2,129 research outputs found
A note on exact solutions for nonlinear integral equations by a modified homotopy perturbation method
In the paper "Exact solutions for nonlinear integral equations by a modified homotopy perturbation method" by A. Ghorbani and J. Saberi-Nadjafi, Computers and Mathematics with Applications, 56, (2008) 1032-1039, the authors introduced a new modification of the homotopy perturbation method to solve nonlinear integral equations.We discuss here the restrictions on their method for solving nonlinear integral equations. We also prove analytically that the method given by Ghorbani and Saberi-Nadjafi is equivalent to the series solution method when selective functions are polynomials
On the Comparison of Perturbation-Iteration Algorithm and Residual Power Series Method to Solve Fractional Zakharov-Kuznetsov Equation
In this paper, we present analytic-approximate solution of a fractional
Zakharov-Kuznetsov equation by means of perturbation-iteration algorithm (PIA)
and residual power series method (RSPM). Basic definitions of fractional
derivatives are described in the Caputo sense. Several examples are given and
the results are compared to exact solutions. The results show that both methods
are competitive, effective, convenient and simple to use
Numerical Study of Astrophysics Equations by Meshless Collocation Method Based on Compactly Supported Radial Basis Function
In this paper, we propose compactly supported radial basis functions for
solving some well- known classes of astrophysics problems categorized as
non-linear singular initial ordinary dif- ferential equations on a
semi-infinite domain. To increase the convergence rate and to decrease the
collocation points, we use the compactly supported radial basis function
through the integral operations. Afterwards, some special cases of the equation
are presented as test examples to show the reliability of the method. Then we
compare the results of this work with some results and show that the new method
is efficient and applicableComment: 24 pages, 8 figures. arXiv admin note: text overlap with
arXiv:1008.2063, arXiv:1008.231
RBF DQ Method for Solving Nonlinear Differential Equations of Lane Emden type
The Lane-Emden type equations are employed in the modelling of several
phenomena in the areas of mathematical physics and astrophysics . In this paper
a new numerical method is applied to investigate some well-known classes of
Lane-Emden type equations which are nonlinear ordinary differential equations
on the semi-infinite domain. We will apply a mesh-less method based on radial
basis function differential quadrature method. In RBFs-DQ the derivative value
of function with respect to a point is directly approximated by a linear
combination of all functional values in the global domain . The main aim of
this method is the determination of weight coefficients. Here we concentrate on
Gaussian(GS) as a radial function for approximating the solution of the
mentioned equation. The comparison of the results with the other numerical
methods shows the efficiency and accuracy of this method.Comment: 9 figures, 29 pages. arXiv admin note: substantial text overlap with
arXiv:1509.0432
Homotopy Perturbation Method for Solving a Spatially Flat FRW Cosmological Model
In the present paper, we study a homogeneous cosmological model in
Friedmann-Robertson-Walker (FRW) space-time by means of the so-called Homotopy
Perturbation Method (HPM). First, we briefly recall the main equations of the
cosmological model and the basic idea of HPM. Next we consider the test example
when the exact solution of the model is known, in order to approbate the HPM in
cosmology and present the main steps in solving by this method. Finally, we
obtain a solution for the spatially flat FRW model of the universe filled with
the dust and quintessence when the exact solution cannot be found. A comparison
of our solution with the corresponding numerical solution shows that it is of a
high degree of accuracy.Comment: 5 pages, 3 figure
A hybrid natural transform homotopy perturbation method for solving fractional partial differential equations
A hybrid analytical method for solving linear and nonlinear fractional
partial differential equations is presented. The proposed analytical method is
an elegant combination of the Natural Transform Method (NTM) and a well-known
method, Homotopy Perturbation Method (HPM). In this analytical method, the
fractional derivative is computed in Caputo sense and the nonlinear terms are
calculated using He's polynomials. The proposed analytical method reduces the
computational size, avoids round-off errors. Exact solutions of linear and
nonlinear fractional partial differential equations is successfully obtained
using the analytical method.Comment: 8 page
Asymptotic Methods in Non Linear Dynamics
This paper features and elaborates recent developments and modifications in
asymptotic techniques in solving differential equation in non linear dynamics.
These methods are proved to be powerful to solve weakly as well as strongly non
linear cases. Obtained approximate analytical solutions are valid for the whole
solution domain. In this paper, limitations of traditional perturbation methods
are illustrated with various modified techniques. Mathematical tools such as
variational approach, homotopy and iteration technique are discussed to solve
various problems efficiently. Asymptotic methods such as Variational Method,
modified Lindstedt-Poincare method, Linearized perturbation method, Parameter
Expansion method, Homotopy Perturbation method and Perturbation-Iteration
methods(singular and non singular cases) have been discussed in various
situations. Main emphasis is given on Singular perturbation method and WKB
method in various numerical problems.Comment: submit, to appear in Journal of Non Linear Science and Applications,
201
The application of the exact operational matrices for solving the Emden-Fowler equations, arising in astrophysics
The objective of this paper is to apply the well-known exact operational
matrices (EOMs) idea for solving the Emden-Fowler equations, illustrating the
superiority of EOMs versus ordinary operational matrices (OOMs). Up to now, a
few studies have been conducted on EOMs and the differential equations solved
by them do not have high-degree nonlinearity and the reported results are not
regarded as appropriate criteria for the excellence of the new method. So, we
chose Emden-Fowler type differential equations and solved them by this method.
To confirm the accuracy of the new method and to show the preeminence of EOMs
versus OOMs, the norm1 of the residual and error function of both methods are
evaluated for multiple values, where is the degree of the Bernstein
polynomials. We reported the results in form of plots to illustrate the error
convergence of both methods to zero and also to show the primacy of the new
method versus OOMs. The obtained results have demonstrated the increased
accuracy of the new method
Approximate Analytical Solution for the Dynamic Model of Large Amplitude Non-Linear Oscillations Arising in Structural Engineering
In this work we obtain an approximate solution of the strongly nonlinear
second order differential equation , describing the large amplitude free vibrations of a uniform
cantilever beam, by using a method based on the Laplace transform, and the
convolution theorem. By reformulating the initial differential equation as an
integral equation, with the use of an iterative procedure, an approximate
solution of the nonlinear vibration equation can be obtained in any order of
approximation. The iterative approximate solutions are compared with the exact
numerical solution of the vibration equation.Comment: 8 pages, 1 figure, accepted for publication in Journal of Applied
Mathematics and Engineerin
Rational Chebyshev of Second Kind Collocation Method for Solving a Class of Astrophysics Problems
The Lane-Emden equation has been used to model several phenomenas in
theoretical physics, mathematical physics and astrophysics such as the theory
of stellar structure. This study is an attempt to utilize the collocation
method with the Rational Chebyshev of Second Kind function (RSC) to solve the
Lane-Emden equation over the semi-infinit interval [0; +infinity). According to
well-known results and comparing with previous methods, it can be said that
this method is efficient and applicable.Comment: arXiv admin note: text overlap with arXiv:1008.206
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