Homotopy perturbation method for solving time-fractional nonlinear Variable-Order Delay Partial Differential Equations

The homotopy perturbation method is extend to derive the approximate solution of the variable order fractional partial differential equations with time delay. The variable order fractional derivative is taken in the Caputo sense. An approximation formula of the Caputo derivative of fractional variable order is presented in terms of standard (integer order) derivatives only. Then the original problem will be transformed into a systems of partial differential equations with delay. By employing the homotopy perturbation method the explicit approximate solutions are found. The error and convergence analysis of the homotopy perturbation method has been discussed for the applicability of the method. The absolute errors and the approximate solutions are presented graphically and by tables at the values of various variable fractional order. From the results of the illustrated examples, we can Judge that the homotopy perturbation method is very effective, and simple accelerates the rapid convergence of the solution.