Class Of Exact Solutions Research Articles

On Marangoni shear convective flows of inhomogeneous viscous incompressible fluids in view of the Soret effect

A new exact solution is obtained for the Oberbeck-Boussinesq equations describing the steady-state layered (shear) Marangoni convection of a binary viscous incompressible fluid with the Soret effect. When layered (shear) flows are considered, the Oberbeck-Boussinesq system is overdetermined. For it to be solvable, a class of exact solutions is constructed, which allows one to satisfy identically the “superfluous” equation (the incompressibility equation). The found exact solution allows the Oberbeck-Boussinesq system of equations to be reduced to a system of ordinary differential equations by the generalized method of separation of variables. The resulting system of ordinary differential equations has an analytical solution, which is polynomial. The polynomial velocity field describes counterflows in the case of a convective fluid flow. It is demonstrated that the components of the velocity vector can have one stagnant (zero) point inside the region under study. In this case, the corresponding component of the velocity vector can be stratified into two zones, in which the fluid flows in opposite directions. The exact solution describing the velocity field for the Marangoni convection of a binary fluid has non-zero helicity, the flow itself being almost everywhere vortex.

Symmetries and special solutions of a parabolic chemotaxis system

In this paper, we consider a class of chemotaxis models with two arbitrary constitutive functions g(u) and f(v). After having performed a complete symmetry group classification with respect to them the reduced systems are derived. By considering g(u) of the logistic form, wide classes of exact solutions are found.

Exact Solutions of the Equation of a Nonlinear Conductor Model

We consider a nonlinear Sobolev equation of the third order describing the electric field potential in a conductor. Six classes of exact solutions are constructed, and their qualitative behavior is analyzed.

A NEW PERSPECTIVE TO STUDY THE THIRD-ORDER MODIFIED KDV EQUATION ON FRACTAL SET

In this paper, we construct the Bäcklund transformations and the super-position formulas to the constant coefficients local fractional Riccati equation for the first time. Next, by means of the Bäcklund transformations and seed solutions which have been known in [X. J. Yang et al., Non-differentiable solutions for local fractional nonlinear Riccati differential equations, Fundam. Inform. 151(1–4) (2017) 409–417], we can get a class of exact solutions to the third-order modified KdV equation on the fractal set. These new type solutions can assist us to review different nonlinear phenomena better, which had been modeled via local fractional derivative.

On classes of exact solutions of quasi-hydrodynamic system

On classes of exact solutions of quasi-hydrodynamic system

Effects of rotation on planetary inertial gravity waves initiated by the stratospheric polar vortex

We examine stability of the polar vortex that initiates planetary-scale westerly (west to east) atmospheric flow that encircles the pole in middle or high latitude of a planet. In our studies, the polar vortex is modeled as an annular flow (zonal flow) extending radially from a rigid inner boundary to a free outer boundary on a Ek− plane, which is parallel to the equatorial plane but generally displaced in latitude. The vortex is associated with a particular class of exact solution of a nonlinear two-dimensional free boundary problem on the nonstationary motion of a perfect fluid propagating around a solid circle with a sufficiently large radius with included Coriolis effect. The intensity of the polar vortex is parametrized by means of the “average” (in vertical) zonal flow that can be associated with atmospheric flow patterns that are parallel or nearly parallel to the lines of latitude. Perturbations are allowed in the zonal and nominally vertical (inward–outward) direction. A linear stability analysis purportedly suggests that increasing latitude decreases the maximum strength of the vortex that is permitted for stability. Three essential parameters that control stability of the vortex are identified.

Stationary cylindrical anisotropic fluid and new purely magnetic GR solutions

The properties of interior spacetimes sourced by stationary cylindrical anisotropic fluids are here analytically studied for both nonrigid and rigid rotation. As regards nonrigid rotation, this is, to our knowledge, the first work dedicated to such a study. We give here a complete equation set describing these spacetime properties. In particular, we focus our attention on both nonrigid and rigid rotation gravito-electromagnetic features and are thus led to display strong hints in favor of conjecturing purely electric Weyl tensor existence in this framework. We have also been able to characterize new purely magnetic physically consistent spacetimes and have found new rigidly rotating exact solution classes to the five Einstein's field equations pertaining to the issue and the two purely magnetic constraints we have derived for this purpose. This should be considered as a prominent result, since extremely few purely magnetic exact solutions are available in the literature.

