Solutions Of Certain Type Research Articles

On meromorphic solutions of certain type of non-linear differential equations

We consider meromorphic solutions of non-linear differential equation of the form f n + Qd(z;f) = p1(z)e 1(z) + p2(z)e 2(z) ; where Qd(z;f) is a differential polynomial in f of degree d ≤ n − 2 with rational functions as its coefficients, p1;p2 are rational functions and 1; 2 are polynomials. More precisely and mainly we have shown the conditions concerning ' = ' that will ensure the existence and forms of the possible meromorphic solutions of the above equation. These results have extended and improved some known results obtained most recently.

Extremal solutions for certain type of fractional differential equations with maxima

In this article, we employ the Tarski's fixed point theorem to establish the existence of extremal solutions for fractional differential equations with maxima.

Uniqueness and value distribution of differences of entire functions

We consider the existence of transcendental entire solutions of certain type of non-linear difference equations. As an application, we investigate the value distribution of difference polynomials of entire functions. In particular, we are interested in the existence of zeros of fn(z)(λfm(z+c)+μfm(z))−a, where f is an entire function, n, m are two integers such that n⩾m>0, and λ, μ are non-zero complex numbers. We also obtain a uniqueness result in the case where shifts of two entire functions share a small function.

Inflating baby-Skyrme branes in six dimensions

We consider a six-dimensional brane world model, where the brane is described by a localized solution to the baby-Skyrme model extending in the extra dimensions. The branes have a cosmological constant modeled by inflating four-dimensional slices, and we further consider a bulk cosmological constant. We construct solutions numerically and present evidence that the solutions cease to exist for large values of the brane cosmological constant in some particular case. Then we study the stability of the model by considering perturbation of the gravitational part (resp. baby Skyrmion) with fixed matter fields (resp. gravitational background). Our results indicate that the perturbation equations do not admit localized solutions for certain type of perturbation. The stability analysis can be alternatively seen as leading to a particle spectrum; we give mass estimations for the baby-Skyrme perturbation and for the graviton.

Entire solutions of certain type of differential equations II

We analyze the transcendental entire solutions of the following type of nonlinear differential equations: f n ( z ) + P ( f ) = p 1 e α 1 z + p 2 e α 2 z in the complex plane, where p 1 , p 2 and α 1 , α 2 are nonzero constants, and P ( f ) denotes a differential polynomial in f of degree at most n − 1 with small functions of f as the coefficients.

Exact solutions of variable coefficient nonlinear diffusion–reaction equations with a nonlinear convective term

Attempts have been made to look for the exact solutions of certain types of nonlinear diffusion–reaction equations which involve not only the quadratic and quartic nonlinearities but also a time-dependent nonlinear convective flux term. In particular, the solitary wave solutions are found. Such equations arise in a variety of contexts in physical and biological problems.

Entire solutions of certain type of differential equations

By utilizing Nevanlinna's value distribution theory of meromorphic functions, we solve the transcendental entire solutions of the following type of nonlinear differential equations in the complex plane: f n ( z ) + P ( f ) = p 1 e α 1 z + p 2 e α 2 z , where p 1 and p 2 are two small functions of e z , and α 1 , α 2 are two nonzero constants with some additional conditions, and P ( f ) denotes a differential polynomial in f and its derivatives (with small functions of f as the coefficients) of degree no greater than n − 1 .

Some new solitary and travelling wave solutions of certain nonlinear diffusion–reaction equations using auxiliary equation method

Attempts are made to look for the soliton content in the exact solutions of certain types of nonlinear diffusion–reaction (DR) equations with the quadratic and cubic nonlinearities. Such equations may arise in a variety of contexts in physical problems. In this Letter using the auxiliary equation method, some new solitary and travelling wave solutions of such nonlinear DR equations are obtained in a very general form. Several interesting special cases of these general solutions are also discussed.

Soliton-like solutions of certain types of nonlinear diffusion–reaction equations with variable coefficient

Abstract With a view to exploring new soliton-like solutions of certain types of nonlinear diffusion–reaction (DR) equations with a variable coefficient, we demonstrate the viability of a method which is the combination of both the symbolic computation technique of Gao and Tian [Y.T. Gao, B. Tian, Comput. Phys. Commun. 133 (2001) 158] and auxiliary equation method of Sirendaoreji [Sirendaoreji, Phys. Lett. A 356 (2006) 124] and used recently for the KdV equation. In particular, the DR equations with quadratic and cubic nonlinearities with a time-dependent velocity in the convective flux term are studied and the existence of soliton-like solutions is shown.

