Double exponential quadrature for fractional diffusion.

Area of Science:

• Numerical Analysis
• Partial Differential Equations
• Fractional Calculus

Background:

• Fractional diffusion equations model complex phenomena but pose numerical challenges.
• Existing discretization methods often require problem-specific parameter tuning.
• Adaptive and efficient numerical schemes are crucial for solving these problems.

Purpose of the Study:

• Introduce a novel discretization technique for elliptic and parabolic fractional diffusion problems.
• Develop a method that offers faster convergence and requires fewer parameters.
• Demonstrate the scheme's ability to leverage inherent data smoothness.

Main Methods:

• Utilized double exponential quadrature formulas for discretization.
• Applied the Riesz-Dunford functional calculus.
• Proved rigorous convergence for data with finite regularity and Gevrey-type classes.

Main Results:

• The novel method achieves faster convergence compared to existing schemes.
• The technique requires fewer problem-dependent parameters for tuning.
• The scheme effectively utilizes additional data smoothness without a-priori knowledge.
• Rigorous convergence proofs were established for different data regularity conditions.

Conclusions:

• The proposed discretization technique is efficient and robust for fractional diffusion problems.
• This method offers a significant improvement over traditional numerical approaches.
• The findings pave the way for more accessible and accurate solutions in fractional calculus applications.