An efficient technique based on cubic B-spline functions for solving time-fractional advection diffusion equation...

Area of Science:

• Numerical analysis
• Computational mathematics
• Partial differential equations

Background:

• Fractional calculus extends the concept of derivatives to non-integer orders.
• The time fractional advection diffusion equation models various physical phenomena.
• Accurate numerical solutions are crucial for understanding these phenomena.

Purpose of the Study:

• To develop and validate a novel numerical scheme for the time fractional advection diffusion equation.
• To utilize the Atangana-Baleanu derivative with a non-singular kernel.
• To employ cubic B-spline approximation for spatial discretization.

Main Methods:

• Finite difference scheme for discretizing the Atangana-Baleanu time derivative.
• Cubic B-spline functions for spatial discretization.
• Weighted scheme for enhancing numerical accuracy.

Main Results:

• The proposed numerical scheme is unconditionally stable.
• The convergence order of the scheme is found to be $$.
• Numerical examples demonstrate the feasibility and accuracy of the method.

Conclusions:

• The developed cubic B-spline approximation with a weighted scheme provides an effective numerical solution.
• The Atangana-Baleanu derivative and finite difference discretization yield a stable and accurate method.
• The scheme is suitable for solving time fractional advection diffusion problems.