An efficient collocation method for a class of boundary value problems arising in mathematical physics and geometry (Q1724102)

Summary: 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 \(\alpha, \beta \in(- 1, \infty)\), \(r \in(0,1)\) 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.