Number Of Exact Solutions Research Articles

Functional and generalized separable solutions to unsteady Navier–Stokes equations

The paper studies unsteady Navier–Stokes equations with two space variables. It shows that the non-linear fourth-order equation for the stream function with three independent variables admits functional separable solutions described by a system of three partial differential equations with two independent variables. The system is found to have a number of exact solutions, which generate new classes of exact solutions to the Navier–Stokes equations. All these solutions involve two or more arbitrary functions of a single argument as well as a few free parameters. Many of the solutions are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary; such solutions, having relatively simple form and presenting significant arbitrariness, can be especially useful for solving certain model problems and testing numerical and approximate analytical hydrodynamic methods. The paper uses the obtained results to describe some model unsteady flows of viscous incompressible fluids, including flows through a strip with permeable walls, flows through a strip with extrusion at the boundaries, flows onto a shrinking plane, and others. Some blow-up modes, which correspond to singular solutions, are discussed.

A generalized simplest equation method and its application to the Boussinesq-Burgers equation.

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.

A Particular Solutions for a Two-Phase Model with a Sharp Interface

Two-phase models can be used to describe the dynamics of mixed materials and can be applied to many physical and biological phenomena. For example, these types of models have been used to describe the dynamics of cancer, biofilms, cytoplasm, and hydrogels. Frequently the physical domain separates into a region of mixed material immersed in a region of pure fluid solvent. Previous works have found a perturbation solution to capture the front velocity at the initial time of contact between the polymer network and pure solvent, then approximated the solution to the sharp-interface at other points in time. The primary purpose of this work is to use a symmetry transformation to capture an exact solution to this two-phase problem with asharp-interface. This solution is useful for a variety of reasons. First, the exact solution replicates the numeric results, but it also captures the dynamics of the volume profile at the boundary between phases for arbitrary time scales. Also, the solution accounts for dispersion of the network further away from the boundary. Further, our findings suggest that an infinite number of exact solutions of various classes exist for the two-phase system, which may give further insights into the behaviors of the general two-phase model.

A Novel Analytical Technique to Obtain Kink Solutions for Higher Order Nonlinear Fractional Evolution Equations

We use the fractional derivatives in Caputo’s sense to construct exact solutions to fractional fifth order nonlinear evolution equations. A generalized fractional complex transform is appropriately used to convert this equation to ordinary differential equation which subsequently resulted in a number of exact solutions.

Nonlinear noise waves in soft biological tissues

The study of intense waves in soft biological tissues is necessary both for diagnostics and therapeutic aims. Tissue represents an inherited medium with frequency-dependent dissipative properties, in which waves are described by nonlinear integro-differential equations. The equations for such waves are well known. Their group analysis has been performed, and a number of exact solutions have been found. However, statistical problems for nonlinear waves in tissues have hardly been studied. As well, for medical applications, both intense noise waves and waves with fluctuating parameters can be used. In addition, statistical solutions are simpler in structure than regular solutions; they are useful for understanding the physics of processes. Below a general approach is described for solving nonlinear statistical problems applied to the considered mathematical models of biological tissues. We have calculated the dependences of the intensities of the narrowband noise harmonics on distance. For wideband noise, we have calculated the dependence of the spectral integral intensity on distance. In all cases, wave attenuation is determined both by the specific dissipative properties of the tissue and the nonlinearity of the medium.

Boundary Value Problems for a Super-Sublinear Asymmetric Oscillator: The Exact Number of Solutions

Properties of asymmetric oscillator described by the equation (i), where and , are studied. A set of such that the problem (i), (ii), and (iii) have a nontrivial solution, is called α-spectrum. We give full description of α-spectra in terms of solution sets and solution surfaces. The exact number of nontrivial solutions of the two-parameter Dirichlet boundary value problem (i), and (ii) is given.

The recursion operator method for generalized symmetries and Bäcklund transformations of the Burgers’ equations

In this paper, the generalized symmetries of the second-order Burgers’ equation are obtained through the symmetry transformation method. The Backlund transformations (BTs) of the two equations are constructed by the recursion operator method. Then, the infinite number of exact solutions to these equations are investigated in terms of the generalized symmetries and Backlund transformations. Furthermore, the Backlund transformations and conservation law of the general Burgers’ equations are discussed.

Nonclassical Symmetry Analysis of Boundary Layer Equations

The nonclassical symmetries of boundary layer equations for two‐dimensional and radial flows are considered. A number of exact solutions for problems under consideration were found in the literature, and here we find new similarity solution by implementing the SADE package for finding nonclassical symmetries.

A note on the interplay between symmetries, reduction and conservation laws of Stokes’ first problem for third-grade rotating fluids

We investigate the invariance properties, nontrivial conservation laws and interplay between these notions that underly the equations governing Stokes’ first problem for third-grade rotating fluids. We show that a knowledge of this leads to a number of different reductions of the governing equations and, thus, a number of exact solutions can be obtained and a spectrum of further analyses may be pursued.

Exact soliton solutions and their stability control in the nonlinear Schrödinger equation with spatiotemporally modulated nonlinearity

We put forward a generic transformation which helps to find exact soliton solutions of the nonlinear Schrödinger equation with a spatiotemporal modulation of the nonlinearity and external potentials. As an example, we construct exact solitons for the defocusing nonlinearity and harmonic potential. When the soliton's eigenvalue is fixed, the number of exact solutions is determined by energy levels of the linear harmonic oscillator. In addition to the stable fundamental solitons, stable higher-order modes, describing array of dark solitons nested in a finite-width background, are constructed too. We also show how to control the instability domain of the nonstationary solitons.

Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge–Ampère Equation in Dimensions Two and Higher

The elliptic Monge–Ampère equation is a fully nonlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing, and image registration. Solutions can be singular, in which case standard numerical approaches fail. Novel solution methods are required for stability and convergence to the weak (viscosity) solution. In this article we build a wide stencil finite difference discretization for the Monge–Ampère equation. The scheme is monotone, so the Barles–Souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation. Solutions of the scheme are found using a damped Newton's method. We prove convergence of Newton's method and provide a systematic method to determine a starting point for the Newton iteration. Computational results are presented in two and three dimensions, which demonstrates the speed and accuracy of the method on a number of exact solutions, which range in regularity from smooth to nondifferentiable.

Sobolev Inequality and the Exact Multiplicity of Solutions and Positive Solutions to a Second-Order Neumann Boundary Value Problem

In this paper, the exact number of solutions and positive solutions for a second-order Neumann boundary value problem is considered. The exact multiplicity results for this problem are established by Sobolev inequality, comparison theorem, maximum principle and lower and upper solutions method.

The exact number of solutions for the second order nonlinear boundary value problem

Abstract A second order nonlinear differential equation with homogeneous Dirichlet boundary conditions is considered. An explicit expression for the root functions for an autonomous nonlinear boundary value problem is obtained using the results of the paper [SOMORA, P.: The lower bound of the number of solutions for the second order nonlinear boundary value problem via the root functions method, Math. Slovaca 57 (2007), 141–156]. Other assumptions are supposed to prove the monotonicity of root functions and to get the exact number of solutions. The existence of infinitely many solutions of the boundary value problem with strong nonlinearity is obtained by the root function method as well.

Exact solution of a problem on heat exchange in channels under unsteady conditions at low biot number values

A number of exact solutions of a problem on heat exchange in channels under unsteady conditions at the low Biot numbers and sufficiently low Brune numbers with and without consideration for a heat carrier stay period in a pipeline are obtained.

Exact Solutions in 5-D Brane Models With Scalar Fields

In this talk we consider the problem of a scalar field, non-minimally coupled to gravity through a −ξϕ2Rterm, in the presence of a Brane. A number of exact solutions, for a wide range of values of the coupling parameter ϕ for both Á-dependent and Á-independent Brane tension, are presented. The behavior and general features of these solutions are discussed. We derive solutions for the scalar field compatible with the Randall-Sundrum metric and also geometries which can accomodate a folded kink-like scalar. Analytic and numerical results are provided for the case of a Brane or for smooth geometries, where the scalar field acts as a thick Brane. We also discuss briefly the graviton localization in our setup and demonstrate the characteristic volcano-like localizing potential for gravitons.

Two interacting spins in external fields. Four-level systems

In the present article, we consider the so-called two-spin equation that describes four-level quantum systems. Recently, these systems attract attention due to their relation to the problem of quantum computation. We study general properties of the two-spin equation and show that the problem for certain external backgrounds can be identified with the problem of one spin in an appropriate background. This allows one to generate a number of exact solutions for two-spin equations on the basis of already known exact solutions of the one-spin equation. Besides, we present some exact solutions for the two-spin equation with an external background different for each spin but having the same direction. We study the eigenvalue problem for a time-independent spin interaction and a time-independent external background. A possible analogue of the Rabi problem for the two-spin equation is defined. We present its exact solution and demonstrate the existence of magnetic resonances in two specific frequencies, one of them coinciding with the Rabi frequency, and the other depending on the rotating field magnitude. The resonance that corresponds to the second frequency is suppressed with respect to the first one.

Mass distribution exponents for growing trees

We investigate the statistics of trees grown from some initial tree by attaching links topre-existing vertices, with attachment probabilities depending only on the valence of thesevertices. We consider the asymptotic mass distribution that measures the repartition of themass of large trees between their different subtrees. This distribution is shown to be abroad distribution and we derive explicit expressions for scaling exponents thatcharacterize its behaviour when one subtree is much smaller than the others.We show in particular the existence of various regimes with different values ofthese mass distribution exponents. Our results are corroborated by a number ofexact solutions for particular solvable cases, as well as by numerical simulations.

MULTIPLE SOLUTIONS OF NONLINEAR BOUNDARY VALUE PROBLEMS WITH OSCILLATORY SOLUTIONS

We consider two second order autonomous differential equations with critical points, which allow the detection of an exact number of solutions to the Dirichlet boundary value problem. Non‐autonomous equations with similar behaviour of solutions also are considered. Estimations from below of the number of solutions to the Dirichlet boundary value problem are given.

New exact solutions to some difference differential equations

In this paper, we use our method to solve the extended Lotka–Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of sinh and cosh, periodic solutions presented by trigonometric functions of sin and cos, and rational solutions. This method can be used to solve some other nonlinear difference–differential equations.

Global structure of solutions for a class of two-point boundary value problems involving singular and convex or concave nonlinearities

This paper investigates the following two-point singular boundary value problems (BVP): { − x ″ ( t ) = λ x q + x p , t ∈ ( 0 , 1 ) , x ( t ) > 0 , t ∈ ( 0 , 1 ) , x ( 0 ) = x ( 1 ) = 0 , where q < 0 and p > 0 are fixed given numbers; λ ∈ R + = [ 0 , + ∞ ) is a parameter. The results obtained are the global structure of solutions and exact number of solutions when p ⩽ 1 or p > 1 and q is sufficiently small.