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Fractional-order mathematical modelling of physical phenomena is a hot topic among various researchers due to its many advantages over positive integer mathematical modelling. In this context, the appropriate solutions of such fractional-order physical modelling become a challenging task among scientists. This paper presents a study of unsteady free convection fluid flow and heat transfer of Maxwell fluids with the presence of Clay nanoparticle modelling using fractional calculus. The obtained model was transformed into a set of linear nondimensional, partial differential equations (PDEs). The finite difference scheme is proposed to discretize the obtained set of nondimensional PDEs. The Maple code was developed and executed against the physical parameters and fractional-order parameter to explain the behavior of the velocity and temperature profiles. Some limiting solutions were obtained and compared with the latest existing ones in literature. The comparative study witnesses that the proposed scheme is a very efficient tool to handle such a physical model and can be extended to other diversified problems of a complex nature.