Applied Mathematics, 2014, 5, 2360-2369 Published Online August 2014 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2014.515228 How to cite this paper: Khader, M.M., Mahdy, A.M.S. and Shehata, M.M. (2014) An Integral Collocation Approach Based on Legendre Polynomials for Solving Riccati, Logistic and Delay Differential Equations. Applied Mathematics, 5, 2360-2369. http://dx.doi.org/10.4236/am.2014.515228 An Integral Collocation Approach Based on Legendre Polynomials for Solving Riccati, Logistic and Delay Differential Equations M. M. Khader 1,2 , A. M. S. Mahdy 3 , M. M. Shehata 3 1 Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, KSA 2 Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt 3 Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt Email: mohamedmbd@yahoo.com , amr_mahdy85@yahoo.com , dr.maha_32@hotmail.com Received 28 May 2014; revised 4 July 2014; accepted 16 July 2014 Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract In this paper, we propose and analyze some schemes of the integral collocation formulation based on Legendre polynomials. We implement these formulae to solve numerically Riccati, Logistic and delay differential equations with variable coefficients. The properties of the Legendre polynomials are used to reduce the proposed problems to the solution of non-linear system of algebraic equa- tions using Newton iteration method. We give numerical results to satisfy the accuracy and the applicability of the proposed schemes. Keywords Integral Collocation Formulation, Spectral Method, Riccati, Logistic and Delay Differential Equations 1. Introduction It is well known that the ordinary differential equations (ODEs) have been the focus of many studies due to their frequent appearance in various applications, such as in fluid mechanics, viscoelasticity, biology, physics and engineering applications, for more details, for example [1]-[5]. Consequently, considerable attention has been given to the efficient numerical solutions of ODEs of physical interest, because it is difficult to find exact solutions. Different numerical methods have been proposed in the literature for solving ODEs [6]-[13]. The Riccati differential equation (RDE) is named after the Italian Nobleman Count Jacopo Francesco Riccati