Main aim of the current study is to examine the outcomes of nonlinear partial modified Degasperis-Procesi equation of arbitrary order by using two analytical methods. Both methods are based on homotopy and a novel adjustment with generalized Laplace transform operator. Nonlinear terms are handled by using He's polynomials. The fractional order modified Degasperis-Procesi (FMDP) equation, is capable to describe the nonlinear aspects of dispersive waves. The Katugampola derivative of fractional order in the caputo type is employed to model this problem. The numerical results and graphical representation demonstrate the efficiency and accuracy of applied techniques.

In this paper, we present a computational algorithm, namely, local fractional natural homotopy analysis method (LFNHAM) to explore the solutions of local fractional coupled Helmholtz and local fractional coupled Burgers' equations (LFCHEs and LFCBEs). This work also investigates the uniqueness and convergence of the solution of a general local fractional partial differential equation (LFPDE) obtained by the suggested method in view of theory of fixed point and Banach spaces. Furthermore, the error analysis of the LFNHAM solution is also discussed. Moreover, the numerical simulations are presented for each of the local fractional coupled equations on the Cantor set. The computational procedure clearly illustrates the validity and reliability of the proposed method for achieving the solutions of local fractional coupled Helmholtz and coupled Burgers' equations. The proposed method also minimizes the computational work unlike other conventional methods while still giving extremely precise results. The implemented combination supplies a more general solution as compared to other methods and assimilates their consequences as a special case. In addition, the acquired solutions are also in excellent match with previously determined solutions.