A novel Chebyshev neural network approach for solving singular arbitrary order Lane-Emden equation arising in astrophysics.

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.