TITLE:
Applications of Non-Classical Equations and Their Approaches to the Solution of Some of Classes Equations Arise in the Kelvin-Helmholtz Mechanism and Instability
AUTHORS:
Mahammad A. Nurmammadov
KEYWORDS:
Kelvin-Helmholtz Mechanism and Instability, Ordinary Differential Equations, Weighted Space, Degenerating, Planetary, Jupiter, Non-Classical Approaches
JOURNAL NAME:
Open Journal of Applied Sciences,
Vol.12 No.11,
November
22,
2022
ABSTRACT: In the presented work, we consider applications of non-classical
equations and their approaches to the solution of some classes of equations
that arise in the Kelvin-Helmholtz Mechanism (KHM) and instability. In all
areas where the Kelvin-Helmholtz instability (KHI) problem is investigated with the corresponding
data unchanged, the solution can be taken directly in a specific form (for
example, to determine the horizontal structure of a perturbation in a
barotropic rotational flow, which is a boundary condition taken, as well as
other types of Kelvin-Helmholtz instability problems). In another example, the
shear flow along the magnetic field in the Z direction, which is the width of
the contact layer between fast and slow flows, has a velocity gradient along
the X axis with wind shear. The most difficult problems arise when the above
unmentioned equation has singularities simultaneously at points and in this
case, our results also remain valid. In the case of linear wave analysis of
Kelvin-Helmholtz instability (KHI) at a tangential discontinuity (TD) of ideal
magneto-hydro-dynamic (MHD) plasma, it can be attributed to the presented
class, and in this case, as far as we know, solutions for eigen modes of
instability KH in MHD plasma that satisfy suitable homogeneous boundary
conditions. Based on the above mentioned area of application for degenerating
ordinary differential equations in this work, the method of functional analysis
in order to prove the generalized solution is used. The investigated equation
covers a class of a number of difficult-to-solve problems, namely, generalized solutions are found for classes
of problems that have analytical and mathematical descriptions. With the aid of
lemmas and theorems, the existence and uniqueness of generalized solutions in
the weight space are proved, and then general and particular exact solutions
are found for the considered problems that are expressed analytically
explicitly. Obtained our results may be used for all the difficult-to-solve
processes of KHM and instabilities and instabilities, which cover widely
studied areas like galaxies, Kelvin-Helmholtz instability in the atmospheres of
planets, oceans, clouds and moons, for example, during the formation of the
Earth or the Red Spot on Jupiter, as well as in the atmospheres of the Sun and
other stars. In this paper, also, a fairly common class of equations and
examples are indicated that can be used directly to enter data for the use of
the studied suitable tasks.