Abstract
This paper is devoted to proposing numerical algorithms based on the use of the tau and collocation procedures, two widely used spectral approaches for the numerical treatment of the initial high-order linear and non-linear equations of the singular type, especially those of the high-order Emden-Fowler type. The class of modified Chebyshev polynomials of the third-kind is constructed. This class of polynomials generalizes the class of the third-kind Chebyshev polynomials. A new formula that expresses the first-order derivative of the modified Chebyshev polynomials in terms of their original modified polynomials is established. The establishment of this essential formula is based on reducing a certain terminating hypergeometric function of the type F-5(4)(1). The development of our suggested numerical algorithms begins with the extraction of a new operational derivative matrix from this derivative formula. Expansion's convergence study is performed in detail. Some illustrative examples of linear and non-linear Emden-Flower-type equations of different orders are displayed. Our proposed algorithms are compared with some other methods in the literature. This confirms the accuracy and high efficiency of our presented algorithms.