126 research outputs found
Periodic solutions with nonconstant sign in Abel equations of the second kind
The study of periodic solutions with constant sign in the Abel equation of
the second kind can be made through the equation of the first kind. This is
because the situation is equivalent under the transformation ,
and there are many results available in the literature for the first kind
equation. However, the equivalence breaks down when one seeks for solutions
with nonconstant sign. This note is devoted to periodic solutions with
nonconstant sign in Abel equations of the second kind. Specifically, we obtain
sufficient conditions to ensure the existence of a periodic solution that
shares the zeros of the leading coefficient of the Abel equation. Uniqueness
and stability features of such solutions are also studied.Comment: 10 page
Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)
We provide a new mathematical technique leading to the construction of the exact parametric or closed form solutions of the classes of Abel's nonlinear differential equations (ODEs) of the first kind. These solutions are given implicitly in terms of Bessel functions of the first and the second kind (Neumann functions), as well as of the free member of the considered ODE; the parameter being introduced furnishes the order of the above Bessel functions and defines also the desired solutions of the considered ODE as one-parameter family of surfaces. The nonlinear initial or boundary value problems are also investigated. Finally, introducing a relative mathematical methodology, we construct the exact parametric or closed form solutions for several degenerate Abel's equation of the first kind
Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in non-linear mechanics
Self-confined light waves in nematic liquid crystals
The study of light beams propagating in the nonlinear, dispersive, birefringent and nonlocal medium of nematic liquid crystals has attracted widespread interest in the last twenty years or so. We review hereby the underlying physics, theoretical modelling and numerical approximations for nonlinear beam propagation in planar cells filled with nematic liquid crystals, including bright and dark solitary waves, as well as optical vortices. The pertinent governing equations consist of a nonlinear Schrödinger-type equation for the light beam and an elliptic equation for the medium response. Since the nonlinear and coupled nature of this system presents difficulties in terms of finding exact solutions, we outline the various approaches used to resolve them, pinpointing the good agreement obtained with numerical solutions and experimental results. Measurement and material details complement the theoretical narration to underline the power of the modelling
Classes of solutions in the problem of stability analysis in bars with varying cross-section and axial distributed loading
Nonlinear and buckling analysis of continuous bars lying on elastic supports, based on the theory of elastica
Exact analytic solutions of unsolvable classes of first and second order nonlinear ODEs (Part I: Abel's equations)
AbstractWe provide a mathematical technique leading to the construction of exact analytic solutions of the classes of Abel’s nonlinear ordinary differential equations (ODEs) of the second kind, as well as of the restricted form of the Abel nonlinear ODE of the first kind
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