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

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

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 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

Wronskian perturbation theoryWronskian perturbation theory

We develop a perturbation method that generalizes an approach proposed recently to treat velocity--dependent quantum--mechanical models. In order to test present approach we apply it to some simple trivial and nontrivial examples.Comment: 5 page

Periodic Maximal surfaces in the Lorentz-Minkowski space \l^3

A maximal surface \sb with isolated singularities in a complete flat Lorentzian 3-manifold $\N$ is said to be entire if it lifts to a (periodic) entire multigraph \tilde{\sb} in \l^3. In addition, \sb is called of finite type if it has finite topology, finitely many singular points and \tilde{\sb} is finitely sheeted. Complete and proper maximal immersions with isolated singularities in $\N$ are entire, and entire embedded maximal surfaces in $\N$ with a finite number of singularities are of finite type. We classify complete flat Lorentzian 3-manifolds carrying entire maximal surfaces of finite type, and deal with the topology, Weierstrass representation and asymptotic behavior of this kind of surfaces. Finally, we construct new examples of periodic entire embedded maximal surfaces in \l^3 with fundamental piece having finitely many singularities.Comment: 27 pages, corrected typos, Lemma 2.5 and Theorem 4.1 change

Relative parabolicity of zero mean curvature surfaces in R3R^3 and R13R_1^3

If the Lorentzian norm on a maximal surface in the 3-dimensional Lorentz-Minkowski space $R_1^3$ is positive and proper, then the surface is relative parabolic. As a consequence, entire maximal graphs with a closed set of isolated singularities are relative parabolic. Furthermore, maximal and minimal graphs over closed starlike domains in $R_1^3$ and $R^3,$ respectively, are relative parabolic

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