, difficult to approximate, are proposed. The square residual error (S.R.E) of the proposed solutions will result to be of hundredths and tenths, requiring only a first order approximation of MLT-HPM, unlike LT-HPM, which will require more iterations for the same cases study. [ABSTRACT FROM AUTHOR]

Abstract: This paper is an elementary introduction to the concepts of the homotopy perturbation method. Particular attention is paid to giving an intuitive grasp for the solution procedure throughout the paper. [Copyright &y& Elsevier]

In this article, we focus on linear and nonlinear fuzzy Volterra integral equations of the second kind and we propose a numerical scheme using homotopy perturbation method (HPM) to obtain fuzzy approximate solutions to them. To facilitate the benefits of this proposal, an algorithmic form of the HPM is also designed to handle the same. In order to illustrate the potentiality of the approach, two test problems are offered and the obtained numerical results are compared with the existing exact solutions and are depicted in terms of plots to reveal its precision and reliability. [ABSTRACT FROM AUTHOR]

Imbibitions process is very important process in oil recovery. In this paper mathematical model for imbibition phenomenon in one dimensional homogeneous porous media is derived. Governing nonlinear partial differential equation is solved by using homotopy perturbation method. Homotopy perturbation method is a new perturbation method is proposed. The initial approximation is chosen using initial and boundary conditions in the form of quadratic equation. [ABSTRACT FROM AUTHOR]

This paper applies a new modification of the Homotopy Perturbation Method that is called Rational Homotopy Perturbation Method (RHPM) to obtain an analytic approximation of stiff systems of ordinary differential equations. The procedure of the method will be explained briefly and some examples are presented to illustrate the method. The results of RHPM, are compared with those of traditional Homotopy Perturbation Method. The results show the accuracy and fast convergence of the proposed method. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for two dimensional time-fractional wave equation (TFWE) with the boundary conditions. The fractional derivative is described in the Caputo sense. The initial approximation can be determined by imposing the boundary conditions. The method provides approximate solutions in the form of convergent series with easily computable components. The obtained results shown that the technique introduced here is efficient and easy to implement. [ABSTRACT FROM AUTHOR]

The present paper presents the comparison of analytical techniques. We establish the existence of the phenomena of the noise terms in the perturbation series solution and find the exact solution of the nonlinear problems. If the noise terms exist, the Homotopy perturbation method gives the same series solution as in Adomian Decomposition Method and we get the exact solution using two iterations only. [ABSTRACT FROM AUTHOR]

In this paper, the Homotopy Perturbation Method (HPM) has been implemented for solving the extended Fisher-Kolomogrov-Petrovskii-Piskunov (FKPP) equation. The HPM is an approximate method which yields an analytical solution in the form of a convergent power series with components, which can be easily computed. We show that the HPM can generate highly accurate solutions for the FKPP equation. [ABSTRACT FROM AUTHOR]

In this paper, the Homotopy Perturbation Method (HPM) which is an approximate analytical method has been used to solve the nonlinear coupled viscous Burgers equations with fractional-time derivative. With the HPM, the solution can be obtained in the form of a convergent power series with terms that can be easily computed. The method does not need linearization and provides an analytical solution by utilizing only the initial condition. We also show that the HPM can generate highly accurate analytical solutions. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

In this study, homotopy perturbation method (HPM) is used to obtain analytic and approximate solutions of the space- and time-fractional telegraph equations. The space- and time-fractional derivatives are considered in the Caputo sense. The analytic solutions are calculated in the form of a series with easily computable terms. Some examples are given. The results reveal that HPM is very effective and convenient. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the homotopy perturbation method is used for calculation of axial secular frequencies of a nonlinear ion trap with only hexapole superposition. The motion of the ion in a rapidly oscillating field is transformed to the motion in an effective potential. The equation of ion motion in the effective potential is the equation of an anharmonic oscillator with quadratic nonlinearity. The homotopy perturbation method is used for solving the resulted nonlinear equation and obtaining the expression for ion secular frequency as a function of nonlinear field parameter. The calculated secular frequencies are compared with the results of L.–P. method and the exact results. [ABSTRACT FROM AUTHOR]

In this study, we present a framework to obtain analytical approximate solutions to the nonlinear fractional convection-diffusion equation. The fractional derivative is considered in the Caputo sense. The applications of the homotopy perturbation method were extended to derive analytical solutions in the form of a series with easily computed terms for this equation. Some examples are tested and the results reveal that the technique introduced here is very effective and convenient for solving nonlinear partial differential equations of fractional order. [ABSTRACT FROM AUTHOR]

Abstract: The paper presents the transportation of viscoelastic fluid with fractional Maxwell model by peristalsis through a channel under long wavelength and low Reynolds number approximations. The propagation of wall of channel is taken as sinusoidal wave propagation (contraction and relaxation). Homotopy perturbation method (HPM) and Adomian decomposition method (ADM) are used to obtain the analytical approximate solutions of the problem. The expressions of axial velocity, volume flow rate and pressure gradient are obtained. The effects of fractional parameters , relaxation time and amplitude on the pressure difference and friction force across one wavelength are calculated numerically for different particular cases and depicted through graphs. [Copyright &y& Elsevier]

Abstract: In this paper, two methods for solving a nonlinear differential equation known as He’s variational iteration and homotopy perturbation methods are applied to derive the approximate kink-type soliton solutions for a new (2+1)-dimensional simplified generalized Broer–Kaup system. Furthermore, the solutions obtained are compared with the corresponding exact solution to show the applicability, accuracy and, finally, efficiency of the present methods in solving a large class of nonlinear physics and engineering problems without substantial noisy sensitivity to the nonlinear terms or any restrictive assumptions or transformations that may change the physical behavior of the problems. [Copyright &y& Elsevier]

