Finite Difference Method for Time-Space Fractional Advection-Diffusion Equations with Riesz Derivative

Area of Science:

• Numerical analysis
• Computational mathematics
• Partial differential equations

Background:

• Fractional calculus extends traditional calculus to non-integer orders, enabling modeling of complex phenomena.
• Advection-diffusion equations describe transport processes in various scientific fields.
• Time-space fractional differential equations capture anomalous diffusion and complex transport behaviors.

Purpose of the Study:

• To develop and analyze a numerical scheme for the time-space fractional advection-diffusion equation.
• To approximate Riesz spatial derivatives and Caputo temporal derivatives.
• To rigorously investigate the stability and convergence properties of the proposed numerical method.

Main Methods:

• Approximation of the Riesz space derivative using a second-order fractional weighted and shifted Grünwald-Letnikov formula.
• Transformation of the fractional differential equation into an equivalent integral equation.
• Approximation of the resulting integral equation using the trapezoidal formula.

Main Results:

• A numerical scheme is formulated and analyzed for the time-space fractional advection-diffusion equation.
• The scheme is proven to have second-order accuracy in both temporal and spatial directions.
• Stability and convergence analyses are rigorously performed.

Conclusions:

• The developed numerical scheme accurately solves the time-space fractional advection-diffusion equation.
• The theoretical findings on accuracy, stability, and convergence are supported by numerical experiments.
• This work provides an efficient and reliable tool for simulating fractional advection-diffusion processes.