A Novel Fractional Analytical Technique for the Time-space Fractional Equations Appearing in Oil Pollution

Oil spills in the seas and oceans cause pollution and have many destructive environmental effects. The diffusion (parabolic) equations are the most reasonable option to model the propagation of this leakage and contamination. These equations allow statistics regarding the amount of oil that has outreached the ocean outlet, to be used as initial and boundary conditions for a mathematical model of oil diffusion and alteration in seas. As it involves the hyperbolic (advection/wave) component of the equation, the most reasonable choices are diffusion and Allen–Cahn (AC) equations, which are difficult to solve numerically. Equations of diffusion and Allen-Cahn were solved with different degrees of fractional derivatives (α= 0.25, α=0.5, α=0.75 and α=0.75), and the oil pollution concentration was obtained at a specific time and place. This study adopts the homotopy perturbation method (HPM) for nonlinear Allen–Cahn (AC) equation and time fractional diffusion equation to express oil pollution in the water. Fractional derivatives are portrayed in the sense of Caputo. Two presented examples illustrate the applicability and validity of the proposed method. Pollution concentrations in flow field over an interval of time and space for different degrees of fractional derivation are shown. At lower fraction derivative degrees, the pollution concentration behavior is nonlinear, and as the degree of fraction derivation increases to one, the nonlinear behavior of the pollution concentration decreases. The results produced by the suggested technique compared to the exact solutions shows that it is efficient and convenient; it is also reduces computational time.