On the Generalized Fractional Convection–Diffusion Equation with an Initial Condition in Rn

Time-fractional convection–diffusion equations are significant for their ability to model complex transport phenomena that deviate from classical behavior, with numerous applications in anomalous diffusion, memory effects, and nonlocality. This paper derives, for the first time, a unique series solution to a multiple time-fractional convection–diffusion equation with a non-homogenous source term, based on an inverse operator, a newly-constructed space, and the multivariate Mittag–Leffler function. Several illustrative examples are provided to show the power and simplicity of our main theorems in solving certain fractional convection–diffusions equations. Additionally, we compare these results with solutions obtained using the AI model DeepSeek-R1, highlighting the effectiveness and validity of our proposed methods and main theorems.