A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass...

Area of Science:

• Applied Mathematics
• Numerical Analysis
• Computational Physics

Background:

• Fractional order differential equations advance dynamical phenomena modeling.
• Diffusion processes are key in heat transfer, fluid flow, and pattern formation in porous media.
• The modified time-fractional diffusion equation offers deeper insights into dynamic phenomena.

Purpose of the Study:

• To develop an efficient meshless technique for approximating the modified time-fractional diffusion problem.
• To address problems formulated in the Riemann-Liouville sense.
• To provide a robust numerical method for complex diffusion scenarios.

Main Methods:

• Temporal discretization achieved through integration of the fractional diffusion model, ensuring unconditional stability and optimal convergence.
• Spatial derivatives discretized using a local hybridization of cubic and Gaussian radial basis functions for an improved system matrix condition.
• A meshless approach utilizing local support domains with a near-constant number of data points.

Main Results:

• The numerical procedure exhibits good accuracy and applicability across complex domains with diverse node distributions.
• Demonstrated accuracy, efficiency, and validity on both regular and irregular domains.
• Validation through three illustrative examples.

Conclusions:

• A local hybrid kernel meshless approach is adopted for solving the modified time-fractional diffusion problem.
• The research highlights a novel numerical technique suitable for non-uniform distributions in irregular grids.
• The method offers an accurate and efficient solution for complex diffusion modeling.