19 research outputs found
Abstract kinetic equations with positive collision operators
We consider "forward-backward" parabolic equations in the abstract form , , where and are
operators in a Hilbert space such that , , and
. The following theorem is proved: if the operator is
similar to a self-adjoint operator, then associated half-range boundary
problems have unique solutions. We apply this theorem to corresponding
nonhomogeneous equations, to the time-independent Fokker-Plank equation , , , as well as to
other parabolic equations of the "forward-backward" type. The abstract kinetic
equation , where is injective and
satisfies a certain positivity assumption, is considered also.Comment: 20 pages, LaTeX2e, version 2, references have been added, changes in
the introductio
Transmutation operators as a solvability concept of abstract singular equations
One of the methods of studying differential equations is the transmutation operators method. Detailed study of the theory of transmutation operators with applications may be found in the literature. Application of transmutation operators establishes many important results for different classes of differential equations including singular equations with Bessel operato
Interphase mass transfer between liquid-liquid counter-current flows. I. Velocity distribution
A theoretical analysis of liquid–liquid counter-current flow in laminar boundary layers with a flat interphase
based on the similarity-variables method has been made. The numerical results for the velocity distribution in
both phases are obtained. The dissipation energy in a boundary layer is found and the results corresponding
to counter-current and co-current flows are compared. The comparison shows significant differences in the
dissipation energy values in the cases of co-current and counter-current flows