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The primary purpose of the present study is to investigate the unsteady flow and heat transfer characteristics of non-Newtonian power-law fluid over an oscillating flat plate. The power-law fluid model is employed to distinguish the non-Newtonian fluid behavior, which has wide applications in nature and technology, such as crude oil exploitation, polymer processing and some bio applications. In this work, the governing equations are set up using the non-Newtonian fluid model. The radial basis function collocation method (RBFCM) and the finite element method (FEM) are used to solve the coupled equations. The numerical results of the fluid velocity and fluid temperature fields are analyzed in detail. For authentication of the proposed schemes, the results are compared with earlier research works. To further verify the validity of the FEM and the RBFCM results, the flow behavior of crude oil was studied and the obtained results were found to be consistent with each other using both methods. Our results provide further guidance for evaluation aspects dealing with crude oil in a pipe, which are basic tools in oil and gas explorations. [ABSTRACT FROM AUTHOR]