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1 – 10 of 120Manel Labidi and Khaled Omrani
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
To assess the accuracy of the solutions, the comparison of the obtained results with the exact solutions reveals that both methods are tremendously effective.
Originality/value
The paper shows that VIM and HPM can be implemented to solve mCH and mDP equations.
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Keywords
Praveen Kumar Gupta, A. Yildirim and K.N. Rai
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…
Abstract
Purpose
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.
Design/methodology/approach
By using initial values, the explicit solutions of the equation are solved with powerful mathematical tools such as He's homotopy perturbation method (HPM).
Findings
This result reveals that the HPM demonstrates the effectiveness, validity, potentiality and reliability of the method in reality and gives the exact solution.
Originality/value
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.
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S. Karimi Vanani, A. Yildirim, F. Soleymani, M. Khan and S. Tutkun
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
This research presents, for the first time, a new modification of the proposed technique, for aforementioned problems and some interesting results are obtained.
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A. Barari, B. Ganjavi, M. Ghanbari Jeloudar and G. Domairry
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
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.
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The purpose of this paper is to solve an unsteady nonlinear convective‐radiative equation and a nonlinear convective‐radiative‐conduction equation containing two small parameters…
Abstract
Purpose
The purpose of this paper is to solve an unsteady nonlinear convective‐radiative equation and a nonlinear convective‐radiative‐conduction equation containing two small parameters of ε1 and ε2 by variational homotopy perturbation method.
Design/methodology/approach
The heat transfer equations are described. The variational homotopy perturbation method as a powerful method for solving linear and nonlinear equations is applied to find the solutions of our model equations.
Findings
The solutions of heat transfer equations are calculated in the form of convergent series with easily computable components. Two examples are solved as illustrations, using symbolic computation.
Originality/value
The results show that the suggested method is easy to implement and has high level of accuracy. The method introduces a reliable tool for solving many linear and nonlinear differential equations.
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Sefa Anıl Sezer, Ahmet Yıldırım and Syed Tauseef Mohyud‐Din
The purpose of this paper is to directly extend the homotopy perturbation method (HPM) that was developed for integer‐order differential equation, to derive explicit and numerical…
Abstract
Purpose
The purpose of this paper is to directly extend the homotopy perturbation method (HPM) that was developed for integer‐order differential equation, to derive explicit and numerical solutions of the fractional KdV‐Burgers‐Kuramoto equation.
Design/methodology/approach
The authors used Maple Package to calculate the functions obtained from the HPM.
Findings
The fractional derivatives are described in the Caputo sense. HPM performs extremely well in terms of accuracy, efficiently, simplicity, stability and reliability.
Originality/value
The paper describes how the HPM has been successfully applied to find the solution of fractional KdV‐Burgers‐Kuramoto equation.
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This study aims to purpose the idea of a new hybrid approach to examine the approximate solution of the fourth-order partial differential equations (PDEs) with time fractional…
Abstract
Purpose
This study aims to purpose the idea of a new hybrid approach to examine the approximate solution of the fourth-order partial differential equations (PDEs) with time fractional derivative that governs the behaviour of a vibrating beam. The authors have also demonstrated the physical representations of the problem in different fractional order.
Design/methodology/approach
Mohand transform is a new technique that the authors use to reduce the order of fractional problems, and then the homotopy perturbation method can be used to handle the further series solution in the form of convergence. The formulation of Mohand transform and the homotopy perturbation method is known as Mohand homotopy perturbation transform (MHPT). The fractional order in this paper is considered in the Caputo sense.
Findings
The results are formulated in the shape of iterative series and predict the solution close to the exact solution. This successive iteration demonstrates the authenticity and reliability of this scheme.
Research limitations/implications
This paper presents the significance of MHPT such that, firstly, Mohand transform is coupled with homotopy perturbation method and, secondly, the fractional order a is used to show the physical behaviour of the graphical solution.
Practical implications
This study presents the consistency and authenticity of the graphical solution with the exact solutions.
Social implications
This study demonstrates that Mohand transform is capable to handle the fractional order problem without any constraints and assumptions.
Originality/value
A new integral transform has been introduced without any restriction of variables that produces the results in a series form and confirms the validity of the proposed algorithm by graphical illustrations.
Details
Keywords
– The purpose of this paper is to present a general framework of Homotopy perturbation method (HPM) for analytic inverse heat source problems.
Abstract
Purpose
The purpose of this paper is to present a general framework of Homotopy perturbation method (HPM) for analytic inverse heat source problems.
Design/methodology/approach
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.
Findings
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.
Originality/value
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.
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Keywords
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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
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.
Originality/value
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.
Details
Keywords
Hüseyin Koçak, Turgut Öziş and Ahmet Yıldırım
This paper aims to apply He's homotopy perturbation method (HPM) to obtain solitary solutions for the nonlinear dispersive equations with fractional time derivatives.
Abstract
Purpose
This paper aims to apply He's homotopy perturbation method (HPM) to obtain solitary solutions for the nonlinear dispersive equations with fractional time derivatives.
Design/methodology/approach
The authors choose as an example the nonlinear dispersive and equations with fractional time derivatives to illustrate the validity and the advantages of the proposed method.
Findings
The paper extends the application of the HPM to obtain analytic and approximate solutions to the nonlinear dispersive equations with fractional time derivatives.
Originality/value
This paper extends the HPM to the equation with fractional time derivative.
Details