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In this paper, we develop two finite difference weighted essentiallynon-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving theDegasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, whichcontain nonlinear high order derivatives, and possibly peakon solutions orshock waves. By introducing auxiliary variable(s), we rewrite the DP equationas a hyperbolic-elliptic system, and the \mdp equation as a first order system.Then we choose a linear finite difference scheme with suitable order ofaccuracy for the auxiliary variable(s), and two finite difference WENO schemeswith unequal-sized sub-stencils for the primal variable. One WENO scheme usesone large stencil and several smaller stencils, and the other WENO scheme isbased on the multi-resolution framework which uses a series of unequal-sizedhierarchical central stencils. Comparing with the classical WENO scheme whichuses several small stencils of the same size to make up a big stencil, bothWENO schemes with unequal-sized sub-stencils are simple in the choice of thestencil and enjoy the freedom of arbitrary positive linear weights. Anotheradvantage is that the final reconstructed polynomial on the target cell is apolynomial of the same degree as the polynomial over the big stencil, while theclassical finite difference WENO reconstruction can only be obtained forspecific points inside the target interval. Numerical tests are provided todemonstrate the high order accuracy and non-oscillatory properties of theproposed schemes.

High order finite difference WENO methods with unequal-sized sub-stencils for the Degasperis-Procesi type equations