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The main aim of book is to determine the efficiency of two numerical methods (Adomian decomposition method (ADM) and the method of lines (MOL) which depends on finite difference) for solving nonlinear systems of partial differential equations (PDEs). We modified the method of computing Adomian's polynomial to find the numerical solutions for these systems with less number of components and faster formula convergence when compared with the standard ADM while the other aim of this book is a comparison between the numerical results for ADM(standard and modified ADM) and MOL which does not require large computer memory and it has more effect with respect to the accuracy and convergence rapidity compared with the standard and modified ADM.