Numerical approximations for Thomas-Fermi model using radial basis functions

In this paper, an efficient meshless method for solving the Thomas-Fermi model is presented. We consider a generalized Thomas-Fermi equation y″ + (b/x) y′ = cx^py^q, where the constants b, c, p and q satisfy the following conditions: 0 ≤ b < 1, c > 0, p > -2 and q > 1. Problems involving such an equation have been solved by various approximation methods and numerical integration schemes, but most of these methods are either complicated in mathematical formulation or resulted in poor approximations. To overcome these, we employ a class of radial basis functions (RBFs), called multiquadric functions (MQ-RBF), which possess a truly mesh free algorithm and a simple mathematical formulation, to approximate the spatial derivatives of the Thomas-Fermi equation. Since MQ-RBF are continuously differentiate, positive definite and integrable functions, it can easily be used to solve high order differential equations and complicated problems. The results of the given equation are computed iteratively by vising the modified Picard's method. The RBFs scheme has a high degree of accuracy and a fast rate of convergence in the computations.