Research Article Modified Decomposition Method with New Inverse Differential Operators for Solving Singular Nonlinear IVPs in First- and Second-Order PDEs Arising in Fluid Mechanics Nemat Dalir Department of Mechanical Engineering, Salmas Branch, Islamic Azad University, Salmas, Iran Correspondence should be addressed to Nemat Dalir; dalir@aut.ac.ir Received 15 February 2014; Accepted 6 June 2014; Published 19 June 2014 Academic Editor: Onesimo Hernandez-Lerma Copyright © 2014 Nemat Dalir. his 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. Singular nonlinear initial-value problems (IVPs) in irst-order and second-order partial diferential equations (PDEs) arising in luid mechanics are semianalytically solved. To achieve this, the modiied decomposition method (MDM) is used in conjunction with some new inverse diferential operators. In other words, new inverse diferential operators are developed for the MDM and used with the MDM to solve irst- and second-order singular nonlinear PDEs. he results of the solutions by the MDM together with new inverse operators are compared with the existing exact analytical solutions. he comparisons show excellent agreement. 1. Introduction Singular nonlinear partial diferential equations (PDEs) arise in various physical phenomena in applied sciences and engineering from such areas as luid mechanics and heat transfer, Riemannian geometry, applied probability, math- ematical physics, and biology. he Adomian decomposi- tion method (ADM) and modiied decomposition method (MDM) are semianalytical methods that give approximate analytical solutions for the diferential equations. MDM was irst developed by Wazwaz and El-Seyed [1] who applied it to solve the ordinary diferential equations (ODEs). Since then the MDM has been used for solving various equations in mathematics and physics [2–4], boundary value problems [5– 9], various problems in engineering [10–13], and initial-value problems [14–17]. Adomian et al. [14] solved the Lane-Emden equation using the MDM. Wazwaz [15] investigated singular initial-value problems, linear and nonlinear, homogeneous and nonhomogeneous, by using the ADM. Hasan and Zhu [16] reported the solution of singular nonlinear initial-value problems in ordinary diferential equations (ODEs) by the ADM. Wu [17] extended the ADM for the calculations of the nondiferentiable functions in nonsmooth initial-value prob- lems. His iteration procedure was based on Jumarie Taylor series. Abassy [18] introduced a qualitative improvement in the ADM for solving nonlinear nonhomogenous initial-value problems. Lin et al. [19], based on a new deinition of the Ado- mian polynomials and the two-step Adomian decomposition method combined with the Pade technique, proposed a new algorithm to construct accurate analytical approximations of nonlinear diferential equations with initial conditions. Wazwaz et al. [20] used the ADM to handle the integral form of the Lane-Emden equations with initial values and boundary conditions. To the best knowledge of author, till now, no one has attempted the modiied decomposition method on solv- ing singular nonlinear partial diferential equations. Our motivation in the present study is to improve the MDM by new developed inverse diferential operators to obtain approximate analytical solutions to the singular nonlinear initial-value problems in irst- and second-order PDEs. 2. Application of MDM for Solving Singular Nonlinear PDEs 2.1. General First-Order Singular Nonlinear PDEs. Consider the following general irst-order (in ) singular nonlinear PDE:   +   =(,,   ), (1) Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2014, Article ID 793685, 7 pages http://dx.doi.org/10.1155/2014/793685