The numerical solution of high dimensional variable-order time fractional diffusion equation via the singular...

Area of Science:

• Multidisciplinary applications in cell biology, computer graphics, image processing, and fluid dynamics.

Background:

• Diffusion processes often exhibit complex behaviors not fully described by integer-order differential equations.
• Understanding and modeling these diffusion mechanisms is crucial across various scientific disciplines.

Purpose of the Study:

• To present a novel meshless numerical method for solving variable-order time fractional diffusion equations (VO-TFDEs).
• To address the challenge of modeling diffusion on surfaces with arbitrary geometries.

Main Methods:

• Development of a meshfree method combining the Singular Boundary Method (SBM) and Dual Reciprocity Method (DRM).
• Utilizes the Caputo derivative for the temporal term and a finite difference scheme for discretization.
• Employs an inhomogeneous Helmholtz-type transformation for SBM approximation and DRM for particular solutions.

Main Results:

• Numerical investigation confirms the stability and convergence of the proposed method in high-dimensional domains.
• Validation through several examples on complex geometries demonstrates the approach's reliability and accuracy.

Conclusions:

• The study provides an efficient and accurate meshless scheme for simulating diffusion processes.
• The developed method offers a rapid and practical tool for capturing diffusion behavior in diverse applications.