An efficient technique for solving fractional-order diffusion equations arising in oil pollution

• Fractional Reduced Differential Method (FRDTM) is established and applied to find the closed-form solution of time-fractional diffusion equation and Allen-Cahn equation arising in oil pollution. • The obtained results using FRDTM has been compared with the exact solution, MVIA-I, MVIA-II, MQM, LLWM, and ADM for integer order. • The simulation results indicate an excellent accordance with the exact solution as compared to any other available method in the literature. • FRDTM gives fast convergence and provides highly accurate numerical results. • The main advantage of this method is its implementation on time-fractional order nonlinear PDEs without discretization and linearization. In this article, non-linear time-fractional diffusion equations are considered to describe oil pollution in the water. The latest technique, fractional reduced differential transform method (FRDTM), is used to acquire approximate solutions of the time fractional-order diffusion equation and two cases of Allen–Cahn equations. The acquired results are collated with the exact solutions and other results from literature for integer-order α , which reveal that the proposed method is effective. Hence, FRDTM can be employed to obtain solutions for different types of nonlinear fractional-order IVPs arising in engineering and science.