2014/06/08 by Iftikhar Ali, Ali, Iftikhar, Nadeem A. Malik +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Computational Physics (physics.comp-ph) #Differential Equations and Numerical Methods #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fractional Differential Equations Solutions #Nanofluid Flow and Heat Transfer
paper · pdf · doi:10.48550/arxiv.1406.2024
openalex publication_date 2014/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a Hilfer advection-diffusion equation of order 0<\α<1 and\ntype 0\≤\β\≤1, and find the power series solution by using variational\niteration method. Power series solutions are expressed in a form that is easy\nto implement numerically and in some particular cases, solutions are expressed\nin terms of Mittag-Leffler function. Absolute convergence of power series\nsolutions is proved and the sensitivity of the solutions is discussed with\nrespect to changes in the values of different parameters. For power law initial\nconditions it is shown that the Hilfer advection-diffusion PDE gives the same\nsolutions as the Caputo and Riemann-Liouville advection-diffusion PDE. To\nleading order, the fractional solution compared to the non-fractional solution\nincreases rapidly with \α for \α > 0.7 at a given time t; but for\n\α<0.7 this factor is weakly sensitive to \α. We also show that the\ntruncation errors, arising when using the partial sum as approximate solutions,\ndecay exponentially fast with the number of terms n used. We find that for\n\α< 0.7 the number of terms needed is weakly sensitive to the accuracy\nlevel and to the fractional order, n\≈ 20; but for \α>0.7 the\nrequired number of terms increases rapidly with the accuracy level and also\nwith the fractional order \α.\n