A fast characteristic finite difference method for fractional advection–diffusion equations

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A characteristic difference method for the variable-order fractional advection-diffusion equation (2013, methodology): “(1) are combined to form a material derivative as follows [33]: ∂u(x, t) ∂t + ν(x, t) t) ∂x =ψ(x, t) t) ∂ρ .”
Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain (2012, methodology): “[42] developed a fast characteristic finite difference method for the efficient solution of space-fractional transient advection–diffusion equations in one space dimension.”
Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients (2015, background): “Because of the nonlocal property of fractional derivative, they can utilize for modeling of memory-dependent phenomena and complex media such as porous media and anomalous diffusion [1–4].”
The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation (2013, methodology): “Similar to the numerical methods for the classical differential equations, there also exist finite difference methods [3, 10, 12, 13, 20, 24, 26, 27, 31, 34, 51, 52, 54, 55, 62], finite element methods (FEMs) [16, 48, 59, 61], and spectral methods [6, 25, 28] for numerically solving the FDEs.”