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Recent Development of the Homotopy Perturbation Method
J.-H. He
| ABSTRACT.
The homotopy perturbation method is extremely accessible to non-mathematicians
and engineers. The method decomposes a complex problem under study into
a series of simple problems that are easy to be solved.
This note gives an elementary introduction to the basic solution
procedure of the homotopy perturbation method.
Particular attention is paid to constructing a suitable homotopy equation.
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205
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Algorithms for Nonlinear Fractional PDE
Sh. Momani, Z. Odibat and I. Hashim
| ABSTRACT.
Fractional order partial differential equations, as
generalization of classical integer order partial differential
equations, are increasingly used to model problems in fluid flow,
finance and other areas of applications. In this paper we present
a collection of numerical algorithms for the solution of nonlinear
partial differential equations with space- and time-fractional
derivatives. The fractional derivatives are considered in the Caputo
sense. Two numerical examples are given to demonstrate the
effectiveness of the present methods. Results show that
the numerical schemes are very effective and convenient
for solving nonlinear partial differential equations of fractional
order.
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211 |
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Applications of VIM and HPM
Z. Odibat and Sh. Momani
| ABSTRACT.
In this paper, variational iteration and homotopy
perturbation methods that developed for integer-order differential
equations are directly extended to derive explicit and numerical
solutions of various evolution equations with time-fractional
derivatives. The results reveal that the two methods are very
effective and convenient for solving nonlinear partial differential equations of fractional
order.
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227
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New Application of HPM to ZK-MEW equation
J.-Ch. Lan, J.-M. Zhu and Zh.-Y. Ma
| ABSTRACT.
The work presents a derivation of solitary solutions of
the two-dimensional Zakharov--Kuznetsov Modified Equal Width (ZK-MEW)
equation using the homotopy perturbation method.
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235
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Application of HPM to the Bratu-Type Equations
X. Feng, Y. He and J. Meng
| ABSTRACT.
A new algorithm is presented for
solving the Bratu-type equations. The numerical scheme based on
the homotopy perturbation method is deduced. Two boundary value
problems and an initial value problem are given to illustrate
effectiveness and convenience of the proposed scheme. Our results
agree very well with the numerical solutions showing that the
homotopy perturbation method is a promising method.
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243
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He's HPM for the Temperature Distribution
L. Xu
| ABSTRACT.
This paper applies J. H. He's homotopy perturbation method
(HPM) to calculate the temperature distribution in convective
straight fins with temperature-dependent thermal conductivity. The
temperature distribution of straight fins is obtained as a function
of thermo-geometric fin parameter. Comparison with the exact
solution shows that the method is very effective and convenient,
only one iteration leads to an accurate solution.
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253
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Simulation of the Predator-Prey Problem by the HPM
M. S. H. Chowdhury, I. Hashim and R. Roslan
| ABSTRACT.
In this paper, the predator-prey problem is revisited. Previous solution by
homotopy-perturbation method (HPM) is improved by treating the
homotopy-perturbation method as an algorithm in a sequence of
intervals (time steps) called the multistage
homotopy-perturbation method (shortly MHPM). Numerical results
show that the multistage homotopy-perturbation method and the
classical fourth-order Rungge-Kutta (RK4) methods are in complete
agreement.
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263
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Large Deflection of a Cantilever Beam under Point Load
D. D. Ganji, A. Sadighi, H. Tari, M. Gorji and N. Haghparast
| ABSTRACT.
In this study, the homotopy-perturbation method (HPM) is used
to investigate the large deflection of a cantilever beam under
point load at the free end. The vertical and horizontal displacements
of the cantilever beam are conveniently obtained in explicit analytical
forms The main objective of this study is to propose an alternative method
of solution, which does not require small parameters and avoid linearization
and physically unrealistic assumptions. The results show that this method is
very efficient and convenient and can be applied to a large class of practical
problems.
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271
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Frequency-Amplitude Relationship of the Duffing-Harmonic Oscillator
Zh.-L. Tao
| ABSTRACT.
The variational iteration method, the variational method and the
parameter-expanding method are applied to obtain the frequency-amplitude
relationship of the Duffing-harmonic oscillator. The obtained results reveal
that all the three methods are very effective and convenient.
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279
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Analytical Approach to Kawahara Equation
J. Lu
| ABSTRACT.
Variational iteration method and homotopy perturbation
method are introduced to solve the Kawahara equation. Comparison of
the obtained results with the numerical solution shows that both
methods lead to remarkably accurate solutions. The main property of
the both methods is its flexibility and ability to solve nonlinear
equations accurately and conveniently.
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287
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HPM for Multi-Dimensional Nonlinear Coupled System
N. H. Sweilam, M. M. Khader and R. F. Al-Bar
| ABSTRACT.
