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The purpose of this paper is to present a weighted algorithm based on the homotopy perturbation method for solving the heat transfer equation in the cast‐mould heterogeneous domain.

A weighted algorithm based on the homotopy perturbation method is used to minimize the volume of computations. The authors show that this technique yields the analytical solution of the desired problem in the form of a rapidly convergent series with easily computable components.

The authors illustrate that the proposed method produces satisfactory results with respect to Adomian decomposition method and standard homotopy perturbation method. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability.

This research presents, for the first time, a new modification of the proposed technique, for aforementioned problems and some interesting results are obtained.

The purpose of this paper is to determine the fin efficiency of convective straight fins with temperature dependent thermal conductivity by using Homotopy Perturbation Method.

Most engineering problems, especially heat transfer equations are in nonlinear form. Homotopy Perturbation Method (HPM) has been applied to solve a wide series of nonlinear differential equations. In this paper, HPM is used for obtaining the fin efficiency of convective straight fins with temperature‐dependent thermal conductivity. Comparison of the results with those of Homotopy Perturbation Method, exact solution, numerical results and Adomian's decomposition method (ADM) were been done by Cihat Arslanturk.

Results show that both Homotopy Perturbation Method and ADM applied to the nonlinear equations were capable of solving them with successive rapidly convergent approximations without any restrictive assumptions or transformations causing changes in the physical properties of the problem. Moreover, adding up the number of iterations leads to explicit solution for the problem. The results are just obtained with two iterations. This shows the accuracy and great potential of this method. Finally, it can be seen that, with increase of thermo‐geometric fin parameter (v), the fin efficiency increases too.

The results demonstrate good validity and great potential of the HPM for Heat Transfer equations in engineering problems.

The purpose of this paper is to introduce a new modified version of the homotopy perturbation method (NMHPM) and Adomian decomposition method (ADM) for solving the nonlinear ordinary differential equation arising in MHD non-Newtonian fluid flow over a linear stretching sheet.

The governing equation is solved analytically by applying a newly developed optimal homotopy perturbation approach and ADM. This optimal approach contains convergence-control parameter and is computationally rather efficient. The results of numerical example are presented and only a few terms are required to obtain accurate solutions.

A new modified optimal and ADM methods accelerate the rapid convergence of the series solution. These methods dramatically reduce the size of work. The obtained series solution is combined with the diagonal Padé approximants to handle the boundary condition at infinity. Results derived from these methods are shown graphically and in tabulated forms to study the efficiency and accuracy.

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. The formulation of mathematical model for these processes can be based on the equations of non-Newtonian fluid mechanics. The flow of an electrically conducting fluid in the presence of a magnetic field is of importance in various areas of technology and engineering such as MHD power generation, MHD flow meters, MHD pumps, etc. It is generally admitted that a number of astronomical bodies (e.g. the sun, Earth, Jupiter, Magnetic stars, Pulsars) posses fluid interiors and (or least surface) magnetic fields.

The present results are original and new for the MHD non-Newtonian fluid flow over a linear stretching sheet. The results attained in this paper confirm the idea that NMHPM and ADM are powerful mathematical tools and that can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

This purpose of this paper is to find the approximate analytical solutions of a multidimensional partial differential equation such as Helmholtz equation with space fractional derivatives α,β,γ (1<α,β,γ≤2). The fractional derivatives are described in the Caputo sense.

By using initial values, the explicit solutions of the equation are solved with powerful mathematical tools such as He's homotopy perturbation method (HPM).

This result reveals that the HPM demonstrates the effectiveness, validity, potentiality and reliability of the method in reality and gives the exact solution.

The most important part of this method is to introduce a homotopy parameter (p), which takes values from [0,1]. When p=0, the equation usually reduces to a sufficiently initial form, which normally admits a rather simple solution. When p→1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation. Here, we also discuss the approximate analytical solution of multidimensional fractional Helmholtz equation.

The purpose of this paper is to provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. Combination of the Laplace transform and homotopy perturbation methods (LTHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.

