Abstract
This study reports novel hybrid computational methods for the solutions of nonlinear singular Lane–Emden type differential equation arising in astrophysics models by exploiting the strength of unsupervised neural network models and stochastic optimization techniques. In the scheme the neural network, sub-part of large field called soft computing, is exploited for modelling of the equation in an unsupervised manner. The proposed approximated solutions of higher order ordinary differential equation are calculated with the weights of neural networks trained with genetic algorithm, and pattern search hybrid with sequential quadratic programming for rapid local convergence. The results of proposed solvers for solving the nonlinear singular systems are in good agreements with the standard solutions. Accuracy and convergence the design schemes are demonstrated by the results of statistical performance measures based on the sufficient large number of independent runs.
Highlights
Mathematical models based on Lane–Emden type equations (LEEs) have been studied in diverse fields of applied sciences, in the domain of astrophysics
In order to evaluate small differences, we presented a statistical analysis, fitting of normal distribution based on absolute errors (AEs) of Sequential quadratic programming (SQP), Pattern search (PS) and Genetic algorithms (GAs) algorithms as shown in Figs. 9, 10 and 11 for problems I, II and III, respectively
The multi-runs of each algorithm independently provide a strong evidence for the accuracy of the proposed method
Summary
Introduction Mathematical models based on Lane–Emden type equations (LEEs) have been studied in diverse fields of applied sciences, in the domain of astrophysics. Singular second order nonlinear initial value problem (IVP) of LEEs describes various real life phenomena. The most of the problems arising in astrophysics are modelled by second order nonlinear ordinary differential equations (ODEs) (Lane 1870; Emden 1907; Fowler 1914, 1931). The general form of LFE is represented mathematically as: d2y α dy dx2 + x dx + f (x, y) = g(x), for α, x ≥ 0, having initial conditions as: y(0) = a, dy(0) = 0, dx. Ahmad et al SpringerPlus (2016) 5:1866 here a is constant and f(x, y) is a nonlinear function. LEE has a singularity at the origin, i.e., x = 0, for the above conditions.
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