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The Adomian Decomposition Method (ADM) is a very effective approach forsolving broad classes of nonlinear partial and ordinary differential equations,with important applications in different fields of applied mathematics,engineering, physics and biology. It is the goal of the present paper toprovide a clear and pedagogical introduction to the Adomian DecompositionMethod and to some of its applications. In particular, we focus our attentionto a number of standard first-order ordinary differential equations (thelinear, Bernoulli, Riccati, and Abel) with arbitrary coefficients, and presentin detail the Adomian method for obtaining their solutions. In each case wecompare the Adomian solution with the exact solution of some particulardifferential equations, and we show their complete equivalence. The secondorder and the fifth order ordinary differential equations are also considered.An important extension of the standard ADM, the Laplace-Adomian DecompositionMethod is also introduced through the investigation of the solutions of aspecific second order nonlinear differential equation. We also present theapplications of the method to the Fisher-Kolmogorov second order partialnonlinear differential equation, which plays an important role in thedescription of many physical processes, as well as three important applicationsin astronomy and astrophysics, related to the determination of the solutions ofthe Kepler equation, of the Lane-Emden equation, and of the generalrelativistic equation describing the motion of massive particles in thespherically symmetric and static Schwarzschild geometry.

A Brief Introduction to the Adomian Decomposition Method, with Applications in Astronomy and Astrophysics