The authors present the solution of nonlinear Boussinesq equation by combination of Laplace transform and new homotopy perturbation methods. An important property of the proposed method, which is clearly demonstrated in example, is that spectral accuracy is accessible in solving specific nonlinear nonlinear Boussinesq equation which has analytic solution functions.

The authors proposed a combination of Laplace transform method and homotopy perturbation method to solve the one-dimensional Boussinesq equation. The results are found to be in excellent agreement. The results show that the LTNHPM is an effective mathematical tool which can play a very important role in nonlinear sciences.

The authors provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. In this work combination of Laplace transform and new homotopy perturbation methods (LTNHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.

The purpose of this paper is to implement variational iteration method (VIM) and homotopy perturbation method (HPM) to solve modified Camassa‐Holm (mCH) and modified Degasperis‐Procesi (mDP) equations.

Perturbation method is a traditional method depending on a small parameter which is difficult to be found for real‐life nonlinear problems. To overcome the difficulties and limitations of the above method, two new ones have recently been introduced by He, i.e. VIM and HPM. In this paper, mCH and mDP equations are solved through these methods.

To assess the accuracy of the solutions, the comparison of the obtained results with the exact solutions reveals that both methods are tremendously effective.

The paper shows that VIM and HPM can be implemented to solve mCH and mDP equations.

In the last two decades with the rapid development of nonlinear science, there has appeared ever‐increasing interest of scientists and engineers in the analytical techniques for nonlinear problems. This paper considers linear and nonlinear systems that are not only regarded as general boundary value problems, but also are used as mathematical models in viscoelastic and inelastic flows. The purpose of this paper is to present the application of the homotopy‐perturbation method (HPM) and variational iteration method (VIM) to solve some boundary value problems in structural engineering and fluid mechanics.

Two new but powerful analytical methods, namely, He's VIM and HPM, are introduced to solve some boundary value problems in structural engineering and fluid mechanics.

Analytical solutions often fit under classical perturbation methods. However, as with other analytical techniques, certain limitations restrict the wide application of perturbation methods, most important of which is the dependence of these methods on the existence of a small parameter in the equation. Disappointingly, the majority of nonlinear problems have no small parameter at all. Furthermore, the approximate solutions solved by the perturbation methods are valid, in most cases, only for the small values of the parameters. In the present study, two powerful analytical methods HPM and VIM have been employed to solve the linear and nonlinear elastic beam deformation problems. The results reveal that these new methods are very effective and simple and do not require a large computer memory and can also be used for solving linear and nonlinear boundary value problems.

The results revealed that the VIM and HPM are remarkably effective for solving boundary value problems. These methods are very promoting methods which can be wildly utilized for solving mathematical and engineering problems.

This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense.

Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique.

HPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability.

The essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplied form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

The purpose of this paper is to present the problem of three‐dimensional flow of a fluid of constant density forced through the porous bottom of a circular porous slider moving laterally on a flat plate.

The transformed nonlinear ordinary differential equations are solved via the homotopy perturbation method (HPM) for small as well as moderately large Reynolds numbers. The convergence of the obtained HPM solution is carefully analyzed. Finally, the validity of results is verified by comparing with numerical methods and existing numerical results.

Close agreement of the two sets of results is observed, thus demonstrating the accuracy of the HPM approach for the particular problem considered.

Interesting conclusions which can be drawn from this study are that HPM is very effective and simple compared to the existing solution method, able to solve problems without using Padé approximants and can therefore be considered as a clear advantage over the N.M. Bujurke and Phan‐Thien techniques.

The purpose of this paper is to present a general framework of Homotopy perturbation method (HPM) for analytic inverse heat source problems.

The proposed numerical technique is based on HPM to determine a heat source in the parabolic heat equation using the usual conditions. Then this shows the pertinent features of the technique in inverse problems.

Using this HPM, a rapid convergent sequence which tends to the exact solution of the problem can be obtained. And the HPM does not require the discretization of the inverse problems. So HPM is a powerful and efficient technique in finding exact and approximate solutions without dispersing the inverse problems.

The essential idea of this method is to introduce a homotopy parameter p which takes values from 0 to 1. When p=0, the system of equations usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.