NONLINEAR NON-LOCAL BOUNDARY-VALUE PROBLEMS AND PERTURBED HAMMERSTEIN INTEGRAL EQUATIONS

AbstractMotivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the form$$ u(t)=\gamma(t)\alpha[u]+\int_{G}k(t,s)f(s,u(s))\,\mathrm{d}s, $$where $G$ is a compact set in $\mathbb{R}^{n}$. Here $\alpha[u]$ is a positive functional and $f$ is positive, while $k$ and $\gamma$ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

Loss of positivity in a nonlinear scalar heat equation

Using the theory of fixed point index, we discuss the existence of nontrivial (multiple) solutions of a nonlinear scalar heat equation with nonlocal boundary conditions depending on a positive parameter. Solutions lose positivity as the parameter decreases. For a certain parameter range, not all solutions can be positive but there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

On the nonexistence of entire solutions of certain type of nonlinear differential equations

By utilizing Nevanlinna's value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations: f n ( z ) + P n − 3 ( f ) = p 1 e α 1 z + p 2 e α 2 z has no nonconstant entire solutions, where n is an integer ⩾4, p 1 and p 2 are two polynomials ( ≢ 0 ) , α 1 , α 2 are two nonzero constants with α 1 / α 2 ≠ rational number, and P n − 3 ( f ) denotes a differential polynomial in f and its derivatives (with polynomials in z as the coefficients) of degree no greater than n − 3 . It is conjectured that the conclusion remains to be valid when P n − 3 ( f ) is replaced by P n − 1 ( f ) or P n − 2 ( f ) .

Correlation Moment Analysis and the Time Dependence of Coherence in Systems Described by Nonlinear Partial Differential Equations

Abstract The work presented introduces correlation moment analysis. This technique can be employed to explore the growth of determinism from stochastic initial conditions in physical systems described by non-linear partial differential equations (PDEs) and is also applicable to wholly deterministic situations. Correlation moment analysis allows the analytic determination of the time dependence of the spatial moments of the solutions of certain types of non-linear partial differential equations. These moments provide measures of the growth of processes defined by the PDE, furthermore the results are obtained without requiring explicit solution of the PDE. The development is presented via case studies of the linear diffusion equation and the non-linear Kortweg de-Vries equation which indicate strategies for exploiting the various properties of correlation moments developed in the text. In addition, a variety of results have been developed which show how various classes of terms in PDEs affect the structure of a sequence of correlation moment equations. This allows results to be obtained about the behavior of the PDE solution, in particular how the presence of certain types of terms affects integral measures of the solution. It is also demonstrated that correlation moments provide a very simple, natural approach to determining certain subsets of conserved quantities associated with the PDEs.

The diffusion–reaction (D–R) Hamiltonian and the solutions of certain types of linear and nonlinear D–R equations in one dimension

With a view to having further insight into the mathematical content of the non-Hermitian Hamiltonian associated with the diffusion–reaction (D–R) equation in one dimension, we investigate (a) the solitary wave solutions of certain types of its nonlinear versions, and (b) the problem of real eigenvalue spectrum associated with its linear version or with this class of non-Hermitian Hamiltonians. For case (a) we use the standard techniques to handle the quadratic and cubic nonlinearities in the D–R equation whereas for case (b) a newly proposed method, based on an extended complex phase space, is employed. For a particular class of solutions, an Ermakov system of equations is also found for the linear case. Further, corresponding to the 'classical' version of the above one-dimensional complex Hamiltonian, an equivalent integrable system of two, two-dimensional, real Hamiltonians is suggested.

