In this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection–diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c*) and weight parameter (ϵ) that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method is tested for numerical solutions of some fractional Black–Sholes and diffusion models. Numerical simulations performed for several benchmark problems verified the proposed method accuracy and efficiency. The quantitative analysis is made in terms of L_∞, L₂, L_rms, and L_rel error norms as well as number of nodes N over space domain and time‐step δt. Numerical convergence in space and time is also studied for the proposed method. The unconditional stability of the proposed HRBFs scheme is obtained using the von Neumann methodology. It is observed that the HRBFs method circumvented the ill‐conditioning problem greatly, a major issue in the Kansa method.

中文翻译：

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径向基函数混合搭配技术可数值求解分数维对流扩散模型

在这项工作中，我们提出了分数阶对流扩散模型数值解的混合径向基函数（RBFs）配置技术。在混合RBFs（HRBFs）的公式中，存在控制数值精度和稳定性的形状参数（c *）和重量参数（ϵ）。对于这些参数，开发并验证了一种自适应算法。所提出的HRBFs方法已针对某些分数Black-Sholes和扩散模型的数值解进行了测试。对几个基准问题进行的数值模拟验证了所提方法的准确性和效率。定量分析中的方面取得了大号_∞，大号₂，大号_有效值，L _rel误差范数以及空间域上的节点数N和时间步长δt。该方法还研究了时空的数值收敛性。拟议的HRBFs方案的无条件稳定性是使用von Neumann方法获得的。可以看出，HRBFs方法极大地规避了病态问题，这是Kansa方法中的一个主要问题。