The purpose of this work is to analytically simulate the mutual impact for the existence of both temporal and spatial Caputo fractional derivative parameters in higher-dimensional physical models. For this purpose, we employ the γ̅-Maclaurin series along with an amendment of the power series technique. To supplement our idea, we present the necessary convergence analysis regarding the γ̅-Maclaurin series. As for the application side, we solved versions of the higher-dimensional heat and wave models with spatial and temporal Caputo fractional derivatives in terms of a rapidly convergent γ̅-Maclaurin series. The method performed extremely well, and the projections of the obtained solutions into the integer space are compatible with solutions available in the literature. Finally, the graphical analysis showed a possibility that the Caputo fractional derivatives reflect some memory characteristics.