6,335 research outputs found
On Some Perturbation Approaches to Population Dynamics
We show that the Adomian decomposition method, the time--series expansion,
the homotopy--perturbation method, and the variational--iteration method
completely fail to provide a reasonable description of the dynamics of the
simplest prey--predator system
Homotopy perturbation method: when infinity equals five
I discuss a recent application of homotopy perturbation method to a heat
transfer problem. I show that the authors make infinity equal five and analyze
the consequences of that magic
About homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients
We analyze a recent application of homotopy perturbation method to some
heat-like and wave-like models and show that its main results are merely the
Taylor expansions of exponential and hyperbolic functions. Besides, the authors
require more boundary conditions than those already necessary for the solution
of the problem by means of power series
Perturbation approaches and Taylor series
We comment on the new trend in mathematical physics that consists of
obtaining Taylor series for fabricated linear and nonlinear unphysical models
by means of homotopy perturbation method (HPM), homotopy analysis method (HAM)
and Adomian decomposition method (ADM). As an illustrative example we choose a
recent application of the HPM to a dynamic system of anisotropic elasticity
Perturbation Theory for Population Dynamics
We prove that a recently proposed homotopy perturbation method for the
treatment of population dynamics is just the Taylor expansion of the population
variables about initial time. Our results show that this perturbation method
fails to provide the global features of the ecosystem dynamics
Rational Approximation for Two-Point Boundary value problems
We propose a method for the treatment of two--point boundary value problems
given by nonlinear ordinary differential equations. The approach leads to
sequences of roots of Hankel determinants that converge rapidly towards the
unknown parameter of the problem. We treat several problems of physical
interest: the field equation determining the vortex profile in a
Ginzburg--Landau effective theory, the fixed--point equation for Wilson's exact
renormalization group, a suitably modified Wegner--Houghton's fixed point
equation in the local potential approximation, a Riccati equation, and the
Thomas--Fermi equation. We consider two models where the approach does not
apply in order to show the limitations of our Pad\'{e}--Hankel approach.Comment: 13 pages, 1 figur
Harmonic oscillator in a one-dimensional box
We study a harmonic molecule confined to a one--dimensional box with
impenetrable walls. We explicitly consider the symmetry of the problem for the
cases of different and equal masses. We propose suitable variational functions
and compare the approximate energies given by the variation method and
perturbation theory with accurate numerical ones for a wide range of values of
the box length. We analyze the limits of small and large box size
Alternative perturbation approaches in classical mechanics
We discuss two alternative methods, based on the Lindstedt--Poincar\'{e}
technique, for the removal of secular terms from the equations of perturbation
theory. We calculate the period of an anharmonic oscillator by means of both
approaches and show that one of them is more accurate for all values of the
coupling constant.Comment: 10 pages, 1 figur
Iterative solution of differential equations
We discuss alternative iteration methods for differential equations. We
provide a convergence proof for exactly solvable examples and show more
convenient formulas for nontrivial problems.Comment: 12 page
Comment to: "The quantum square well with moving boundaries: a numerical analysis"
In this comment we show that the approach presented by Foj\'on et
al~\cite{Fojon10} is not as accurate as they claim. A straightforward
calculation using the models considered buy those authors clearly shows that
the spectral method, which the authors criticize, proves to be considerably
better.Comment: 12 pages, 11 figure
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