Gegenbauer Approximation and Its Applications to Differential Equations on the Whole Line

A Gegenbauer approximation is discussed. Several imbedding inequalities and inverse inequalities are obtained. Some approximation results are given. By variable transformation, differential equations on the whole line are changed to certain equations on a finite interval. Gegenbauer polynomials are used for their numerical solutions. The stabilities and convergences of proposed schemes are proved. The main idea and techniques used in this paper are also applicable to other multiple-dimensional problems in unbounded domains