Modified Homotopy Perturbation Method Research Articles

Analytical Investigation of Jeffery–Hamel Flow by Modified Adomian Decomposition Method

Abstract The objective of this paper is to present a reliable approach to compute an approximate analytical solution of magneto-hydrodynamic (MHD) Jeffery–Hamel flow by using a new modification of Adomian decomposition method. The approximate solution of this problem is calculated here in the form of a rapidly convergent series at one and both grid points. Results for velocity profiles in divergent and convergent channels are presented for different values of Hartmann and Reynolds numbers. The validity and applicability of this new technique are illustrated through a comparison between reproducing Kernel Hilbert Space Method (RKHSM) (Inc et al., 2013), Modified Homotopy Perturbation Method (MHPM) (Singh, 2014) and numerical solutions.

Modified homotopy perturbation method for optimal control problems using the Padé approximant

In this study, we propose a hybrid method that combines the homotopy perturbation method (HPM) and Padé technique to obtain the approximate analytic solution of the Hamilton–Jacobi–Bellman equation. The truncated series solution for the HPM is suitable but only in a small domain when the exact solution is not obtained. To improve the accuracy and enlarge the convergence domain, the Padé technique is applied to the series solution for the HPM. Three examples are given to illustrate the applicability, simplicity, and efficiency of the proposed method. The results obtained are then compared with the exact solution and basic HPM. We demonstrate that this hybrid method provides an approximate analytic solution with higher accuracy than the classic HPM.

A modified homotopy perturbation method for singular time dependent Emden–Fowler equations with boundary conditions

In this paper, we propose a new modification of the homotopy perturbation method (HPM) for solving nonlinear and singular time-dependent Emden–Fowler-types equations with the Neumann and Dirichlet boundary conditions. We first transform the singular problem into an equivalent integral equation, and we then apply the HPM to obtain approximate series solution. This new modified HPM will be used without unknown constants while computing the successive solution components, and we also avoid solving a sequence of transcendental equations for the determination of the unknown constants. Moreover, the proposed technique is reliable enough to overcome the difficulty of the singular point at \(x=0\). Four illustrative examples are examined to demonstrate the accuracy and applicability of the proposed method.

Modified homotopy perturbation method for solving linear second-order Fredholm integro-differential equations

In this paper, we propose a modification to homotopy perturbation method and improve to accelerate the rate of convergence in solving linear second-order Fredholm integro-differential equations. Some examples are given to show that this method is easy to apply and the results is obtained very fast.

On systems of nonlinear equations: some modified iteration formulas by the homotopy perturbation method with accelerated fourth- and fifth-order convergence

This paper is concerned with the development of efficient algorithms for solving the system of nonlinear equations based on the modified homotopy perturbation method. The analysis of convergence shows that the proposed methods are fourth and fifth-order convergence. Several numerical examples are presented. Comparisons are made to confirm the reliability and effectiveness of the proposed schemes.

Radiation Effects on Two-Dimensional MHD Falkner-Skan Wedge Flow

Radiation effects on two-dimensional MHD Falkner-Skan boundary layer wedge have been studied. Analytical solution of nonlinear boundary-layer equations is obtained by modified homotopy perturbation method. It is observed that the magnetic field tends to decelerate fluid flow whereas radiations and thermal diffusion tend to increase fluid temperature.

A theoretical and experimental investigation on large amplitude free vibration behavior of a pretensioned beam with clamped–clamped ends using modified homotopy perturbation method

In this study, large amplitude free vibration behavior of a pretensioned Euler–Bernoulli beam is investigated both theoretically and experimentally. The Hamilton's principle is used to derive the beam governing equation of motion. By implementing the Galerkin's method and assuming the clamped–clamped boundary condition, the partial differential equation is converted to an ordinary nonlinear differential equation. Because of the large coefficient of the nonlinear term, the new homotopy perturbation method proposed by He, is modified to solve the governing nonlinear equation. Comparing the first- and the second-order approximate solutions of the modified homotopy perturbation method (MHPM) and those available in the literature demonstrates that the second-order MHPM leads to a more accurate solution which is valid for a wide range of the vibration amplitudes. The results have been validated by the experimental tests and the MHPM method. Also, the results show that an increase in the vibration amplitude and/or the pretension load increases the fundamental resonance frequency ratio. Besides, it would decrease with increasing the beam slenderness ratio.

Approximate solutions of multi-order fractional advection-dispersion equation with non-polynomial conditions

Purpose – The purpose of this paper is to apply a new modified homotopy perturbation method, which is effective to solve multi-order fractional equations with non-polynomial initial and boundary conditions. Design/methodology/approach – The proposed algorithm is tested on multi-order fractional advection-dispersion equations. The fractional derivatives described in this paper are in the Caputo sense. Findings – Approximate results explicitly reveal the complete reliability, efficiency and accuracy of the new modified technique. Originality/value – It is observed that the approach may be implemented to other multi-fractional models with non-polynomial initial and boundary conditions.

A Simple Modification of Homotopy Perturbation Method for the Solution of Blasius Equation in Semi-Infinite Domains

A simple modification of the homotopy perturbation method is proposed for the solution of the Blasius equation with two different boundary conditions. Padé approximate is used to deal with the boundary condition at infinity. The results obtained from the analytical method are compared to Howarth’s numerical solution and fifth order Runge-Kutta Fehlberg method indicating a very good agreement. The proposed method is a simple and reliable modification of homotopy perturbation method, which does not require the existence of a small parameter, linearization of the equation, or computation of Adomian’s polynomials.