Abstract: Continuum hypothesis on nanoscales is invalid, and a differential–difference model is considered as an alternative approach to describing discontinued problems. This paper applies the homotopy perturbation method to a nonlinear differential–difference equation arising in nanotechnology. Comparison of the approximate solution with the exact one reveals that the method is very effective. [Copyright &y& Elsevier]

Abstract: This paper investigates the forced Duffing equation with integral boundary conditions. Its approximate solution is developed by combining the homotopy perturbation method (HPM) and the reproducing kernel Hilbert space method (RKHSM). HPM is based on the use of the traditional perturbation method and the homotopy technique. The HPM can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully linear boundary value problems. Therefore, the forced Duffing equation with integral boundary conditions can be solved using advantages of these two methods. Two numerical examples are presented to illustrate the strength of the method. [Copyright &y& Elsevier]

Abstract: In the paper, we implement relatively new analytical techniques, the variational iteration method, the Adomian decomposition method and the homotopy perturbation method, for obtaining a rational approximation solution of the fractional Sharma–Tasso–Olever equation. The three methods in applied mathematics can be used as alternative methods for obtaining an analytic and approximate solution for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The numerical results demonstrate the significant features, efficiency and reliability of the three approaches. [Copyright &y& Elsevier]

Abstract: In this paper, we present the variational iteration method and homotopy perturbation method to solve the modified Kawahara equations. Both methods provide remarkable accuracy for the approximate solutions when compared to the exact solutions. Numerical results demonstrate that the methods provide efficient approaches to solving the modified Kawahara equation. [Copyright &y& Elsevier]

Abstract: In this paper, we apply a new modified homotopy perturbation method to find exact solutions for nonlinear integral equations. A reliable modification of the homotopy perturbation method is proposed, and the modified method is employed to solve the nonlinear integral equations; the results are compared with those obtained by the original homotopy perturbation method. Some examples are given to illustrate the ability and reliability of the modified method. The results reveal that the modified method is very simple and effective. [Copyright &y& Elsevier]

Abstract: In this paper, the solution of the generalized Huxley equation is obtained by means of homotopy perturbation method and Adomian decomposition method. The comparison reveals that the former method is more effective than the later. In this method, a homotopy is constructed for the equation.The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Moreover, we will show that He’s homotopy perturbation method overcome the difficulties arising in calculating Adomian polynomials. It is predicted that the HPM can be found wide application in engineering problems. [Copyright &y& Elsevier]

Abstract: In this paper, an application of homotopy perturbation method is applied to solve the nonlinear two-dimensional wave equation. The analytic solution of the nonlinear wave equation is calculated in the form of a series with easily computable components. The non-homogenous equation is effectively solved by employing the phenomena of the self-canceling “noise” terms, where sum of components vanishes in the limit. Comparing the methodology with some known techniques shows that the present approach is powerful and reliable. Its remarkable accuracy properties are finally demonstrated by an example. [Copyright &y& Elsevier]

Abstract: In this paper, we present an efficient numerical algorithm for solving nonlinear algebraic equations based on Newton–Raphson method and homotopy analysis method. Also, we compare homotopy analysis method with Adomian’s decomposition method and homotopy perturbation method. Some numerical illustrations are given to show the efficiency of algorithm. [Copyright &y& Elsevier]

Recently some semi-analytical methods have been introduced for solving a class of nonlinear optimal control problems such as the Adomian decomposition method, homotopy perturbation method and modified variational iteration method. In this manuscript we compare these methods for solving a type of nonlinear optimal control problem. We prove that these methods are equivalent, which means that they use the same iterative formula to obtain the approximate/analytical solution. [ABSTRACT FROM AUTHOR]

There are some methods to obtain approximate solutions of functional equations. One of them is the homotopy perturbation method. However, computing coefficients of pj, is time-consuming and challenging. In order to deal with this problem, a new modified homotopy perturbation method is introduced, which doesn't need computations of the coefficients of pj. Here, the method has been applied to solve some examples and the results have been compared with those obtained from the homotopy perturbation method. Moreover, convergence of the method has been discussed. [ABSTRACT FROM AUTHOR]

Homotopy perturbation method is used to extend the approximate analytical solutions of non-linear reaction equations describing enzyme kinetics for combinations of parameters for which solutions obtained in previous works are not valid. Also, by constructing a new homotopy, alternative approximate analytical expressions for substrate, substrate-enzyme complex and product concentrations are found. These first-order approximate solutions give more accurate results than the second-order approximations derived in previous works. [ABSTRACT FROM AUTHOR]

We implemented homotopy perturbation method for approximating the solution to the nonlinear dispersive K(m,n,1) type equations. By using this scheme, the explicit exact solution is calculated in the form of a quickly convergent series with easily computable components. To illustrate the application of this method, numerical results are derived by using the calculated components of the homotopy perturbation series. [ABSTRACT FROM AUTHOR]

We combined the Homotopy perturbation method (HPM) and Padé techniques to solve the well-known Blaszak-Marciniak lattice. Blaszak-Marciniak lattice has rich mathematical structures and many important applications in physics, engineering and mathematics. Generally, the truncated series solution of HPM is adequate only in a small region when the exact solution is not reached. We overcame this limitation by using the Padé techniques, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. Using this combined technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the HPM alone. An example of computed calculation is followed by comparative plots of the solutions obtained through the (HPM) and (HPM-Padé) methods. Graphics are gathered for discussion of convergence. The same protocol is applied to Blaszak-Marciniak four-field lattice. [ABSTRACT FROM AUTHOR]