Class Of Exact Solutions Research Articles

ON THE STRUCTURE OF HELICAL AXISYMMETRIC SOLUTIONS OF THE NAVIER-STOKES SYSTEM FOR INCOMPRESSIBLE FLUIDS

A class of exact solutions to the Navier-Stokes equations for axisymmetric vortex flows of incompressible fluids is obtained. Invariant manifolds of flows with rotational symmetry relative to a given axis in three-dimensional coordinate space are identified, and the structure of the solutions is described. It is established that typical invariant regions of such flows are rotational figures homeomorphic to a torus, forming a structure of topological fibration, such as in a sphere, cylinder, and more complex configurations composed of such figures. The results are extended to similar solutions of the magnetohydrodynamics (MHD) system and Maxwell’s electrodynamics equations, which possess ℝ3 analogous properties. Examples of axisymmetric vortex vector fields and the topological fibrations they generate on manifolds invariant ℝ3 under the dynamical systems defined by these fields are provided.

Analogous Hawking Radiation in Dispersive Media

In the framework of the analogous Hawking effect, we significantly improve our previous analysis of the master equation that encompasses very relevant physical systems, like Bose–Einstein condensates (BECs), dielectric media, and water. In particular, we are able to provide two significant improvements to the analysis. As our main result, we provide a complete set of connection formulas for both the subluminal and superluminal cases without resorting to suitable boundary conditions, first introduced by Corley, but simply on the grounds of a rigorous mathematical setting. Moreover, we provide an extension to the four-dimensional case, showing explicitly that, apart from obvious changes, adding transverse dimensions does not substantially modify the Hawking temperature in the dispersive case. Furthermore, an important class of exact solutions of the so-called reduced equation that governs the behavior of non-dispersive modes is also provided.

Multiple soliton solutions and other travelling wave solutions to new structured (2+1)-dimensional integro-partial differential equation using efficient technique

The Ito equation belongs to the Korteweg–de Vries (KdV) family and is commonly employed to predict how ships roll in regular seas. Additionally, it characterizes the interaction between two internal long waves. In the 1980s, Ito extended the bilinear KdV equation, resulting in the well-known (1+1)-dimensional and (2+1)-dimensional Ito equations. In this study finds numerous classes of exact solutions for a new structured (2 + 1)-dimensional Ito integro-differential equation using the help of the Mathematica software. The Improved Modified Extended Tanh Function Scheme (IMETFS) is utilised to address the aforementioned equation analytically. Bright, dark, and singular soliton solutions are produced. Additionally, periodic, exponential, rational, singular periodic, and Weierstrass elliptic doubly periodic results are achieved. The method employed includes the nonlinear evolution equations that arise in a variety of real-world situations, and it is efficient, applicable, and simple to handle. For certain obtained solutions, specific options of free constants are presented in 3D, 2D, and contour graphical depictions.

Evaluating the effects of nonlocality and numerical discretization in peridynamic solutions for quasi-static elasticity and fracture

The difference between the computational peridynamic results and the corresponding exact classical solutions is contributed by the error induced by numerical discretization and the nonlocality-induced difference. To evaluate and compare these contributions and investigate their dependence on the peridynamic influence functions, in this paper, we apply different peridynamic influence functions and different horizon factors (m, the ratio between the horizon size and the grid spacing) to conduct static (or quasi-static) tensile numerical tests. We calculate the difference between the peridynamic solutions and the corresponding classical solutions for static uniaxial tension of thin plates with or without hole, the J-integral of a single crack under Mode I loading condition, and the quasi-static fracture in a perforated thin plate. For the case of uniaxial tension in a thin, homogeneous plate, we separate the effects of nonlocality and numerical discretization by implementing the boundary conditions in different ways. The numerical results show that both the effects of nonlocality and numerical discretization correspond to the nonlocal constants of the influence functions (with few exceptions). For problems with the presence of the peridynamic surface effect, such as holes, influence function with weaker nonlocality is a better choice to obtain more accurate results.

