Hilfer fractional advection-diffusion equations with power-law initial condition; a Numerical study using variational iteration method

We propose a Hilfer advection-diffusion equation of order $0<\alpha<1$ and type $0\leq\beta\leq1$, and find the power series solution by using variational iteration method. Power series solutions are expressed in a form that is easy to implement numerically and in some particular cases, solutions are expressed in terms of Mittag-Leffler function. Absolute convergence of power series solutions is proved and the sensitivity of the solutions is discussed with respect to changes in the values of different parameters. For power law initial conditions it is shown that the Hilfer advection-diffusion PDE gives the same solutions as the Caputo and Riemann-Liouville advection-diffusion PDE. To leading order, the fractional solution compared to the non-fractional solution increases rapidly with $\alpha$ for $\alpha > 0.7$ at a given time $t$; but for $\alpha<0.7$ this factor is weakly sensitive to $\alpha$. We also show that the truncation errors, arising when using the partial sum as approximate solutions, decay exponentially fast with the number of terms $n$ used. We find that for $\alpha< 0.7$ the number of terms needed is weakly sensitive to the accuracy level and to the fractional order, $n\approx 20$; but for $\alpha>0.7$ the required number of terms increases rapidly with the accuracy level and also with the fractional order $\alpha$.