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

Abstract EN

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 ( α , β ) ( r ) with α , β ∈ ( - 1 , ∞ ) , r ∈ ( 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.