The asymptotic behavior of the solutions of the Cauchy problem generated by ϕ-accretive operators

The purpose of this paper is to study the asymptotic behavior of the solutions of certain type of differential inclusions posed in Banach spaces. In particular, we obtain the abstract result on the asymptotic behavior of the solution of the boundary value problem { u t − Δ p ( u ) + | u | γ − 1 u = f on ] 0 , ∞ [ × Ω , − ∂ u ∂ η ∈ β ( u ) on [ 0 , ∞ [ × ∂ Ω , u ( 0 , x ) = u 0 ( x ) in Ω , where Ω is a bounded open domain in R n with smooth boundary ∂ Ω, f ( t , x ) is a given L 1 -function on ] 0 , ∞ [ × Ω , γ ⩾ 1 and 1 ⩽ p < ∞ . Δ p represents the p-Laplacian operator, ∂ ∂ η is the associated Neumann boundary operator and β a maximal monotone graph in R × R with 0 ∈ β ( 0 ) .

Reachability analysis for controlled systems subjected to stochastic disturbances

Among the recent problems of control of special interest are those of reachability under disturbances and resets of system dynamics. The present paper deals with the problem of reachability under stochastic disturbances. Considered is a linear system subjected to perturbations generated by Brownian noise which may or may not be dependent on the values of the control. The latter in its turn may be either unbounded or bounded by hard bounds. The reachabilty sets introduced here are deterministic. They consist of all points whose mean -square deviation from some controlled trajectory is small. These sets are presented in terms of level sets to solutions of certain types of the Hamilton-Jacobi-Bellman equation. The respective controls are explicitly calculated when the controls are unbounded and are presented in terms of solutions to some dual optimization problems when the controls are bounded.

ASYMPTOTIC EQUILIBRIUM FOR CERTAIN TYPE OF DIFFERENTIAL EQUATIONS WITH MAXIMUM

In this work we obtain asymptotic representations for the solutions of certain type of differential equations with maximum. We deduce the asymptotic equilibrium for this class of differential equations.

Numerical Study of Hyperbolic Equations with Integral Constraints Arising in Semiconductor Theory

An efficient numerical scheme is described for the solution of certain types of nonlinear hyperbolic equations with an integral constraint which are used to model the Gunn effect in semiconductors with impurity capture. We analyze the stability and convergence properties of the scheme and present the results of numerical simulations. Depending on the value of the parameters defining the problem, a great variety of solutions are obtained, including periodic recycling of solitary waves and chaotic regimes.

When an entire function and its linear differential polynomial share two values

In this note, the relationship between a non-constant entire function $f$ and its linear differential polynomial $L(f)$ has been obtained when they share two finite values, ignoring multiplicities, by applying value distribution theory. This confirms Frank's conjecture as a special case. Entire solutions of certain types of non-linear differential equations are also discussed.

Modified linearized Burger's flow for two-dimensional viscous motion

In this expository paper we use a complex ver- sion of the two-dimensional steady flow Navier-Stokes equa- tions, originally due to Legendre (6) and develop a new lin- earization which is similar, but different from Burger's flow. A similarity solution is found where the stream function for the potential flow is a linear combination of the linear shear flow and stagnation-point flow in two dimensions. Such a similarity solution describes a flow impinging on a plane wall at an arbi- trary angle of incidence and reduces the problem to a system of two ordinary differential equations which can be integrated numerically. The technique is similar to a method used by Ranger and Davis (8) and discussed in a nonlinear context by Dorrepaal (1). The existence of the boundary layer near the wall is interpreted from the similarity solution. A promulga- tion of the case of inflow only is carried out for the linearized model. 1. Introduction. Motion in two dimensions is characterized by the streamlines being all continuously equidistant to a fixed plane and the vorticity vector being at right angles to the plane of the motion and therefore fixed in direction. The fact that two-dimensional motion offers opportunities for special mathematical treatment and empowers us to scrutinize the nature of many phenomena which in their full three- dimensional form have so far proved intractable, is of long standing interest in the field of fluid dynamics. In the context of the fluid dynamic literature, the rareness in the exact solutions of the Navier-Stokes equations is due to the analytic nature of the nonlinear boundary-value problems. Since nonlinear problems fail to admit a superposition principle, this raises the issue of the existence of a similarity solution for certain types of problems. One such flow which admits a similarity solution to the full Navier-Stokes equations is the two-dimensional stagnation-point flow in which an incompressible viscous fluid flows steadily towards a plane wall. In such a flow a two- dimensional rigid wall occupies the entire x-axis and the fluid domain is y> 0. At a stagnation point, a streamline splits in order for flow separation to occur. Copyright c