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...

Employing Riccati techniques and the integral averaging method, we establish interval oscillation criteria for the second-order Emden-Fowler neutral delay differential equation [|x′(t)|γ−1x′(t)]′ + q1(t)|y(t− σ)|α−1y(t− σ) + q2(t)|y(t− σ)|β−1y(t− σ) = 0, where t ≥ t0 and x(t) = y(t) + p(t)y(t − τ). The criteria obtained here are different from most known criteria in the sense that they are base...

We consider the following anisotropic Emden-Fowler equation ∇(a(x)∇u) + εa(x)e = 0 in Ω, u = 0 on ∂Ω where Ω ⊂ R is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of bubbling solutions. We show that at given local maximum points of a(x), there exists arbitrarily many bubbles. As a consequence, the quanti...

We prove that the following pointwise inequality holds −∆u ≥ √ 2 (p + 1)− cn |x| a 2 u p+1 2 + 2 n− 4 |∇u|2 u in R where cn := 8 n(n−4) , for positive bounded solutions of the fourth order Hénon equation that is ∆u = |x|u in R where a ≥ 0 and p > 1. Motivated by the Moser iteration argument in the regularity theory, we develop an iteration argument to prove the above pointwise inequality. As fa...

The stability question of the Lane-Emden stationary gaseous star configurations is an interesting problem arising in astrophysics. We establish both linear and nonlinear dynamical instability results for the Lane-Emden solutions in the framework of the Navier-Stokes-Poisson system with adiabatic exponent 6/5 < γ < 4/3.

Numerical continuation calculations for ordinary differential equations (ODEs) are, by now, an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences. Although several excellent standard software packages are available for ODEs, there are for good reasons no standard numerical continuation toolboxes available for parti...