This paper develops an effective semi-discrete method based on the 2D Chelyshkov polynomials (CPs) to provide an approximate solution of the fractal–fractional nonlinear Emden–Fowler equation. In this model, the fractal–fractional derivative in the concept of Atangana–Riemann–Liouville is considered. The proposed algorithm first discretizes the fractal–fractional differentiation by using the finite difference formula in the time direction. Then, it simplifies the original equation to the recurrent equations by expanding the unknown solution in terms of the 2D CPs and using the [Formula: see text]-weighted finite difference scheme. The differentiation operational matrices and the collocation method play an important role to obtaining a linear system of algebraic equations. Last, solving the obtained system provides an approximate solution in each time step. The validity of the formulated method is investigated through a sufficient number of test problems.

中文翻译：

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利用二维 CHELYSHKOV 多项式对非线性分形-分形 2D EMDEN-FOWLER 方程进行数值处理

本文开发了一种基于 2D Chelyshkov 多项式 (CP) 的有效半离散方法，以提供分形-分形非线性 Emden-Fowler 方程的近似解。在该模型中，考虑了 Atangana-Riemann-Liouville 概念中的分形-分数导数。所提出的算法首先通过在时间方向上使用有限差分公式对分形-分数微分进行离散化。然后，它通过在 2D CP 方面扩展未知解并使用 [公式：见文本] 加权有限差分方案，将原始方程简化为循环方程。微分运算矩阵和搭配方法对于获得线性代数方程组具有重要作用。最后的，求解获得的系统提供了每个时间步长的近似解。通过足够数量的测试问题来研究所制定方法的有效性。