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An in‐depth study and analysis of the stability of one‐dimensional Linear System of PDEs via Caputo time fractional derivative (LSCFD) was presented. We proved some stability results for the LSCFD in different Hilbert spaces. Indeed, by using Fourier analysis method and the properties of the Mittag–Leffler Function (MLF), some polynomial stability results for LSCFD have been established. Finally, as an application, we used finite difference methods well suited to integer and fractional order derivatives, and performed some numerical experiments to confirm the theoretical results. [ABSTRACT FROM AUTHOR]