Cosmological constant in coherent quantum gravity

It is argued that quantum states of geometry, like those of particles, should be coherent on light cones of any size. An exact classical solution, the gravitational shock wave of a relativistic point particle, is used to estimate gravitational drag from coherent energy flows, and the expected gravitational effect of virtual transverse vacuum energy fluctuations on surfaces of causal diamonds. It is proposed that the appropriately spacetime-averaged gravitational effect of the Standard Model vacuum state leads to the observed small nonzero value of the cosmological constant, dominated by gravitational drag of virtual gluonic strings at the strong interaction scale.

Exact solutions to steady radial flow in a porous medium with variable permeability

A wide class of exact solutions to equations of steady flow of ideal gas in a porous medium is obtained in a planar annular domain comprising sectors of distinct permeabilities. The formulation involves an unconventional Sturm–Liouville problem. The class also admits solutions to configurations with respective distinct generation rates. The solutions are expected to be applicable to landfill gas or natural gas extraction. The breaking of axial symmetry enables realistic modeling of the radius of influence for both horizontal and vertical wells. The analysis proves that the well’s reach would have an azimuthal dependence in any problem with a heterogeneous medium. Therefore, it is suggested that the concept of radius of influence as perceived today be revised.

A generating function and new exact solutions for geodesic matter

In this paper we study a geodesic radiating stellar model in which the Einstein field equations contain a cosmological constant and electric charge. Firstly, we study the boundary condition by introducing a generating function that is directly related to the horizon function. This generating function transforms the boundary condition into an algebraic equation which is solvable. New classes of exact solutions can be obtained by specifying a form for the generating function. It is possible to express physical quantities such as the mass and compactness factor in terms of the generating function. We also regain earlier results with only electric charge in this process. Secondly, we transformed the boundary condition into a second order differential equation in terms of the generating function. By specifying certain parameters we solved the boundary condition by direct integration and obtained a new special class of exact solutions in terms of confluent hypergeometric functions.

On some exact solutions of heavenly equations in four dimensions

Some new classes of exact solutions (so-called functionally invariant solutions) of the elliptic and hyperbolic complex Monge–Ampère equations and of the second heavenly equation are found. Besides, non-invariance of the found classes of solutions of the hyperbolic complex Monge–Ampère equation and the second heavenly equation is investigated—these classes of solutions determine the new classes of metrics without Killing vectors. A criterion of non-invariance of the solutions belonging to the found classes is also formulated.

Exact Solution of 3D Navier–Stokes Equations

Procedure for constructing exact solutions of 3D Navier–Stokes equations for an incompressible fluid flow is proposed. It is based on the relations representing the previously obtained first integral of the Navier–Stokes equations. A primary generator of particular solutions is proposed. It is used to obtain new classes of exact solutions

Exact solutions and conservation laws in dissipative fluid dynamics

In this paper the 1D Euler system with a source term is considered in the case of isentropic flows. Classes of exact solutions of the equations under interest have been determined within the framework of the differential constraint method and a Riemann problem was solved. Finally Conservation Laws of the system here considered have been determined following the Direct Method.

Photons as solitons

The field model of space-time film is considered. New class of exact solutions is discussed. These solutions are solitons moving with the speed of light or light-like solitons. Subclass of these solutions with approximate helical symmetry or having twist is considered. Determinate conformity of twisted light-like solitons and photons is shown. In particular, the obtained expression of the soliton energy coincides with the appropriate expression for photon in high-frequency approximation. Planck formula is obtained (in some approximation) for the equilibrium energy spectral density of the ideal gaze of the twisted solitons. An experimental check of the modified formula for the energy of photon (in low-frequency region) as the twisted light-like soliton is proposed.

