In this paper, we present an ultraspherical wavelets-Gauss collocation method for obtaining direct solutions of first- and second-order nonlinear differential equations subject to homogenous and nonhomogeneous initial conditions. The properties of ultraspherical wavelets are used to reduce the differential equations with their initial conditions to systems of algebraic equations, which then must be solved by using suitable numerical solvers. The function approximations are spectral and have been chosen in such a way that make them easy to calculate the expansion coefficients of the thought-for solutions. Uniqueness and convergence of the proposed function approximation is discussed. Four illustrative numerical examples are considered and these results are comparing favorably with the analytic solutions and proving more accurate than those discussed by some other existing techniques in the literature.
Youssri, Y. H.; Abd-Elhameed, W. M.; and Doha, E. H.
Accurate spectral solutions of first and second-order initial value problems by the ultraspherical wavelets-Gauss collocation method,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 10,
2, Article 13.
Available at: https://digitalcommons.pvamu.edu/aam/vol10/iss2/13