  In this work, the analytical solutions of the $D$-dimensional Schr\"odinger
equation are studied in great detail for the Wood-Saxon potential by taking
advantage of the Pekeris approximation. Within a novel improved scheme to
surmount centrifugal term, the energy eigenvalues and corresponding radial wave
functions are found for any angular momentum case within the context of the
Nikiforov-Uvarov (NU) and Supersymmetric quantum mechanics (SUSYQM) methods. In
this way, based on these methods, the same expressions are obtained for the
energy eigenvalues, and the expression of radial wave functions transformed
each other is demonstrated. In addition, a finite number energy spectrum
depending on the depth of the potential $V_{0}$, the radial $n_{r}$ and orbital
$l$ quantum numbers and parameters $D, a, R_{0}$ are defined as well.
