We provide a general solution for first order ordinary differential equation with rational right-hand side, which arises in constructing asymptotics large time of simultaneous solutions the Korteweg-de Vries and stationary part its higher non-autonomous symmetry. This symmetry is determined by linear combination autonomous classical Galileo depends on an arbitrary parameter. By implicit functio...

A phase function is an important characteristic of a scattering medium. A method to derive new analytic phase functions is proposed. The relation between a phase function and an angle-averaged single-scattering intensity, derived earlier [M. L. Shendeleva, J. Opt. Soc. Am. A 30, 2169 (2013)], is considered as an integral equation for a phase function. This equation is classified as an Abel inte...

We investigate the concept of Abel continuity. A function f defined on a subset of ℝ, the set of real numbers, is Abel continuous if it preserves Abel convergent sequences. Some other types of continuities are also studied and interesting result is obtained. It turned out that uniform limit of a sequence of Abel continuous functions is Abel continuous and the set of Abel continuous functions is...

In this paper, a homotopy perturbation method (HPM) and modified homotopy perturbation (MHPM) are proposed to solve singular integral equation with generalized Abel's kernel.