Jacobi Approximations in Certain Hilbert Spaces and Their Applications to Singular Differential Equations

Jacobi approximations in certain Hilbert spaces are investigated. Several weighted inverse inequalities and Poincar´e inequalities are obtained. Some approximation results are given. Singular differential equations are approximated by using Jacobi polynomials. This method keeps the spectral accuracy. Some linear problems and a nonlinear logistic equation are considered. The stabilities and the convergences of proposed schemes are proved strictly. The main idea and techniques used in this paper are also applicable to other singular problems in multiple-dimensional spaces