The current research numerically investigates the Marangoni convection in a cylindrical annulus filled with hybrid nanofluid saturated porous media. The interior and exterior walls are subjected to spatially varying sinusoidal thermal distributions with various amplitude ratios and phase deviations. The limits at the top and bottom are adiabatic. To solve the system of non-dimensional governing equations, the finite difference approach is applied. The major goal of the ongoing study is to investigate the impact of the Marangoni convection, amplitude ratio and phase deviation on the fluid flow, thermal characteristics, local and average Nusselt numbers in the hybrid nanofluid filled vertical cylindrical annulus with magnetic effects. The findings indicate that the sinusoidal temperature promotes multicellular flow in the porous annular region. In the annulus with sinusoidal boundaries, the Marangoni number underperforms while the nanoparticle volume fraction outperforms.

In this paper we apply the fractional integrals with arbitrary order depending on the fractional operators of Riemann type (ABR) and Caputo type (ABC) with kernels of Mittag Lefller in three parameters $E_{\alpha,\mu}^\gamma(\lambda,t)$ for solving the time fractional parabolic nonlinear equation. We utilize these operators with homotopy analysis method (HAM) for constructing the new scheme for generating the successive approximations. This procedure are used successfully on two examples for finding the solutions. The effectiveness and accuracy are verified by clarifying the convergence region in the $h$-curves as well as by calculating the residual error and the results were accurate. Depending on this results, this treatment can be used to find the approximate solutions to many fractional differential equations.