New ultraspherical wavelets collocation method for solving 2nth-order initial and boundary value problems

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 studying 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 examples are given for the sake of testing the efficiency and the applicability of the proposed algorithm.