A single layer fractional orthogonal neural network for solving various types of Lane–Emden equation

Abstract

Lane–Emden equation is an important nonlinear singular second order differential equation which can express various phenomena in astrophysics. On the other hand, by growing accessible data and information of physical dynamics, solving differential equations by data-driven algorithms becomes more significant. In this paper, an artificial neural network framework is provided to approximate the solution of different types of Lane–Emden equation such as fractional order or system of Lane–Emden equation. The presented neural network is a single layer orthogonal network. We use fractional order of Legendre functions as active functions of the hidden layer. Moreover, the Levenberg–Marquardt algorithm is used to train this neural network. In order to show the efficiency and the accuracy of the presented algorithm, we test it on several examples and compare with some other numerical methods. The obtained numerical results show that this network is extremely accurate and feasible. For example, in fractional version of Lane–Emden equation the obtained accuracy is about 10−50 while it was about 10−10 in some other methods.