Gegenbauer approximation and its applications to differential equations with rough asymptotic behaviors at infinity Original Research Article

The Gegenbauer approximation with the index View the MathML source is investigated. Several weighted inverse inequalities and imbedding inequalities are given. Some approximation results in certain Hilbert spaces are obtained. By variable transformation, differential equations on the whole line are changed to certain singular equations on a finite interval. Gegenbauer polynomials are used for numerical solutions of some steady linear equations and the nonlinear Klein–Gordon equation with rough asymptotic behaviors at infinity. The stabilities and convergences of suggested schemes are proved. The numerical results show the advantage of this new approach. The main idea and techniques used in this paper are also applicable to other multiple-dimensional problems with rough asymptotic behaviors at infinity.