<p>This research aims to study an important fractional-order model that contains the Caputo–Fabrizio derivative (CFD) in view of its applications to brain metabolite variations in the circadian rhythm. The general form of this model is given, and the existence and uniqueness of its solution are proven. We solve this problem via two analytical methods: the Adomian decomposition method (ADM) and the Picard method (PM). The convergence of the series solution is proven, and the maximum value of the error is estimated. The solution stability is discussed. Given that, fractional-order models represent real-world phenomena better than integer-order models do, we aim to determine the effect of changing orders in this model. Therefore, four different cases of this fractional brain model (FBM) are solved.</p>