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Adomians decomposition method is a useful and powerful method for solving linear and nonlinear ordinary differential equations. In this book we consider singular initial value problems, singular boundary value problems, boundary value problems and nonlinear oscillatory equations. Singular initialvalue problems in secondorder ordinary differential equations of LaneEmden type are investigated.In this study, we shall introduce a further development in the Adomian decomposition method to overcome the difficulty at linear and nonlinear LaneEmdenlike equations; especially when the singularity appears two times on the left hand side of this type of equations. A more general concept of this type of initialvalue problems is introduced with various examples. The numerical results of these examples are compared with exact solutions when available. A fast and accurate algorithm is developed for the solution of higher order boundary value problems with boundary conditions. A modified form of the Adomian decomposition method is applied to construct the numerical solution for such problems.
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