  The biharmonic equation, as well as its nonlinear and inhomogeneous
generalizations, plays an important role in engineering and physics. In
particular the focusing biharmonic nonlinear Schr\"{o}dinger equation, and its
standing wave solutions, have been intensively investigated. In the present
paper we consider the applications of the Laplace-Adomian and Adomian
Decomposition Methods for obtaining semi-analytical solutions of the
generalized biharmonic equations of the type $\Delta ^{2}y+\alpha \Delta
y+\omega y+b^{2}+g\left( y\right) =f$, where $\alpha $, $\omega $ and $b$ are
constants, and $g$ and $f$ are arbitrary functions of $y$ and the independent
variable, respectively. After introducing the general algorithm for the
solution of the biharmonic equation, as an application we consider the
solutions of the one-dimensional and radially symmetric biharmonic standing
wave equation $\Delta ^{2}R+R-R^{2\sigma +1}=0$, with $\sigma = {\rm
constant}$. The one-dimensional case is analyzed by using both the
Laplace-Adomian and the Adomian Decomposition Methods, respectively, and the
truncated series solutions are compared with the exact numerical solution. The
power series solution of the radial biharmonic standing wave equation is also
obtained, and compared with the numerical solution.