New features of circular geodesics in Kalb–Ramond gravity: an autonomous dynamical system approach

Kalb–Ramond gravity is expected to show the signatures of the low energy limit of the Lorentz violating extension of the standard model of particle physics. To understand the theory further, we study the circular geodesics belonging to a class of exact solutions [found recently by Lessa, Silva, Maluf, Almeida (LSMA)] by posing them as a Hamiltonian dynamical autonomous system. The analysis on the phase space for relevant values of the Lorentz violating parameters reveals for the first time a wealth of novel information that include prediction of homoclinic orbits, saddle points, separatrices that are not available from the conventional analysis. Their image on the actual physical space is discussed.

Integrability, Similarity Reductions and New Classes of Exact Solutions for (3+1)-D Potential Yu–Toda–Sasa–Fukuyama Equation

Integrability, Similarity Reductions and New Classes of Exact Solutions for (3+1)-D Potential Yu–Toda–Sasa–Fukuyama Equation

Exact solutions of the Navier–Stokes equations generated by the eigenfunctions of the Stokes operator

We derive a class of exact solutions to the incompressible unsteady Navier–Stokes equations. These solutions are generated by the eigenfunctions of the Stokes operator and can be applied to studying boundary value problems for the Navier–Stokes equations related to pipe and channel flows. We explore the relation between the spectrum of the Stokes operator, generalized Beltrami flows, and exact solutions of the Navier–Stokes equations. Relevant properties of the Stokes eigenfunctions are also discussed in some detail.

Solutions with pure radiation and gyratons in 3D massive gravity theories

We find exact solutions of topologically massive gravity (TMG) and new massive gravity (NMG) in 2 + 1 dimensions (3D) with an arbitrary cosmological constant, pure radiation, and gyratons, i.e. with possibly non-zero T uu and T ux in canonical coordinates. Since any ‘reasonable’ geometry in 3D (i.e. admitting a null geodesic congruence) is either expanding Robinson–Trautman ( Θ≠0 ) or Kundt ( Θ=0 ), we focus on these two classes. Assuming expansions Θ=1/r (‘GR-like’ Robinson–Trautman) or Θ=0 (general Kundt), we systematically integrate the field equations of TMG and NMG and identify new classes of exact solutions. The case of NMG contains an additional assumption of g ux being quadratic in r, which is automatically enforced in TMG as well as in 3D GR. In each case, we reduce the field equations as much as possible and identify new classes of solutions. We also discuss various special subclasses and study some explicit solutions.

Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD, and SIR Compartment Models

The susceptible–infected–recovered–vaccinated–deceased (SIRVD) epidemic compartment model extends the SIR model to include the effects of vaccination campaigns and time-dependent fatality rates on epidemic outbreaks. It encompasses the SIR, SIRV, SIRD, and SI models as special cases, with individual time-dependent rates governing transitions between different fractions. We investigate a special class of exact solutions and accurate analytical approximations for the SIRVD and SIRD compartment models. While the SIRVD and SIRD equations pose complex integro-differential equations for the rate of new infections and the fractions as a function of time, a simpler approach considers determining equations for the sum of ratios for given variations. This approach enables us to derive fully exact analytical solutions for the SIRVD and SIRD models. For nonlinear models with a high-dimensional parameter space, such as the SIRVD and SIRD models, analytical solutions, exact or accurately approximative, are of high importance and interest, not only as suitable benchmarks for numerical codes, but especially as they allow us to understand the critical behavior of epidemic outbursts as well as the decisive role of certain parameters. In the second part of our study, we apply a recently developed analytical approximation for the SIR and SIRV models to the more general SIRVD model. This approximation offers accurate analytical expressions for epidemic quantities, such as the rate of new infections and the fraction of infected persons, particularly when the cumulative fraction of infections is small. The distinction between recovered and deceased individuals in the SIRVD model affects the calculation of the death rate, which is proportional to the infected fraction in the SIRVD/SIRD cases but often proportional to the rate of new infections in many SIR models using an a posteriori approach. We demonstrate that the temporal dependence of the infected fraction and the rate of new infections differs when considering the effects of vaccinations and when the real-time dependence of fatality and recovery rates diverge. These differences are highlighted for stationary ratios and gradually decreasing fatality rates. The case of stationary ratios allows one to construct a new powerful diagnostics method to extract analytically all SIRVD model parameters from measured COVID-19 data of a completed pandemic wave.

