  The Adomian Decomposition Method (ADM) is a very effective approach for
solving 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 to
provide a clear and pedagogical introduction to the Adomian Decomposition
Method and to some of its applications. In particular, we focus our attention
to a number of standard first-order ordinary differential equations (the
linear, Bernoulli, Riccati, and Abel) with arbitrary coefficients, and present
in detail the Adomian method for obtaining their solutions. In each case we
compare the Adomian solution with the exact solution of some particular
differential equations, and we show their complete equivalence. The second
order and the fifth order ordinary differential equations are also considered.
An important extension of the standard ADM, the Laplace-Adomian Decomposition
Method is also introduced through the investigation of the solutions of a
specific second order nonlinear differential equation. We also present the
applications of the method to the Fisher-Kolmogorov second order partial
nonlinear differential equation, which plays an important role in the
description of many physical processes, as well as three important applications
in astronomy and astrophysics, related to the determination of the solutions of
the Kepler equation, of the Lane-Emden equation, and of the general
relativistic equation describing the motion of massive particles in the
spherically symmetric and static Schwarzschild geometry.
