Nikan O, Avazzadeh Z, Machado J A Tenreiro
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran.
Department of Applied Mathematics, Xi'an Jiaotong-Liverpool University, Suzhou 215123, China.
J Adv Res. 2021 Mar 10;32:45-60. doi: 10.1016/j.jare.2021.03.002. eCollection 2021 Sep.
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.
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.
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.
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.
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.
在过去几年中，通过纳入从分数阶微分方程借鉴的概念，动态现象的建模取得了进展。扩散过程不仅在传热和流体流动问题中起着重要作用，而且在多孔介质中出现的图案形成建模中也起着重要作用。修正的时间分数阶扩散方程为深入理解几种动态现象提供了帮助。
本文的目的是开发一种高效的无网格技术，用于逼近在黎曼 - 刘维尔意义下制定的修正时间分数阶扩散问题。
通过对修正的时间分数阶扩散模型的两边进行积分来进行时间离散化。获得了时间离散化方案的无条件稳定性和最优收敛速率。然后，通过三次样条和高斯径向基函数的局部混合来离散空间导数。这种混合核改善了系统矩阵的条件。因此，可以使用直接求解器获得线性系统的解，从而显著降低计算成本。该方法的主要思想是考虑数据点在局部支撑域上的分布，其中点的数量几乎是恒定的。
三个例子表明，该数值方法具有良好的精度，适用于具有各种节点分布的复杂域。在规则和不规则域上的数值结果说明了该技术的准确性、效率和有效性。
本文采用局部混合核无网格方法来解决修正的时间分数阶扩散问题。研究的主要成果是在不规则网格中具有非均匀分布的数值技术。
