Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 542839, 9 pages http://dx.doi.org/10.1155/2013/542839 Research Article New Wavelets Collocation Method for Solving Second-Order Multipoint Boundary Value Problems Using Chebyshev Polynomials of Third and Fourth Kinds W. M. Abd-Elhameed, 1,2 E. H. Doha, 2 and Y. H. Youssri 2 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt Correspondence should be addressed to W. M. Abd-Elhameed; walee 9@yahoo.com Received 7 August 2013; Revised 13 September 2013; Accepted 13 September 2013 Academic Editor: Soheil Salahshour Copyright © 2013 W. M. Abd-Elhameed et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis paper is concerned with introducing two wavelets collocation algorithms for solving linear and nonlinear multipoint boundary value problems. Te principal idea for obtaining spectral numerical solutions for such equations is employing third- and fourth- kind Chebyshev wavelets along with the spectral collocation method to transform the diferential equation with its boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefcients which can be efciently solved. Convergence analysis and some specifc numerical examples are discussed to demonstrate the validity and applicability of the proposed algorithms. Te obtained numerical results are comparing favorably with the analytical known solutions. 1. Introduction Spectral methods are one of the principal methods of dis- cretization for the numerical solution of diferential equa- tions. Te main advantage of these methods lies in their accuracy for a given number of unknowns (see, e.g., [1– 4]). For smooth problems in simple geometries, they ofer exponential rates of convergence/spectral accuracy. In con- trast, fnite diference and fnite-element methods yield only algebraic convergence rates. Te three most widely used spec- tral versions are the Galerkin, collocation, and tau methods. Collocation methods [5, 6] have become increasingly popular for solving diferential equations, also they are very useful in providing highly accurate solutions to nonlinear diferential equations. Many practical problems arising in numerous branches of science and engineering require solving high even-order and high odd-order boundary value problems. Legendre poly- nomials have been previously used for obtaining numerical spectral solutions for handling some of these kinds of prob- lems (see, e.g., [7, 8]). In [9], the author has constructed some algorithms by selecting suitable combinations of Legendre polynomials for solving the diferentiated forms of high- odd-order boundary value problems with the aid of Petrov- Galerkin method, while in the two papers [10, 11], the authors handled third- and ffh-order diferential equations using Jacobi tau and Jacobi collocation methods. Multipoint boundary value problems (BVPs) arise in a variety of applied mathematics and physics. For instance, the vibrations of a guy wire of uniform cross-section composed of  parts of diferent densities can be set up as a multipoint BVP, as in [12]; also, many problems in the theory of elastic stability can be handled by the method of multipoint problems [13]. Te existence and multiplicity of solutions of multipoint boundary value problems have been studied by many authors; see [14–17] and the references therein. For two-point BVPs, there are many solution methods such as orthonormalization, invariant imbedding algorithms, fnite diference, and collocation methods (see, [18–20]). However, there seems to be little discussion about numerical solutions of multipoint boundary value problems. Second-order multipoint boundary value problems (BVP) arise in the mathematical modeling of defection of can- tilever beams under concentrated load [21, 22], deformation