Interaction solutions of (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation via bilinear method

Using the bilinear neural network method (BNNM) and the symbolic computation system Mathematica, this paper explains how to find an exact solution for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation. In terms of activation function and weight coefficient, BNNM is a more appealing option for users than traditional symbolic computation methods. It is possible to develop a wide range of solutions and expand the classes of exact solutions by modifying the activation function. The activation function’s versatility allows it to generate a wide range of solutions with several theoretical and practical uses. The analytical solution is obtained by using a double layer type, while the rogue wave solution and mixed solutions are obtained by using a single layer type. The evolution of these waves is then illustrated using various 3D graphs, 2D graphs, and density plots.

The (2+1)-dimensional generalized Benjamin–Ono equation: Nonlocal symmetry, CTE solvability and interaction solutions

In this paper, the nonlocal symmetry of the (2+1)-dimensional generalized Benjamin–Ono equation are obtained by using the truncated Painlevé expansion. The nonlocal symmetry are localized by introducing auxiliary variables. Furthermore, this equation is also proved to be consistent tanh expansion solvable, and three classes of exact solutions are derived.

A new discretization of the Euler equation via the finite operator theory

We propose a novel discretization procedure for the classical Euler equation, based on the theory of Galois differential algebras and the finite operator calculus developed by G.C. Rota and collaborators. This procedure allows us to define algorithmically a new discrete model which inherits from the continuous Euler equation a class of exact solutions.

Anisotropic compact stellar objects with a slow rotation effect

A new class of exact solutions depicting anisotropic compact objects is presented in the current work. This spherically symmetric matter distribution assumes a specific form of anisotropy to obtain the exact solution for the field equations. The obtained interior solutions are smoothly matched with the Schwarzschild exterior metric over the bounding surface of a compact star and together with the condition that the radial pressure vanishes at the boundary, the form of the model parameters are attained. One of the interesting features of the obtained solutions is the codependency of the metric potentials. We have considered the pulsar 4U1608-52 with its current estimated data (mass =1.57-0.29+0.30M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$=1.57^{+0.30}_{-0.29} ~M\\odot $$\\end{document} and radius =9.8±1.8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$=9.8 \\pm 1.8$$\\end{document} km [Özel in Astrophys J 820(1):28, 2016]) to study the model graphically. Moreover, we have studied the physical features and some important stability conditions for the model. Tabular comparison with other known pulsars infers that the obtained model represents a compact star within a radius of 8–12 km. Finally, we have found the angular momentum that causes the dragging of inertial frames of the slowly rotating equilibrium compact objects.

Two classes of exact solutions in the linear elastodynamics of transversely isotropic solids

Two classes of exact solutions in the linear elastodynamics of transversely isotropic solids

Molecular dynamics analysis of interatomic potentials of vanadium using exact solutions of the equations of atomic motion

This paper describes a special class of exact solutions to the equations of motion of atoms in a bcc lattice, which exist for every chemical element with such a lattice. Such exact solutions are called delocalized nonlinear vibrational modes (DNVMs). We propose to use 14 single-component short-wavelength DNVMs of the bcc lattice to compare their frequency responses calculated for vanadium using the molecular dynamics method with different interatomic potentials. In the harmonic (small-amplitude) approximation, different potentials give a good estimate of the DNVM vibrational frequencies, but for molecular dynamics the anharmonic (nonlinear) vibrations are of key importance. The DNVMs under consideration are derived from the bush theory developed by Chechin and Sakhnenko and correspond in the low amplitude range to phonon modes with a wave vector at highly symmetric points on the boundary of the first Brillouin zone. However, due to the high symmetry of the oscillations, DNVMs do not interact with other phonon modes even at large amplitudes and thus characterize the nonlinear part of the frequency responses. The frequency responses of DNVMs have been calculated for vanadium with three classical and one machine-learned interatomic potential using the LAMMPS software package, and a significant discrepancy of the results in the nonlinear region has been shown. Thus, the need to improve modern approaches to the development and testing of interatomic potentials is demonstrated.

