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This article extends the application of fractional-order time derivatives to replace their integer-order counterparts within a system comprising two singular one-dimensional coupled partial differential equations. The resulting model proves invaluable in representing radially symmetric deformation and temperature distribution within a unit disk. The incorporation of fractional-order derivatives in mathematical models is shown to significantly enhance their capacity for characterizing real-life phenomena in comparison to their integer-order counterparts. To address the studied system numerically, we employ the <i>q</i>-homotopy analysis transform method (<i>q</i>-HATM). We evaluate the efficiency of this method in solving the problem through a series of illustrative examples. The convergence of the derived scheme is assessed visually, and we compare the performance of the <i>q</i>-HATM with that of the Laplace decomposition method (LDM). While both methods excel in resolving the majority of the presented examples, a notable divergence arises in the final example: the numerical solutions obtained using <i>q</i>-HATM converge, whereas those derived from LDM exhibit divergence. This discrepancy underscores the remarkable efficiency of the <i>q</i>-HATM in addressing this specific problem.