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Fractional cable problem in the frame of meshless singular boundary method

机译：无网格奇异边界法框架中的分形电缆问题

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摘要

In this study, singular boundary method (SBM) is employed for solving fractional cable problem in two dimensional space with initial and Dirichlet-type boundary conditions. The process is modeled as a two dimensional time-fractional equation in sense of Riemann-Liouville fractional derivatives. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. We present the numerical operation for calculating the particular solution and homogeneous solution. For achieving approximation particular solution and homogeneous solution, we employ Method of Particular solution (MPS) and SBM, respectively. We use theta-weighted and finite difference method as time discretization for time derivatives. A comparison between the present method and other methods is given to show the accuracy of SBM applying on this equation. Consequently, some numerical examples with different domains are tested and compared with the exact analytical solutions to display the validity and accuracy of the numerical method in comparison with other methods.
机译：在这项研究中，奇异边界法（SBM）被用于解决二维空间中具有初始边界和Dirichlet型边界条件的分数电缆问题。就Riemann-Liouville分数阶导数而言，该过程被建模为二维时间分数方程。应用拆分方案将不均匀控制方程的解拆分为齐次解和特定解。我们提出了用于计算特定解和齐次解的数值运算。为了获得近似的特定解和均匀解，我们分别采用了“特殊解法”（MPS）和“ SBM”。我们使用theta加权有限差分法作为时间导数的时间离散化。给出了本方法与其他方法的比较结果，表明了该方法应用于SBM的精度。因此，对一些具有不同域的数值示例进行了测试，并与精确的解析解进行了比较，以显示数值方法与其他方法相比的有效性和准确性。

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