New Modification of Homotopy Perturbation Method and the Fourth - Order Parabolic Equations with Variable Coefficients

In this paper, the exact solution of the fourth - order parabolic equations with variable coefficients is obtained by using a new homotopy perturbation method (NHPM), theoretical consideration are discussed. Finally, three examples are illustrated to show the validity and applicability of the proposed method.

Comparative Solution of Nonlinear Quintic Cubic Oscillator Using Modified Homotopy Perturbation Method

We use modified homotopy perturbation method to find out the solution of nonlinear cubic quintic equation. Besides this method solution of the problem with the following methods is discussed, Energy Balance Method and He’s Frequency Formulation method and then compare the results with each other and Global Error Method. The results show that these three methods are effective as global error method for nonlinear cubic quintic oscillator equation with multiple nonlinear terms and have different effects on the solution. In particular, the homotopy perturbation solution is quite surprising. A cubic quintic nonlinear oscillator is used as an example to compare the results.

A Novel Effective Approach for Solving Fractional Nonlinear PDEs.

The present work introduces an effective modification of homotopy perturbation method for the solution of nonlinear time-fractional biological population model and a system of three nonlinear time-fractional partial differential equations. In this approach, the solution is considered a series expansion that converges to the nonlinear problem. The new approximate analytical procedure depends only on two iteratives. The analytical approximations to the solution are reliable and confirm the ability of the new homotopy perturbation method as an easy device for computing the solution of nonlinear equations.

Study on motion of rigid rod on a circular surface using MHPM

In this paper motion of rigid rod on a circular surface is studied analytically. A new analytical method called modified homotopy perturbation method (MHPM) is applied for solving this problem in different initial conditions to show capability of this method. The governing equation for motion of a rigid rod on the circular surface without slipping have been solved using MHPM. The efficacy of MHPM for handling nonlinear oscillation systems with various small and large oscillation amplitudes are presented in comparison with numerical benchmarks. Outcomes reveal that MHPM has an excellent agreement with numerical solution. The results show that by decreasing the oscillation amplitude, the velocity of rigid rod decreases and for A=π/3 the velocity profile is maximum.

On Solutions of the Nonlinear Oscillators by Modified Homotopy Perturbation Method

The nonlinear oscillator problem has extensive applications. In this paper, we examined the modified homotopy perturba tion method to obtain the solution of the problem and to reduce the computational efforts. For the accuracy purpose the Laplace transform and Pade approximation have also been used. We construct the deformation equations by two components of the homotopy series, then by means of the Laplace transform the obtain solution is converted to series form and Pade approximation is exploited to increase the accuracy of the achieved result. The computational efforts are significantly reduced. And, the calculated analytic solutions are in excellent agreement with the numerical solutions.

A new modification to homotopy perturbation method combined with Fourier transform for solving nonlinear Cauchy reaction diffusion equation

A hybrid of Fourier transform and new modified homotopy perturbation method based on the Adomian method is developed to solve linear and nonlinear partial differential equations. The Taylor series expansion is used to expand nonlinear term of partial differential equation and the Adomian polynomial incorporated into homotopy perturbation method combined with Fourier transform, is used to solve partial differential equations. Three case study problems, partial differential equations, are handled using homotopy perturbation method and Fourier transform modified homotopy perturbation method (FTMHPM). Results obtained are compared with exact solution. The comparison reveals that for same components of recursive sequences, errors associated with Fourier transform modified method are much less than the other and are valid for a large range of x-axis coordinates.

Numerical computation of fractional multi-dimensional diffusion equations by using a modified homotopy perturbation method

The main aim of the present work is to present a numerical algorithm for solving fractional multi-dimensional diffusion equations which describes density dynamics in a material undergoing diffusion by using a modified homotopy perturbation method with the help of the sumudu transform. The modified homotopy perturbation method is not limited to the small parameter, such as in the classical perturbation method. The method gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive.

Modified Homotopy Perturbation Technique for the Approximate Solution of Nonlinear Equations

We use a new modified homotopy perturbation method to suggest and analyze some new iterative methods for solving nonlinear equations. This new modification of the homotopy method is quite flexible. Various numerical examples are given to illustrate the efficiency and performance of the new methods. These new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations.

A comparison between numerical methods for solving Fuzzy fractional differential equations

In this paper, numerical simulation of linear and nonlinear fuzzy initial value problems of fractional order has been done. A modificaton in the improved Euler’s method for fuzzy fractional order is proposed and utilized. Also, the two component homotopy perturbation method also known as a modified homotopy perturbation method (MHPM) is employed to solve the considered problems. The results obtained from the considered techniques are compared. These methods are illustrated by solving several examples. Effciency and exactness of results worked out from MHPM is examined from the tables and comparatively it is found that MHPM is more accurate and closer to the exact solutions than the existing methods.

Modified Homotopy Perturbation Method for Solving Fractional Differential Equations

The modified homotopy perturbation method is extended to derive the exact solutions for linear (nonlinear) ordinary (partial) differential equations of fractional order in fluid mechanics. The fractional derivatives are taken in the Caputo sense. This work will present a numerical comparison between the considered method and some other methods through solving various fractional differential equations in applied fields. The obtained results reveal that this method is very effective and simple, accelerates the rapid convergence of the series solution, and reduces the size of work to only one iteration.

The use of homotopy methods for solving nonlinear foam drainage equation

Foaming occurs in many distillation and absorption processes. The drainage of liquid foams involves the interplay of gravity, surface tension, and viscous forces. In this paper, the nonlinear foam drainage equation is solved by using the Adomian's decomposition method, modified Adomian's decomposition method, variational iteration method, modified variational iteration method, homotopy perturbation method, modified homotopy perturbation method and homotopy analysis method. The existence and uniqueness of the solution and convergence of the proposed methods are proved in details. Finally an example shows the accuracy of these methods.