An efficient collocation method for a class of boundary value problems arising in mathematical physics and geometry
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.
scientific article; zbMATH DE number 7022369
scientific article