Pseudospectral techniques are known as powerful tools and highly accurate solvers to deal with partial differential equations. In the current work, a local meshless technique in Pseudospectral mode is employed to deal with an interesting and general mathematical model which describes anomalous electrodiffusion of ions in spiny dendrites, the two‐dimensional variable‐order time fractional nonlinear cable equation. For this purpose, at first step a second‐order implicit difference method and a modified second‐order weighted and shifted Grünwald difference scheme are used to discretize the appearing integer and variable order fractional time derivatives, respectively. Then a local Pseudospectral meshless method based on the sufficiently smooth compactly supported radial point interpolation basis functions is formulated for solving the semidiscretized problem. The main advantage of the proposed computational technique is that it leads to a sparse and better‐conditioned system of algebraic equations. Finally, some numerical experiments are presented to demonstrate and verify the performance and accuracy of the method.

中文翻译：

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伪谱模式下局部无网格方法对神经细胞中广义异常电扩散过程的数值模拟

伪光谱技术被称为功能强大的工具和高度精确的求解器，用于处理偏微分方程。在当前工作中，采用伪光谱模式的局部无网格技术来处理一个有趣且通用的数学模型，该模型描述了棘状树枝状结构中离子的异常电扩散，二维变数时间分数非线性电缆方程。为此，第一步使用二阶隐式差分方法和改进的二阶加权和移位的Grünwald差分方案来离散化出现的整数和可变阶分数时间导数。然后，提出了基于充分光滑的紧支撑径向点插值基函数的局部伪谱无网格方法，以解决半离散问题。所提出的计算技术的主要优点是它导致了一个稀疏且条件更好的代数方程组。最后，通过一些数值实验证明和验证了该方法的性能和准确性。