Introduction

This study describes a novel meshless technique for solving one of common problem within cell biology, computer graphics, image processing and fluid flow. The diffusion mechanism has extremely depended on the properties of the structure.

Objectives

The present paper studies why diffusion processes not following integer-order differential equations, and present novel meshless method for solving. diffusion problem on surface numerically.

Methods

The variable- order time fractional diffusion equation (VO-TFDE) is developed along with sense of the Caputo derivative for $(0<\alpha (t)<1)$. An efficient and accurate meshfree method based on the singular boundary method (SBM) and dual reciprocity method (DRM) in concomitant with finite difference scheme is proposed on three-dimensional arbitrary geometry. To discrete of the temporal term, the finite diffract method (FDM) is utilized. In the spatial variation domain; the proposal method is constructed two part. To evaluating first part, fundamental solution of (VO-TFDE) is transformed into inhomogeneous Helmholtz-type to implement the SBM approximation and other part the DRM is utilized to compute the particular solution.

Results

The stability and convergent of the proposed method is numerically investigated on high dimensional domain. To verified the reliability and the accuracy of the present approach on complex geometry several examples are investigated.

Conclusions

The result of study provides a rapid and practical scheme to capture the behavior of diffusion process.

中文翻译：

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高维变阶时间分数阶扩散方程的奇异边界法数值解

介绍

这项研究描述了一种新的无网格技术，用于解决细胞生物学、计算机图形学、图像处理和流体流动中的一个常见问题。扩散机制极其依赖于结构的特性。

目标

本文研究了扩散过程不遵循整数阶微分方程的原因，并提出了一种新的无网格求解方法。表面扩散问题的数值计算。

方法

变阶时间分数扩散方程 (VO-TFDE) 是随着 Caputo 导数的意义而发展起来的 $(0<\alpha (吨)<1)$. 提出了一种基于奇异边界法(SBM)和对偶互易法(DRM)并结合有限差分格式的三维任意几何结构的高效精确无网格方法。为了离散时间项，使用了有限衍射法（FDM）。在空间变化域；提议方法由两部分构成。为了评估第一部分，将（VO-TFDE）的基本解转化为非齐次亥姆霍兹型以实现 SBM 近似，而其他部分则利用 DRM 来计算特定解。

结果

在高维域上对所提出方法的稳定性和收敛性进行了数值研究。为了验证本方法在复杂几何形状上的可靠性和准确性，我们研究了几个例子。

结论

研究结果为捕捉扩散过程的行为提供了一种快速实用的方案。