Method of Directly Defining the Inverse Mapping for Nonlinear Ordinary and Partial Fractional-Order Differential Equations

Abstract

We extend the Method of Directly Defining the inverse Mapping (MDDiM) to determine approximate solutions for fractional-order ordinary and partial differential equations. The Riccati, Abel, and time-fractional Rosenau-Hyman equations were solved here. The MDDiM was utilized for the first time to solve fractional-order ordinary and partial differential equations. By considering the sum of the initial three terms of the series solution, we were able to get approximate solutions for the fractional Riccati ordinary differential equation and the time-fractional Rosenau-Hyman equation. We also used the fourth-order series solution to get an approximate solution for the Abel differential equation. By determining the ideal option of the convergence control value for quick convergence, as well as alternative fractional orders on solutions, we were able to achieve solution graphs and minimum errors.