This paper is concerned with the oscillatory behavior of the second-order half-linear advanced dynamic equation r t xΔ t γ Δ p t x g t 0 on an arbitrary time scale T with sup T ∞, where g t ≥ t and ∫∞ to Δs/ r1/γ s < ∞. Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify th...

Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adju...

We consider the half-linear second order differential equation which is viewed as a perturbation of the so-called Riemann-Weber half-linear differential equation. We present a comparison theorem with respect to the power of the half-linearity in the equation under consideration. Our research is motivated by the recent results published by J. Sugie, N.Yamaoka, Acta Math. Hungar. 111 (2006), 165–...

We study the generalized half-linear second order differential equation and the associated Riccati type differential equation. We introduce the concepts of minimal and principal solutions of these equations and using these concepts we prove a new conjugacy criterion for the generalized half-linear equation.

In this study‎, ‎an efficient method is presented for solving infinite boundary integro-differential equations (IBI-DE) of the second kind with degenerate kernel in terms of Laguerre polynomials‎. ‎Properties of these polynomials and operational matrix of integration are first presented‎. ‎These properties are then used to transform the integral equation to a matrix equation which corresponds t...