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• Novel stability result, of 1D fractional-order differential advection-diffusion system in infinite time, is proved on the space L 2 (Ω) × L 2 (Ω). • Novel stability result, of 1D fractional-order differential advection-diffusion system in infinite time, is proved on the space H 1 (Ω) × H 1 (Ω). • Some numerical methods have been implemented and some important numerical experiments have been established to confirm the theoretical results obtained. In this paper, we consider a one dimensional advection-diffusion system in Caputo fractional order derivative. Using a Fourier decomposition and the Mittag-Leffler Function (MLF), we prove a new stability results for the solution of a such system. Numerical experiments were carried out at the end of this work to confirm the theoretical results obtained. [ABSTRACT FROM AUTHOR]