A highly accurate numerical method for solving nonlinear time-fractional differential difference equation

摘要

This work is based on the implementation of an iterative perturbation method to attain the series solutions of nonlinear fractional differential difference equation (NFD Delta E). Perturbation-iteration algorithm (PIA) assigns a perturbation parameter epsilon to all nonlinear terms and converts it into a simple fractional differential difference equation (FD Delta E). By simply solving this FD Delta E, series solutions can be obtained. To show the efficacy and accuracy of this method, three famous NFD Delta E, i.e, fractional Lotka-Volterra equation, fractional discrete KdV equation and discretized fractional mKdV lattice equation, will be solved numerically via PIA. Also, comparison of numerical results for alpha=1 will be done with exact solutions, and their absolute error will also be provided. Graphical illustrations for different values of alpha will be given to establish the certainty of results. Also, to prove the proficiency among other methods, comparison with different numerical methods is given. Advantage of PIA is that nonlinear terms get vanished during the calculations of Taylor series expansion; therefore, less calculation effort can obtain comprehensive accurate solutions.