21 research outputs found
Applications of Group Analysis to the Three-Dimensional Equations of Fluids with Internal Inertia
Group classification of the three-dimensional equations describing flows of fluids with internal inertia, where the potential function W = W(ρ,ρ·), is presented. The given equations include such models as the non-linear one-velocity model of a bubbly fluid with incompressible liquid phase at small volume concentration of gas bubbles, and the dispersive shallow water model. These models are obtained for special types of the function W(ρ,ρ·). Group classification separates out the function W(ρ,ρ·) at 15 different cases. Another part of the manuscript is devoted to one class of partially invariant solutions. This solution is constructed on the base of all rotations. In the gas dynamics such class of solutions is called the Ovsyannikov vortex. Group classification of the system of equations for invariant functions is obtained. Complete analysis of invariant solutions for the special type of a potential function is given
Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - II: Partial Differential Equations
The linearization of complex ordinary differential equations is studied by
extending Lie's criteria for linearizability to complex functions of complex
variables. It is shown that the linearization of complex ordinary differential
equations implies the linearizability of systems of partial differential
equations corresponding to those complex ordinary differential equations. The
invertible complex transformations can be used to obtain invertible real
transformations that map a system of nonlinear partial differential equations
into a system of linear partial differential equation. Explicit invariant
criteria are given that provide procedures for writing down the solutions of
the linearized equations. A few non-trivial examples are mentioned.Comment: This paper along with its first part ODE-I were combined in a single
research paper "Linearizability criteria for systems of two second-order
differential equations by complex methods" which has been published in
Nonlinear Dynamics. Due to citations of both parts I and II these are not
replaced with the above published articl
Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source
A new approach to group classification problems and more general
investigations on transformational properties of classes of differential
equations is proposed. It is based on mappings between classes of differential
equations, generated by families of point transformations. A class of variable
coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the
general form () is studied from the
symmetry point of view in the framework of the approach proposed. The singular
subclass of the equations with is singled out. The group classifications
of the entire class, the singular subclass and their images are performed with
respect to both the corresponding (generalized extended) equivalence groups and
all point transformations. The set of admissible transformations of the imaged
class is exhaustively described in the general case . The procedure of
classification of nonclassical symmetries, which involves mappings between
classes of differential equations, is discussed. Wide families of new exact
solutions are also constructed for equations from the classes under
consideration by the classical method of Lie reductions and by generation of
new solutions from known ones for other equations with point transformations of
different kinds (such as additional equivalence transformations and mappings
between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica
Generalized Contour Dynamics: A Review
Contour dynamics is a computational technique to solve for the motion of vortices in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours is followed, and the velocity field moving the contours can be computed as integrals along the contours. Its best-known examples are in two dimensions, for which the vorticity between contours is taken to be constant and the vortices are vortex patches, and in axisymmetric flow for which the vorticity varies linearly with distance from the axis of symmetry. This review discusses generalizations that incorporate additional physics, in particular, buoyancy effects and magnetic fields, that take specific forms inside the vortices and preserve the contour dynamics structure. The extra physics can lead to time-dependent vortex sheets on the boundaries, whose evolution must be computed as part of the problem. The non-Boussinesq case, in which density differences can be important, leads to a coupled system for the evolution of both mean interfacial velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities are materially conserved and whose evolution governs the flow
Generalized Riemann waves and their adjoinment through a shock wave
Generalized simple waves of the gas dynamics equations in Lagrangian and Eulerian descriptions are studied in the paper. As in the collision of a shock wave and a rarefaction wave, a flow becomes nonisentropic. Generalized simple waves are applied to describe such flows. The first part of the paper deals with constructing a solution describing their adjoinment through a shock wave in Eulerian coordinates. Even though the Eulerian form of the gas dynamics equations is most frequently used in applications, there are advantages for some problems concerning the gas dynamics equations in Lagrangian coordinates, for example, of being able to be reduced to an Euler–Lagrange equation. Through the technique of differential constraints, necessary and sufficient conditions for the existence of generalized simple waves in the Lagrangian description are provided in the second part of the paper