Meshfree methods with arbitrary order smooth approximation are very attractive for accurate numerical modeling of fractional differential equations, especially for multi-dimensional problems. However, the non-local property of fractional derivatives poses considerable difficulty and complexity for the numerical simulations of fractional differential equations and this issue becomes much more severe for meshfree methods due to the rational nature of their shape functions. In order to resolve this issue, a new weak formulation regarding multi-dimensional Riemann–Liouville fractional diffusion equations is introduced through unequally splitting the original fractional derivative of the governing equation into a fractional derivative for the weight function and an integer derivative for the trial function. Accordingly, a Petrov–Galerkin finite element-meshfree method is developed, where smooth reproducing kernel meshfree shape functions are adopted for the trial function approximation to enhance the solution accuracy, and the discretization of weight function is realized by the explicit finite element shape functions with an analytical fractional derivative evaluation to further reduce the computational complexity and improve efficiency. The proposed method enables a direct and efficient employment of meshfree approximation, and also eliminates the undesirable singular integration problem arising in the fractional derivative computation of meshfree shape functions. A nonlinear extension of the proposed method to the fractional Allen–Cahn equation is presented as well. The effectiveness of the proposed methodology is consistently demonstrated by numerical results.

中文翻译：

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多维分数扩散方程的 Petrov-Galerkin 有限元无网格公式

具有任意阶平滑逼近的无网格方法对于分数阶微分方程的精确数值建模非常有吸引力，特别是对于多维问题。然而，分数阶导数的非局部性质给分数阶微分方程的数值模拟带来了相当大的困难和复杂性，并且由于无网格方法的形状函数的合理性，这个问题变得更加严重。为了解决这个问题，通过将控制方程的原始分数阶导数不等分分裂为权函数的分数阶导数和试函数的整数导数，引入了关于多维黎曼-刘维尔分数阶扩散方程的新弱公式. 因此，发展了Petrov-Galerkin有限元无网格方法，其中试函数近似采用平滑再现核无网格形状函数以提高求解精度，权函数的离散化通过显式有限元形状函数实现分数阶导数求值，进一步降低计算复杂度，提高效率。所提出的方法能够直接有效地使用无网格近似，并且还消除了在无网格形状函数的分数阶导数计算中出现的不良奇异积分问题。还介绍了所提出的方法对分数阶 Allen-Cahn 方程的非线性扩展。