In this paper, the homotopy perturbation method (HPM)
proposed by J. H. He is adopted for solving multi-dimensional
nonlinear coupled system of parabolic and hyperbolic equations. The
numerical results of the present method are compared with the exact
solution of an artificial multi-dimensional nonlinear coupled system
of parabolic and hyperbolic model to show the efficiency of the
method. Moreover, comparison is made between the results obtained
by the present method and that obtained by the Adomian decomposition
method (ADM). It is found that the present method works extremely
well, very efficient, simple and convenient.
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295
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Application of HPM to Regularization of Scalar Images
Q. Ma, R.-Y. Xing and S.-L. Mei
| ABSTRACT.
The homotopy perturbation method is implemented to solve
nonlinear equations. Based on this method, a multi-step scheme is
constructed for a kind of Hamilton-Jacobi formulations by assuming
the homotopy parameter is a linear function of time. Using this
multi-step scheme, a minimal surface regularization equation is
solved, which designates a regularization process that doesn't
smooth the image with the same weight in all the spatial
directions. Some image denoisying examples illustrate its
effectiveness and convenience.
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305 |
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On the Solution of Stochastic Oscillatory Quadratic Nonlinear Equations
M. A. El-Tawil and A. S. Al-Jihany
| ABSTRACT.
In this paper, nonlinear oscillators under quadratic
nonlinearity with stochastic inputs are considered.
Different methods are used to obtain first order approximations,
namely the WHEP technique, the perturbation method, the Pickard
approximations, the Adomian decompositions and the homotopy perturbation
method (HPM). Some statistical moments are computed for the different
methods using Mathematica 5. Comparisons are illustrated by figures
for different case-studies.
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315
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HPM Solution for Peristaltic Flow of a Third Order Fluid
A. M. Siddiqui, Q. A. Azim, A. Ashraf and Q. K. Ghori
| ABSTRACT.
The peristaltic transport of a third order fluid in a planar channel as well
as in an axisymmetric tube having walls that are transversely displaced by
an infinite, harmonic travelling wave of large wavelength and negligibly
small Reynolds number, has been analyzed using homotopy perturbation
technique. Unlike perturbation method, this method does not restrict the
Deborah number $\Gamma $ to be very large or small and works fairly well
for any choice of $\Gamma $. The expressions for stream function and
pressure rise per wavelength have been obtained up to second order of
approximation.
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331
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Application of the HPM
Z. Z. Ganji, D. D. Ganji, H. Jafari and M. Rostamian
| ABSTRACT.
The homotopy perturbation method (HPM) is applied to solve nonlinear partial
differential equations of fractional orders. The corresponding solutions for
integer orders of the fractional derivatives are found to be special cases
of the fractional differential equations. It is predicted that HPM can be
found widely applicable in engineering.
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341
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Determination of Limit Cycles by Iterated Homotopy Perturbation Method
T. \"{O}z\D i\c s and A. Yildirim
| ABSTRACT.
He's Homotopy Perturbation Method which reduced to an
Iterative Scheme is applied to nonlinear oscillators with strong
nonlinearity. With the method, the iteration scheme provides
excellent approximations to the solutions even though the iteration
can only be done to the first stage.
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349
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Solitary Wave Solutions for a Coupled MKdV System
Y.-Q. Jiang and J.-M. Zhu
| ABSTRACT.
The work presents a derivation of solitary solutions of
a coupled MKdV system using the homotopy perturbation method.
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359
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HPM for Two Point Boundary Value Problems
S.-D. Zhu
| ABSTRACT.
The homotopy perturbation method is applied for solving
two point boundary value problems. In this method a trial function
(initial solution) is chosen with some unknown parameters, which
are identified using the method of weighted residuals. An example
is given, the obtained result is compared with the exact solution,
revealing that this method is very efficient and the obtained
solution is of high accuracy.
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369
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HPM for the Nonlinear Relativistic Toda Lattice Equations
J.-M. Zhu
| ABSTRACT.
The work presents a derivation of solitary wave solutions
of the nonlinear relativistic Toda lattice equations using the
homotopy perturbation method. The work presents a derivation of solitary wave solutions
of the nonlinear relativistic Toda lattice equations using the
homotopy perturbation method.
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373
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Chinese Mathematics for Nonlinear Oscillators
L. Zhao
| ABSTRACT.
Ancient Chinese mathematicians made dramatic progress toward answering
one of the oldest, most fundamental problem of how to solve approximately
a real root of a nonlinear algebra equation in about 2nd century BC.
The idea was further extended to nonlinear differential equations by
J. H. He in 2002. In this paper, J. H. He's frequency-amplitude formation
is used to find periodic solution of a pure nonlinear oscillator (without
a linear term). The obtained result is of remarkable accuracy.
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383
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Application of He's Frequency-Amplitude Formulation
J. Fan
| ABSTRACT.
The work presents a derivation of frequency-amplitude of the
Duffing-harmonic oscillator from a formulation suggested
by Ji-Huan He. The obtained result is valid for all amplitudes,
and its maximal error is less than 2.2\%.
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389
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