Analytical Solution of a Generalized Fractional Diffusion-Advection Equation

This study investigates a generalised nonlinear fractional diffusion–advection equation incorporating concentration-dependent diffusion and nonlinear advection terms. The model $\mathcal{D}_t^\beta u(x,t) = \frac{\partial}{\partial x} \left( f(u) \frac{\partial u}{\partial x} \right) - \frac{dK}{du} \frac{\partial u}{\partial x}$, where, \( \mathcal{D}_t^\beta \) represents the Caputo derivative of the fractional order \( \beta \in (0,1] \), and \(f(u), \frac{dK}{du} \) are nonlinear functions. This formulation incorporates a concentration-dependent diffusion coefficient, allowing diffusive behaviour to vary with the state variable, and a nonlinear advection term that more realistically accounts for convective effects. The mathematical model is expressed using the Caputo fractional derivative, which is particularly suitable for problems with physically meaningful initial conditions and memory effects. To establish mathematical well-posedness, sufficient conditions for the existence and uniqueness of solutions are derived by applying the Banach fixed-point theorem. This analysis guarantees that the problem admits a unique solution within an appropriate functional framework, providing a solid theoretical foundation for further investigation. For constructing approximate analytical solutions, the homotopy perturbation method (HPM) is employed. This technique yields solutions in the form of rapidly convergent series without requiring small parameters or linearizing the governing equation. Several illustrative examples demonstrate the effectiveness, accuracy, and simplicity of the proposed approach. The obtained results confirm the applicability of HPM to nonlinear fractional diffusion–advection problems and generalise earlier studies on fractional diffusion equations and Burgers-type equations, offering a unified framework for analysing a wide range of nonlinear fractional models.