Thesis (Ph.D.) - Cairo University - Faculty of Science - Department of Mathematics

The main aim of this thesis is to develop new e{uFB03}cient spectral and wavelets algorithms for solving 2nth-order and (2n+1)th-order linear or nonlinear di{uFB00}erential equations involving some speci{uFB01}c problems such as Lane-Emden, Bratu and Burger type equa- tions. In Chapter 1, we give a brief introduction to the spectral methods and their advantages over the other standard methods. Moreover, an overview on wavelets and their uses in various disciplines is also presented. In Chapter 2, Second kind Cheby- shev operational matrix algorithm is employed for solving di{uFB00}erential equations of Lane-Emden type. In Chapter 3, a novel operational matrix which is expressed in terms of the well-known harmonic numbers and based on shifted Legendre polynomi- als is introduced and used for solving second-order boundary value problems involving singular, singularly perturbed and Bratu type equations. In this chapter, we also dis- cussed the convergence analysis of the proposed method of solution

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