This paper examines the mixed convective heat transfer effect on a non-Darcy porous fluid flowing through a vertical plate considering the concentration and effect of melt on flow and heat transfer. The mathematical models describing the transport phenomena arising from mechanics of the fluid are developed and then analyzed using the homotopy perturbation method. The results of the approximate analytical solutions in this present study are verified with the results of the past works as established in the literature. Thereafter, the developed approximate analytical solutions are used to investigate the effect of rheological parameters such as melting, thermal radiation, and dispersion parameters. From the results, it is established that as the melting parameter increases, the temperature distribution toward center plate increases significantly, while the concentration distribution, it increases toward the upper plate as the melt parameter increases. Also, it was observed that an increase in the R parameter causes a corresponding decrease in velocity distribution. Increasing thermal dispersion parameter (D) decreases temperature distribution. The present study will be useful as it provides useful insight to applications including fossil fuel combustion, astrophysical flows, and geothermal systems amongst others.