non-classical sturm-liouville problems

The aim of this paper is to introduce two new extensions of the Jacobi and Laguerre polynomials as the eigenfunctions of two non-classical Sturm-Liouville problems. We prove some important properties of these operators such as: These sets of functions are orthogonal with respect to a positive de nite inner product de ned over the compact intervals [-1, 1] and [0,1), respectively and also these sequences form two new orthog-onal bases for the corresponding Hilbert spaces. Finally, the spectral and Rayleigh-Ritz methods are carry out using these basis functions to solve some examples. Our nu-merical results are compared with other existing results to con rm the eciency and accuracy of our method.