In this paper, we use the extended Hadamard k-fractional integral to obtain some new fractional integral inequalities by introducing the new parameters s and k. These extended fractional integral inequalities also hold true for usual Hadamard fractional integral when we substitute k is equal to one and s is equal to zero.

This article presents some new inequalities of Simpson's type for differentiable functions by using (alpha, m)-convexity. Some results for concavity are also obtained. These new estimates improve on the previously known ones. Some applications for special means of real numbers are also provided.

A numerical study based on finite difference approximation is attempted to analyze the bulk flow, micro spin flow and heat transfer phenomenon for micropolar fluids dynamics through Darcy porous medium. The fluid flow mechanism is considered over a moving permeable sheet. The heat transfer is associated with two different sets of boundary conditions, the isothermal wall and isoflux boundary. On the basis of porosity of medium, similarity functions are utilized to avail a set of ordinary differential equations. The non-linear coupled ODE’s have been solved with a very stable and reliable numerical scheme that involves Simpson’s Rule and Successive over Relaxation method. The accuracy of the results is improved by making iterations on three different grid sizes and higher order accuracy in the results is achieved by Richardson extrapolation. This study provides realistic and differentiated results with due considerations of micropolar fluid theory. The micropolar material parameters demonstrated reduction in the bulk fluid speed, thermal distribution and skin friction coefficient but increase in local heat transfer rate and couple stress. The spin behavior of microstructures is also exhibited through microrotation vector N ( η ) .

Heat and flow characteristics for bioconvection of nanofluid due to a radially stretching and rotating disk are examined. The nanofluid bioconvection is caused by the combined effects of magnetic field and buoyancy force on the interaction of motile gyrotactic microorganisms and nanoparticles in a dilute base fluid. The doping of suspended gyrotactic microorganisms and nanoparticles produces innovative heat transfer enhancement with provision of higher thermal conductivity through stability of nano particles. The motile microorganisms are self-propelled and they can actively swim in the fluid in response to such stimuli as gravity, light or chemical attraction. The computational results for physical quantities of interest due to influential thermophysical parameters and bioconvection parameters have been evaluated by employing bvp4c solver in Matlab. It is observed that the cooling rate becomes faster when radial stretching, Brownian diffusion and thermophoretic diffusion are increased.

A coupled system of singular fractional differential equations involving Riemann–Liouville integral and Caputo derivative is considered in this paper. The question of existence and uniqueness of solutions is studied using Banach contraction principle. Furthermore, the question of existence of at least one solution is discussed. At the end, an illustrative example is given in details.

In current continuation, we have incorporated the notion of $s- ( {\alpha,m} ) $ -convex functions and have established new integral inequalities. In order to generalize Hermite–Hadamard-type inequalities, some new integral inequalities of Hermite–Hadamard and Simpson type using $s- ( {\alpha,m} ) $ -convex function via Riemann–Liouville fractional integrals are obtained that reproduce the results presented by (Appl. Math. Lett. 11(5):91–95, 1998; Comput. Math. Appl. 47(2–3): 207–216, 2004; J. Inequal. Appl. 2013:158, 2013). Applications to special means are also provided.