A Numerical Approach to Solve Lane-emden-type Equations by the Fractional Order of Rational Bernoulli Functions

In this paper, a numerical method based on the hybrid of the quasilinearization method (QLM) and the collocation method is suggested for solving wellknown nonlinear Lane-Emden-type equations as singular initial value problems, which model many phenomena in mathematical physics and astrophysics. First, by using the QLM method, the nonlinear ordinary differential equation is converted into a sequence of linear differential equations, and then the linear equations by the fractional order of rational Bernoulli collocation (FRBC) method on the semi-infinite interval [0,∞) are solved. This method reduces the solution of these problems to the solution of a system of algebraic equations. Computational results of several problems are presented to demonstrate the viability and powerfulness of the method. Further, the fractional order of rational Bernoulli functions has also been used for the first time. The first zeros of standard Lane-Emden equation and the approximations of y(x) for Lane-Emden-type equations are given with unprecedented accuracy.