Diagonalized Gegenbauer rational spectral methods for second- and fourth-order problems on the whole line

Abstract

Fully diagonalized Gegenbauer rational spectral methods for solving second- and fourth-order differential equations on the whole line are proposed and analyzed. Some Gegenbauer rational Sobolev orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Gegenbauer rational series. Optimal error estimates of the fully diagonalized Gegenbauer rational spectral method for second-order problem are obtained. Finally, some numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized methods.