An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry

Abstract

We present a numerical method for a class of boundary value problems on the unit interval which feature a type of power-law nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is based on shifted Jacobi polynomials $J_n^{(\alpha ,\beta )}(r)$ with $\alpha ,\beta \in (-1,\mathrm{\infty })$, $r\in (\mathrm{0,1})$ and $n$ the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes for the spectral method. After deriving the method for a rather general class of equations, we apply it to several specific examples. One natural example is a nonlinear boundary value problem related to the Yamabe problem which arises in mathematical physics and geometry. A number of specific numerical experiments demonstrate the accuracy and the efficiency of the spectral method. We discuss the extension of the method to account for more complicated forms of nonlinearity.