10,264 research outputs found
A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions
In this paper, the fractional order of rational Bessel functions collocation
method (FRBC) to solve Thomas-Fermi equation which is defined in the
semi-infinite domain and has singularity at and its boundary condition
occurs at infinity, have been introduced. We solve the problem on semi-infinite
domain without any domain truncation or transformation of the domain of the
problem to a finite domain. This approach at first, obtains a sequence of
linear differential equations by using the quasilinearization method (QLM),
then at each iteration solves it by FRBC method. To illustrate the reliability
of this work, we compare the numerical results of the present method with some
well-known results in other to show that the new method is accurate, efficient
and applicable
Fractional Calculus in Wave Propagation Problems
Fractional calculus, in allowing integrals and derivatives of any positive
order (the term "fractional" kept only for historical reasons), can be
considered a branch of mathematical physics which mainly deals with
integro-differential equations, where integrals are of convolution form with
weakly singular kernels of power law type. In recent decades fractional
calculus has won more and more interest in applications in several fields of
applied sciences. In this lecture we devote our attention to wave propagation
problems in linear viscoelastic media. Our purpose is to outline the role of
fractional calculus in providing simplest evolution processes which are
intermediate between diffusion and wave propagation. The present treatment
mainly reflects the research activity and style of the author in the related
scientific areas during the last decades.Comment: 33 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1008.134
Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order
Trigonometric formulas are derived for certain families of associated
Legendre functions of fractional degree and order, for use in approximation
theory. These functions are algebraic, and when viewed as Gauss hypergeometric
functions, belong to types classified by Schwarz, with dihedral, tetrahedral,
or octahedral monodromy. The dihedral Legendre functions are expressed in terms
of Jacobi polynomials. For the last two monodromy types, an underlying
`octahedral' polynomial, indexed by the degree and order and having a
non-classical kind of orthogonality, is identified, and recurrences for it are
worked out. It is a (generalized) Heun polynomial, not a hypergeometric one.
For each of these families of algebraic associated Legendre functions, a
representation of the rank-2 Lie algebra so(5,C) is generated by the ladder
operators that shift the degree and order of the corresponding solid harmonics.
All such representations of so(5,C) are shown to have a common value for each
of its two Casimir invariants. The Dirac singleton representations of so(3,2)
are included.Comment: 44 pages, final version, to appear in Constructive Approximatio
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