On a novel iterative method to compute polynomial approximations to Bessel functions of the first kind and its connection to the solution of fractional diffusion/diffusion-wave problems

We present an iterative method to obtain approximations to Bessel functions of the first kind $J_p(x)$ ($p>-1$) via the repeated application of an integral operator to an initial seed function $f_0(x)$. The class of seed functions $f_0(x)$ leading to sets of increasingly accurate approximations $f_n(x)$ is considerably large and includes any polynomial. When the operator is applied once to a polynomial of degree $s$, it yields a polynomial of degree $s+2$, and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree. We focus on the set of polynomial approximations generated from the seed function $f_0(x)=1$. This set of polynomials is not only useful for the computation of $J_p(x)$, but also from a physical point of view, as it describes the long-time decay modes of certain fractional diffusion and diffusion-wave problems.Comment: 14 pages, 4 figures. To be published in J. Phys. A: Math. Theo

An accurate closed-form approximate solution for the quintic Duffing oscillator equation

An accurate closed-form solution for the quintic Duffing equation is obtained using a cubication method. In this method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude. The replacement of the original nonlinear equation by an approximate cubic Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn, respectively. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is lower than 0.37%.This work was supported by the “Ministerio de Ciencia e Innovación” of Spain, under project FIS2008-05856-C02-02 and by the “Vicerrectorado de Tecnología e Innovación Educativa” of the University of Alicante, Spain (GITE-09006-UA)