ABSTRACT The motivation of this investigation is to develop a single-layer Chebyshev Neural Network (ChNN) model to handle singular fractional (arbitrary)-order Lane-Emden type equations. These equations are well-known application problems of astrophysics and quantum mechanics. Fractional Lane-Emden equations are singular so it is very difficult to solve analytically. Thus, an efficient method is required to handle the above equations. Here, our main aim is to use a single-layer ChNN model for solving fractional Lane-Emden equations. ChNN model is one kind of Functional Link Neural Network (FLNN) in which the hidden layer is replaced by a functional expansion block of the input pattern using orthogonalshifted Chebyshev polynomials (SChP). Thus, the network parameters of ChNN are less than the Multi-Layer Artificial Neural Network (MLANN). We have considered factional-order singular nonlinear problems of astrophysics to show the computational effort of the proposed method. Back Propagation algorithm of the unsupervised version has been considered for minimizing the error function and updating the weights of the ChNN model. Computed results are displayed in terms of tables and graphs.

中文翻译：

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一种用于求解天体物理学中出现的奇异任意阶 Lane-Emden 方程的新型 Chebyshev 神经网络方法

摘要 本研究的动机是开发单层切比雪夫神经网络 (ChNN) 模型来处理奇异分数（任意）阶 Lane-Emden 类型方程。这些方程是天体物理学和量子力学的著名应用问题。分数式 Lane-Emden 方程是奇异的，因此很难解析求解。因此，需要一种有效的方法来处理上述等式。在这里，我们的主要目标是使用单层 ChNN 模型来求解分数式 Lane-Emden 方程。ChNN 模型是一种功能链接神经网络 (FLNN)，其中隐藏层被使用正交移位切比雪夫多项式 (SChP) 的输入模式的功能扩展块替换。因此，ChNN 的网络参数小于多层人工神经网络（MLANN）。我们已经考虑了天体物理学的阶次奇异非线性问题，以显示所提出方法的计算工作量。无监督版本的反向传播算法已被考虑用于最小化误差函数和更新 ChNN 模型的权重。计算结果以表格和图形的形式显示。