An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry
Abbreviated Journal Title
Abstract Appl. Anal.
SPECTRAL-GALERKIN METHOD; NUMERICAL-SOLUTION; YAMABE-EQUATION; JACOBI
We present a numerical method for a class of boundary value problems on the unit interval which feature a type of powerlaw 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 epsilon (-1, infinity), r epsilon (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.
Abstract and Applied Analysis
"An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry" (2014). Faculty Bibliography 2010s. 5075.