In this paper, a new spectral algorithm based on employing ultraspherical wavelets along
with the spectral collocation method is developed. The proposed algorithm is utilized to solve linear
and nonlinear even-order initial and boundary value problems. This algorithm is supported by study-
ing the convergence analysis of the used ultraspherical wavelets expansion. The principle idea for
obtaining the proposed spectral numerical solutions for the above-mentioned problems is actually
based on using wavelets collocation method to reduce the linear or nonlinear differential equations
with their initial or boundary conditions into systems of linear or nonlinear algebraic equations in
the unknown expansion coefficients. Some specific important problems such as Lane–Emden and
Burger's type equations can be solved efficiently with the suggested algorithm. Some numerical ex-
amples are given for the sake of testing the efficiency and the applicability of the proposed algorithm