Mathematical Models and Methods for Calculating the Stress-Strain State of Aircraft Panels from Composite Materials Taking into Account the Production Technology

The relations of the mathematical model for studying the stress-strain state of structurally anisotropic panels made of composite materials are given. The mathematical model of the reinforcing element is clarified under conditions of one-sided contact with the skin. The influence of panels manufacturing technology is taken into account: residual temperature stresses and prestressing of reinforcing fibers. Based on the Lagrange variational principle, an eighth-order resolving equation and natural boundary conditions are obtained. A class of exact solutions of boundary value problems is considered. In the MATLAB operating environment, a software package has been developed. The results of the numerical implementation of the solution and the results of field experiments are presented.

Dynamics and exact solutions of the generalized Harry Dym equation

The Harry Dym equation is the third-order evolutionary partial differential equation. It describes a system in which dispersion and nonlinearity are coupled together. It is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It has an infinite number of conservation laws and does not have the Painleve property. The Harry Dym equation has strong links to the Korteweg – de Vries equation and it also has many properties of soliton solutions. A connection was established between this equation and the hierarchies of the Kadomtsev – Petviashvili equation. The Harry Dym equation has applications in acoustics: with its help, finite-gap densities of the acoustic operator are constructed. The paper considers a generalization of the Harry Dym equation, for the study of which the methods of the theory of finite-dimensional dynamics are applied. The theory of finite-dimensional dynamics is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of evolutionary differential equations. In our case, this approach allows us to obtain some classes of exact solutions of the generalized equation, and also indicates a method for numerically constructing solutions.

A Method for the Construction of Exact Solutions to the Nonlinear Heat Equation Ut = (F(U)Ux)X + G(U)Ux + H(U)

We propose a method for the construction of exact solutions to the nonlinear heat equation based on the classical method of separation of variables and its generalization. We consider substitutions, which reduce the nonlinear heat equation to a system of two ordinary differential equations and construct the classes of exact solutions by the method of generalized separation of variables.

Quantum quench in c = 1 matrix model and emergent space-times

We consider quantum quench in large-N singlet sector quantum mechanics of a single hermitian matrix in the double scaling limit. The time dependent parameter is the self-coupling of the matrix. We find exact classical solutions of the collective field theory of the eigenvalue density with abrupt and smooth quench profiles which asymptote to constant couplings at early and late times, and with the system initially in its ground state. With adiabatic initial conditions we find that adiabaticity is always broken regardless of the quench speed. In a class of quench profiles the saddle point solution for the collective field diverges at a finite time, and a further time evolution becomes ambiguous. However the underlying matrix model expressed in terms of fermions predict a smooth time evolution across this point. By studying fluctuations around the saddle point solution we interpret the emergent space-times. They generically have spacelike boundaries where the couplings of the fluctuations diverge and the semi-classical description fails. Only for very finely tuned quench profiles, the space-time is normal.

A family of exact models for radiating matter

In this paper, the cosmological constant and electric charge are incorporated in the Einstein–Maxwell field equations. Two approaches are used to investigate the problem. First, the boundary condition is expressed as a generalized Riccati equation in one of the gravitational potentials. New classes of exact solutions are found by writing the Riccati equation in linear, Bernoulli, and inhomogeneous forms. Our solutions contain previous results in the absence of the cosmological constant and charge. Second, it is possible to preserve the form of the generalized Riccati equation by introducing a transformation called the horizon function. This transformation simplifies the generalized Riccati equation. We generate new solutions to the transformed Riccati equation when one of the metric functions serves as a generating function. We also obtain other families of new classes of exact solutions, where the horizon function serves as a generating function. Interestingly, new uncharged solutions, not contained in previous studies, arise as special cases of the inhomogeneous Riccati equation in both approaches.

A new class of exact solutions in the planar nonstationary problem of motion of a fluid with a free boundary

We consider the classical problem of potential unsteady flow of an ideal incompressible fluid with a free boundary. It was previously discovered that in the absence of external forces and capillarity, a wide class of exact solutions of the problem can be described by the Hopf equation for a complex velocity. We here obtain a new class of solutions described by the Hopf equation for a quantity that is the inverse of the complex velocity. These solutions describe the evolution of two-dimensional perturbations of the free boundary in compression or expansion of a circular cavity (in the unperturbed state) in the fluid.