Aim of this paper was to appraise the magneto-hydrodynamic boundary layer flow of a Prandtl-Eyring fluid model on the top of stretched sheet. Ancillary to the same, the evolution and process of transfer of heat in the appearance of heat radiation and convective heating are investigated. Then, we solved these phenomena via the method of directly defined inverse mapping (MDDIM) with the approximate analytical solution firstly reported in this paper. We assumed the set of base functions, initial guesses and directly defined inverse mappings for the solution analysis. The present results are compared numerically with those of the Keller-Box method (KBM). We have found that, the velocity generate an upliftment (increase) with an increase in fluid parameter $\lambda$. Moreover, the thermal temperature profile’s adverse and antipodal behaviour can also be seen for access in fluid parameter $\lambda$.

In this paper, the hydromagnetic heat and mass transfer by mixed convection flow due to a vertical flat plate is considered for analysis. The governing equations are solved both analytically and numerically. The analytical solutions are obtained using the Homotopy Analysis Method (HAM) while the numerical solutions are computed using Keller–Box method (K–B). Convergence of the Homotopy solutions for the governing non-dimensional equations are derived. A detailed error analysis is done to compute the average squared residual errors for flow, temperature and concentration. The optimal values of the convergence control parameter are computed for velocity and temperature. This study includes the effects of various parameters such as magnetic parameter, Grashof number, chemical reaction parameter, heat source parameter and Biot number on skin friction, heat and mass transfer rates as well on velocity, temperature and concentration profiles. Comparison of the HAM and K–B methods shows a very good agreement

We derive a series-form solution of the coupled nonlinear equations by the method of directly defined inverse mapping (MDDIM) and spectral relaxation method (SRM). Then, via the method of directly defined inverse mapping with the series-form solution firstly reported in this paper, we can obtain the theoretical and approximate analytical solution that is laid out to explore the features of boundary layer analysis of the nanofluid flow due to a bidirectional stretching sheet under magnetic field along with thermal radiation. Three water-based nanofluids with nanoparticles of Al₂O₃, TiO₂ and CuO play an important role in the solutions. Based on the MDDIM and SRM, the convergence and accuracy of the proposed problems are illustrated, which exhibit meaningful structures. We has been proposed the convergence controlling relaxation parameter ( $\omega$) with the help of which the convergence rate and accuracy of the MDDIM are highly improved.