Homotopy Perturbation Method for Solving Fourth-Order Boundary Value Problems
TL;DR: In this paper, the homotopy perturbation method was applied for solving the fourth-order boundary value problems and the analytical results were obtained in terms of convergent series with easily computable components.
Abstract: We apply the homotopy perturbation method for solving the
fourth-order boundary value problems. The analytical results of the boundary value
problems 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. Homotopy method can be considered an alternative method to Adomian
decomposition method and its variant forms.
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TL;DR: In this article, the authors apply He's polynomials to investigate propagating traveling solitary wave solutions of seventh order generalized KdV (SOG-KdV) equations.
Abstract: In this paper, we apply He's polynomials to investigate propagating traveling solitary wave solutions of seventh order generalized KdV (SOG-KdV) equations which play a very important role in mathematical physics and engineering as well. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The suggested iterative scheme leads to the needed solution without any discretization, linearization or restrictive assumptions. The fact that proposed scheme solves nonlinear problems without using the Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.
159 citations
TL;DR: In this paper, the authors apply the variational iteration method using He's polynomials (VIMHP) for solving the higher-order boundary value problems, which is an elegant combination of variational iterative and the homotopy perturbation methods.
Abstract: In this paper, we apply the variational iteration method using He's polynomials (VIMHP) for solving the higher-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discritization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed VIMHP solves nonlinear problems without using the Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.
155 citations
Cites background or methods from "Homotopy Perturbation Method for So..."
...It is well known that series (8) is convergent for most of the cases and also the rate of convergence is dependent on L (u); see [9-15, 20, 26-31]....
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...The embedding parameter ρ e (0, 1] can be considered as an expanding parameter [9-15, 20, 26-31]....
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...The variational iteration and homotopy perturbation methods have been applied to a wide class of functional equations; see [1, 2, 9-37, 42] and the references therein....
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TL;DR: In this paper, a detailed study of some relatively new techniques which are originated by He for solving diversified nonlinear problems of physical nature is presented, focusing on the variational iteration method (VIM) and its modifications, the homotopy perturbation method (HPM), the parameter expansion method and exp-function method.
Abstract: This paper outlines a detailed study of some relatively new techniques which are originated by He for solving diversified nonlinear problems of physical nature. In particular, we will focus on the variational iteration method (VIM) and its modifications, the homotopy perturbation method (HPM), the parameter expansion method, and exp-function method. These relatively new but very reliable techniques proved useful for solving a wide class of nonlinear problems and are capable to cope with the versatility of the physical problems. Several examples are given to reconfirm the efficiency of these algorithms. Some open problems are also suggested for future research work.
144 citations
TL;DR: Higher order boundary value problems can be written as a system of integral equations, which can be solved by using the variational iteration technique, which is considered as alternative and efficient for finding the approximate solutions of the boundary values problems.
142 citations
Cites methods from "Homotopy Perturbation Method for So..."
...which is exactly the same as obtained in [14] by using the homotopy perturbation method....
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...Homotopy perturbation method has been used in [13,14] for the solution of fourth-order boundary value problems....
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TL;DR: This paper applies the homotopy perturbation method for solving the sixth-order boundary value problems by reformulating them as an equivalent system of integral equations by using a suitable transformation.
Abstract: In this paper, we apply the homotopy perturbation method for solving the sixth-order boundary value problems by reformulating them as an equivalent system of integral equations. This equivalent formulation is obtained by using a suitable transformation. The analytical results of the integral 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 homotopy perturbation method. We have also considered an example where the homotopy perturbation method is not reliable.
118 citations
Cites background or methods from "Homotopy Perturbation Method for So..."
...The embedding parameter p ∈ (0, 1] can be considered as an expanding parameter, see [1–7,10,11,13,26,27]....
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...to obtain the solution of a large variety of non-linear problems, see [1–7,10,12–14,26,27,42–44] and the references therein....
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...It is well known that perturbation methods provide the most versatile tools available in the non-linear analysis of engineering problems, see [1–13]....
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References
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Book•
01 Jan 1981
TL;DR: In this paper, the authors introduce the notion of forced Oscillations of the Duffing Equation and the Mathieu Equation for weakly nonlinear systems with quadratic and cubic nonlinearities.
Abstract: Algebraic Equations. Integrals. The Duffing Equation. The Linear Damped Oscillator. Self-Excited Oscillators. Systems with Quadratic and Cubic Nonlinearities. General Weakly Nonlinear Systems. Forced Oscillations of the Duffing Equation. Multifrequency Excitations. The Mathieu Equation. Boundary-Layer Problems. Linear Equations with Variable Coefficients. Differential Equations with a Large Parameter. Solvability Conditions. Appendices. Bibliography. Index.
3,020 citations
"Homotopy Perturbation Method for So..." refers background in this paper
...It is well known that perturbation methods [1, 2] provide the most versatile tools available in nonlinear analysis of engineering problems....
[...]
TL;DR: In this paper, a variational iteration method for non-linear problems is proposed, where the problems are initially approximated with possible unknowns and a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.
Abstract: In this paper, a new kind of analytical technique for a non-linear problem called the variational iteration method is described and used to give approximate solutions for some well-known non-linear problems. In this method, the problems are initially approximated with possible unknowns. Then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory. Being different from the other non-linear analytical methods, such as perturbation methods, this method does not depend on small parameters, such that it can find wide application in non-linear problems without linearization or small perturbations. Comparison with Adomian’s decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method.
2,371 citations
TL;DR: In this paper, a survey of recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones, is presented.
Abstract: This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.
2,135 citations
"Homotopy Perturbation Method for So..." refers background or methods in this paper
...These facts have motivated to suggest alternative techniques such as the homotopy analysis method [3, 4], decomposition and the variational iteration method [5–8]....
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...The embedding parameter can be considered as an expanding parameter [7]....
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...Such type of boundary value problems arise in the mathematical modeling of the viscoelastic flows, deformation of beams, and plate deflection theory and other branches of mathematical, physical, and engineering sciences, see [10, 11, 7, 12, 8] and the references therein....
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...The homotopy perturbation method uses the homotopy parameter p as an expanding parameter [7] to obtain...
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...7) is convergent for most of the cases and also the rate of convergence is dependent on L(u), see [7]....
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TL;DR: In this paper, a simple non-linear equation is used to describe a kind of analytical technique for nonlinear problems, which is based on both homotopy in topology and the Maclaurin series.
Abstract: One simple, typical non-linear equation is used in this paper to describe a kind of analytical technique for non-linear problems. This technique is based on both homotopy in topology and the Maclaurin series. In contrast to perturbation techniques, the proposed method does not require small or large parameters. The example shows that the proposed method can give much better approximations than those given by perturbation techniques. In addition the proposed method can be used to obtain formulae uniformly valid for both small and large parameters in non-linear problems.
565 citations
"Homotopy Perturbation Method for So..." refers methods in this paper
...These facts have motivated to suggest alternative techniques such as the homotopy analysis method [3, 4], decomposition and the variational iteration method [5–8]....
[...]
TL;DR: In this article, a new iteration method is proposed to solve nonlinear problems with convolution product nonlinearities and the results reveal that the approximations obtained by the proposed method are uniformly valid for both small and large parameters in non-linear problems.
490 citations