English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We propose a Hilfer advection-diffusion equation of order $0<\alpha<1$ andtype $0\leq\beta\leq1$, and find the power series solution by using variationaliteration method. Power series solutions are expressed in a form that is easyto implement numerically and in some particular cases, solutions are expressedin terms of Mittag-Leffler function. Absolute convergence of power seriessolutions is proved and the sensitivity of the solutions is discussed withrespect to changes in the values of different parameters. For power law initialconditions it is shown that the Hilfer advection-diffusion PDE gives the samesolutions as the Caputo and Riemann-Liouville advection-diffusion PDE. Toleading order, the fractional solution compared to the non-fractional solutionincreases 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 thetruncation 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 accuracylevel and to the fractional order, $n\approx 20$; but for $\alpha>0.7$ therequired number of terms increases rapidly with the accuracy level and alsowith the fractional order $\alpha$.

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