  In this paper, we develop two finite difference weighted essentially
non-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving the
Degasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, which
contain nonlinear high order derivatives, and possibly peakon solutions or
shock waves. By introducing auxiliary variable(s), we rewrite the DP equation
as a hyperbolic-elliptic system, and the \mdp equation as a first order system.
Then we choose a linear finite difference scheme with suitable order of
accuracy for the auxiliary variable(s), and two finite difference WENO schemes
with unequal-sized sub-stencils for the primal variable. One WENO scheme uses
one large stencil and several smaller stencils, and the other WENO scheme is
based on the multi-resolution framework which uses a series of unequal-sized
hierarchical central stencils. Comparing with the classical WENO scheme which
uses several small stencils of the same size to make up a big stencil, both
WENO schemes with unequal-sized sub-stencils are simple in the choice of the
stencil and enjoy the freedom of arbitrary positive linear weights. Another
advantage is that the final reconstructed polynomial on the target cell is a
polynomial of the same degree as the polynomial over the big stencil, while the
classical finite difference WENO reconstruction can only be obtained for
specific points inside the target interval. Numerical tests are provided to
demonstrate the high order accuracy and non-oscillatory properties of the
proposed schemes.
