The main goal of this work is to find the solutions of linear and nonlinear fractional differential equations with the Mittag-Leffler nonsingular kernel. An accurate numerical method to search this problem has been constructed. The theoretical results are proved by utilizing two experiments.

The current analysis signifies the Casson nanofluid stream with inclined magnetohydrodynamics, viscous dissipation, joule heating, thermal radiation, particle shape factor effects over a moving flat horizontal surface with entropy analysis. This study presents a novel idea regarding the implementation of Tiwari and Das model on Casson fluid model by considering ethylene glycol as a base fluid. Tiwari and Das model considers nanoparticles volume fraction for heat transfer enhancement instead of the Buongiorno model which heavily relies on thermophoresis and Brownian diffusion effects for heat transfer analysis. The velocity slip and convective slip boundary conditions have been employed at the surface of the sheet. By utilizing the suitable transformations, the modeled PDEs (partial-differential equations) are renewed in ODEs (ordinary-differential equations) and treated these equations numerically with the help of classical Keller box method. Two different classes of nanofluids, Copper-Ethylene Glycol ( C u − EG ) and Molybdenum Disulfide-Ethylene Glycol ( M o S 2 − EG ) have been taken into considering for our analysis. The behaviour of surface drag coefficient and Nusselt number for the varied values of various sundry parameters is designed via tables. Our findings show that an increase in the Reynolds and Brinkman numbers increased the overall entropy of the system.

The objective of the present article is to investigate an incompressible micropolar Prandtl fluid flowing over a porous stretching sheet with the inclusion of effects like exponential temperature-dependent heat source, higher-order chemical reaction viscous dissipation, nonlinear thermal radiation, and multiple convective surface boundary conditions. The nonlinear partial differential equations (PDEs) for momentum, energy, micro-rotation, and concentration are discussed. These PDEs are converted into ordinary differential equations (ODEs) by utilizing suitable variables and then tackled with the utilization of the nonlinear shooting method. For the numerical assessment of upcoming results comparison with already existing literature has been taken. It is canvassed that by augmenting the microrotation parameter, a boost in the angular velocity profile is seen. Moreover, The higher order of reactivity upsurges the heat transmission rate, whilst the mass convection enhances the concentration size. This gives the current model potency as a potential application in clean engine lubricants and coolant industrial liquids under the same constraints.

In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this kind problems. The experiments are discussed and the approximate solutions are obtained to be more correct compared to the other obtained results in the literature.

In this study, we solved the economic models based on market equilibrium with constant proportional Caputo derivative using the Laplace transform. We proved the accuracy and efficiency of the method. We constructed the relations between the solutions of the problems and bivariate Mittag–Leffler functions.