Introduction

During the last years the modeling of dynamical phenomena has been advanced by including concepts borrowed from fractional order differential equations. The diffusion process plays an important role not only in heat transfer and fluid flow problems, but also in the modelling of pattern formation that arises in porous media. The modified time-fractional diffusion equation provides a deeper understanding of several dynamic phenomena.

Objectives

The purpose of the paper is to develop an efficient meshless technique for approximating the modified time-fractional diffusion problem formulated in the Riemann–Liouville sense.

Methods

The temporal discretization is performed by integrating both sides of the modified time-fractional diffusion model. The unconditional stability of the time discretization scheme and the optimal convergence rate are obtained. Then, the spatial derivatives are discretized through a local hybridization of the cubic and Gaussian radial basis function. This hybrid kernel improves the condition of the system matrix. Therefore, the solution of the linear system can be obtained using direct solvers that reduce significantly computational cost. The main idea of the method is to consider the distribution of data points over the local support domain where the number of points is almost constant.

Results

Three examples show that the numerical procedure has good accuracy and applicable over complex domains with various node distributions. Numerical results on regular and irregular domains illustrate the accuracy, efficiency and validity of the technique.

Conclusion

This paper adopts a local hybrid kernel meshless approach to solve the modified time-fractional diffusion problem. The main results of the research is the numerical technique with non-uniform distribution in irregular grids.

中文翻译：

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用于近似传热传质中出现的修正时间分数扩散问题的局部稳定方法

介绍

在过去的几年中，通过引入从分数阶微分方程借用的概念，动力学现象的建模得到了进步。扩散过程不仅在传热和流体流动问题中发挥着重要作用，而且在多孔介质中出现的图案形成建模中也发挥着重要作用。修正的时间分数扩散方程提供了对几种动态现象的更深入的理解。

目标

本文的目的是开发一种有效的无网格技术来近似黎曼-刘维尔意义上的修正时间分数扩散问题。

方法

通过对改进的时间分数扩散模型的两侧进行积分来执行时间离散化。得到了时间离散格式的无条件稳定性和最优收敛速度。然后，通过三次和高斯径向基函数的局部混合来离散空间导数。该混合核改善了系统矩阵的状况。因此，可以使用直接求解器获得线性系统的解，从而显着降低计算成本。该方法的主要思想是考虑数据点在局部支持域上的分布，其中点的数量几乎恒定。

结果

三个例子表明，数值过程具有良好的准确性，并且适用于具有各种节点分布的复杂域。规则域和不规则域的数值结果说明了该技术的准确性、效率和有效性。

结论

本文采用局部混合核无网格方法来解决修正的时间分数扩散问题。研究的主要成果是不规则网格中非均匀分布的数值技术。