Exact solutions of the Oberbeck–Boussinesq equations for describing the creeping flows of multicomponent fluids

A new class of exact solutions of magnetohydrodynamic equations for description of creeping flows of multilayer incompressible fluids with the Rayleigh dissipative function is found. The velocity field is described by linear forms with respect to two spatial (horizontal, longitudinal) coordinates. The coefficients of the linear forms depend on the third (vertical, transverse) coordinate. The pressure field and the temperature field are quadratic forms with similar dependence on coordinates and coefficients as in the Lin-Sidorov-Aristov class.

A New Class of Exact Solutions for Magnetohydrodynamics Equations to Describe Convective Flows of Binary Liquids

A New Class of Exact Solutions for Magnetohydrodynamics Equations to Describe Convective Flows of Binary Liquids

Non-Stationary Helical Flows for Incompressible Couple Stress Fluid

We explored here the case of three-dimensional non-stationary flows of helical type for the incompressible couple stress fluid with given Bernoulli-function in the whole space (the Cauchy problem). In our presentation, the case of non-stationary helical flows with constant coefficient of proportionality α between velocity and the curl field of flow is investigated. In the given analysis for this given type of couple stress fluid flows, an absolutely novel class of exact solutions in theoretical hydrodynamics is illuminated. Conditions for the existence of the exact solution for the aforementioned type of flows were obtained, for which non-stationary helical flow with invariant Bernoulli-function satisfying to the Laplace equation was considered. The spatial and time-dependent parts of the pressure field of the fluid flow should be determined via Bernoulli-function if components of the velocity of the flow are already obtained. Analytical and numerical findings are outlined, including outstanding graphical presentations of various types of constructed solutions, in order to elucidate dynamic snapshots that show the timely development of the topological behavior of said solutions.

Entropy-based investigation of blood flow in elliptical multi-stenotic artery with hybrid nanofluid in a fuzzy environment: Applications as drug carriers for brain diseases

The growing stenosis alters the blood flow characteristics and causes different diseases of arteries such as Carotid artery, Peripheral artery, and Coronary artery diseases. The proposed study aims to investigate the blood-based hybrid nanofluid flow via a multi-stenotic elliptical artery. The nanoparticles are vital as drug carriers in curing arterial diseases with consequences like Carotid artery disease (causes brain diseases), Peripheral artery disease, etc., and other medical applications. The single-wall and multi-wall carbon nanotubes are accounted as nanoparticles in blood. The mathematical equations have been transformed into dimensionless form and reduced non-linearity using mild case assumptions to get the exact solution. The local sensitivity analysis of temperature solution clarifies that the volume fraction of both types of nanoparticles is the most and equally influential parameter. The sensitivity of these parameters causes uncertainty in the temperature profile, and therefore, they are also analyzed in the fuzzy environment by considering them as triangular fuzzy numbers. The exact classical and fuzzy solutions are examined graphically in a comprehensive way. It is found that the non-symmetric stenosis shape and developing stenosis height cause to decline and grow the fluid’s velocity, respectively, but the flow nature reverses near the wall of the stenotic elliptic artery. The temperature profile appeared more accurate in parabolic shape along the major axis, while a quick decrease occurred near the stenosed wall along the minor axis. Further, the temperature profile revealed as triangular fuzzy number of non-uniform shapes at various points inside the elliptic artery.

On One Class of Exact Solutions of the Navier–Stokes System of Equations for an Incompressible Fluid

On One Class of Exact Solutions of the Navier–Stokes System of Equations for an Incompressible Fluid