This contribution proposes two third-order numerical schemes for solving time-dependent linear and non-linear partial differential equations (PDEs). For spatial discretization, a compact fourth-order scheme is deliberated. The stability of the proposed scheme is set for scalar partial differential equation, whereas its convergence is specified for a system of parabolic equations. The scheme is applied to linear scalar partial differential equation and non-linear systems of time-dependent partial differential equations. The non-linear system comprises a set of governing equations for the heat and mass transfer of magnetohydrodynamics (MHD) mixed convective Casson nanofluid flow across the oscillatory sheet with the Darcy–Forchheimer model, joule heating, viscous dissipation, and chemical reaction. It is noted that the concentration profile is escalated by mounting the thermophoresis parameter. Also, the proposed scheme converges faster than the existing Crank-Nicolson scheme. The findings that were provided in this study have the potential to serve as a helpful guide for investigations into fluid flow in closed-off industrial settings in the future.

The article presents a mathematical model for the magnetized nanofluid flow and heat transfer with an exothermic chemical reaction controlled by Arrhenius kinetics. Buongiorno’s model with passive boundary condition is employed to formulate the governing equation for nanoparticles concentration. The momentum equation with slip boundary conditions is modelled with the inclusion of electroosmotic effects which remain inattentive in the study of microchannel flows with electric double layer (EDL) effects. Conclusions are based on graphical and numerical results for the dimensionless numbers representing the features of heat transfer and fluid flow. Frank-Kamenetskii parameter resulting from the chemical reaction showed significant effects on the optimization of heat transfer, leading to increased heat exchangers’ effectiveness. The Hartmann number and slip parameter significantly affect skin friction, demonstrating the notable effects of electroosmotic flow and the exothermic chemical reaction on heat transfer in microchannels. This analysis contributes to prognosticating the convective heat transfer of nanofluids on a micro-scale for accomplishing successful thermal designs.

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The present involvement is the theoretical use of the thermal extrusion structure accompanying with the various industrial progressions. The problem is composed by exploiting the MHD aspect on flow of Maxwell fluid. The properties of chemically reactive flow of magneto Maxwell fluid with effects of viscous dissipation over stretching sheet in stagnation region are elaborated here. The governing equations of phenomena are given in set of partial differential equations, and further these equations are reduced to set of ordinary differential equations using similarity transformations. MATLAB built in solver bvp4c is employed to solve obtained nonlinear boundary value problem. The solver uses the 4th and 5th order discretization scheme, and the outcomes in the form of velocity, temperature, and concentration profiles with variations of magnetic parameter, Maxwell parameter, heat generation parameter, Eckert number, Prandtl number, Schmidt number and reaction rate parameter are deliberated through graphs.

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