Mathematical analysis of fractional-order convection–reaction–diffusion equations under the Caputo fractional derivative

The main objective of this study is to employ two unique methods to approximately solve time-fractional convection–reaction–diffusion equations (CRDEs). The suggested techniques combine the Adomian decomposition approach, the homotopy perturbation method, and the Yang transform. He’s polynomials and Adomian polynomials are employed to address the nonlinearity that develops in our assumed problems. To clarify the efficacy of the employed schemes, three test examples are taken into account. This article considers sophisticated approaches and the fractional operator in this aspect to obtain satisfactory approximations to the provided problems. We first build the Yang transforms of the Caputo fractional derivative and apply them for CRDEs to obtain improved approximations after a finite number of iterations. The two suggested methods are used to generate some extremely accurate analytical approximations. The approximations obtained through these methods are represented as convergent series solutions. The solution obtained using the suggested methods converges at the desired rate to the precise solution. To show the usefulness of the offered techniques, we provide some graphical representations of the precise and analytical results, which are in excellent agreement with one another. In addition, we used several tables for different fractional orders to visually represent the physical behavior of the approximate solution. The convergence of the fractional solutions towards integer-order solutions is examined for the efficacy of the current strategies. The proposed methods are validated by solving four important cases. The suggested solutions are proven to be very effective, straightforward, and appropriate for other nonlinear issues raised in science and engineering.