An equivalence theorem on properties A, B for third order differential equations

Differential equations are often classified according to oscillatory/nonoscillatory properties of their solutions as equations having property A or property B. The aim of the paper is to state an equivalence theorem between property A and property B for third order differential equations. Some applications, to linear as well as to nonlinear equations, are given too. Particularly, we give integral criteria ensuring property A or B for nonlinear equations. Our only assumption on nonlinearity is its superlinearity in neighbourhood of infinity, hence our results apply also to Emden-Fowler type equations.The second author wishes to thank C.N.R. of Italy and Grant Agency of Czech Republic (grant 201/96/0410) which made this research possible.     

Approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic

In 1956 Whitham gave a nonlinear theory for computing the intensity of an acoustic pulse of an arbitrary shape. The theory has been used very successfully in computing the intensity of the sonic bang produced by a supersonic plane. [4.] derived an approximate quasi-linear equation for the propagation of a short wave in a compressible medium. These two methods are essentially nonlinear approximations of the perturbation equations of the system of gas-dynamic equations in the neighborhood of a bicharacteristic curve (or rays) for weak unsteady disturbances superimposed on a given steady solution. In this paper we have derived an approximate quasi-linear equation which is an approximation of perturbation equations in the neighborhood of a bicharacteristic curve for a weak pulse governed by a general system of first order quasi-linear partial differential equations in m + 1 independent variables (t, x₁,…, x_m) and derived Gubkin's result as a particular case when the system of equations consists of the equations of an unsteady motion of a compressible gas. We have also discussed the form of the approximate equation describing the waves propagating upsteam in an arbitrary multidimensional transonic flow.     

On solutions of third order nonlinear differential equations

The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity.     

Families of normalizes equations in the problem of dislocations in a solid body

A classical nonlinear differential equation with deviations in a spatial variable is considered. Solutions with initial conditions from a small neighborhood of equilibrium are studied by constructing multiparameter families of special nonlinear systems of equations, which play the role of normal forms. Systems of Schrödinger-type nonlinear equations in a two-dimensional domain are presented.

     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

Chaotic Behavior in Slowly Varying Systems of Nonlinear Differential Equations

A technique is developed to find parameter regions of chaotic behavior in certain systems of nonlinear differential equations with slowly varying periodic coefficients. The technique combines previous results on how to find branches of periodic solutions which terminate with a homoclinic orbit and results on how to find chaotic trajectories in the neighborhood of homoclinic trajectories of the autonomous system. The technique is applied to the continuous stirred tank reaction A → B, for which it is shown that a slowly varying periodic flow rate can yield aperiodic temperature fluctuations.     

On the convergence of quasi-newton methods for nonsmooth problems

We develop a theory of quasi-New ton and least-change update methods for solving systems of nonlinear equations F(x) = 0. In this theory, no differentiability conditions are necessary. Instead, we assume that Fcan be approximated, in a weak sense, by an affine function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-differentiable functions and to partially differentiable functions.     

On HSS-based iteration methods for weakly nonlinear systems

Based on separable property of the linear and the nonlinear terms and on the Hermitian and skew-Hermitian splitting of the coefficient matrix, we present the Picard-HSS and the nonlinear HSS-like iteration methods for solving a class of large scale systems of weakly nonlinear equations. The advantage of these methods over the Newton and the Newton-HSS iteration methods is that they do not require explicit construction and accurate computation of the Jacobian matrix, and only need to solve linear sub-systems of constant coefficient matrices. Hence, computational workloads and computer memory may be saved in actual implementations. Under suitable conditions, we establish local convergence theorems for both Picard-HSS and nonlinear HSS-like iteration methods. Numerical implementations show that both Picard-HSS and nonlinear HSS-like iteration methods are feasible, effective, and robust nonlinear solvers for this class of large scale systems of weakly nonlinear equations.     

A nonlinear elliptic problem with terms concentrating in the boundary

In this paper, we investigate the behavior of a family of steady‐state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a ε‐neighborhood of a portion Γ of the boundary. We assume that this ε‐neighborhood shrinks to Γ as the small parameter ε goes to zero. Also, we suppose the upper boundary of this ε‐strip presents a highly oscillatory behavior. Our main goal here was to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on Γ, which depends on the oscillating neighborhood. Copyright © 2012 John Wiley & Sons, Ltd.     

On the behavior of solutions of some nonlinear degenerate elliptic inequalities

We consider the behavior of solutions in unbounded domains as |x| → ∞ and in a neighborhood of an isolated singular point for a class of nonlinear elliptic equations and inequalities degenerating with respect to the coordinate variable, the solution, and the solution gradient.     

On smoothness and invariance properties of the gauss-newton method

We consider systems of m nonlinear equations in m + p unknowns which have p-dimensional solution manifolds. It is well-known that the Gauss-Newton method converges locally and quadratically to regular points on this manifold. We investigate in detail the mapping which transfers the starting point to its limit on the manifold. This mapping is shown to be smooth of one order less than the given system. Moreover, we find that the Gauss-Newton method induces a foliation of the neighborhood of the manifold into smooth submanifolds. These submanifolds are of dimension m, they are invariant under the Gauss-Newton iteration, and they have orthogonal intersections with the solution manifold.     

Stochastic control with exit time and constraints,application to small time attainability of sets

We study the existence of a solution of controlled stochastic differential equations remaining in a given set of constraints at any time smaller than the exit time of a given open set. We also investigate the small time attainability of a given closed set K, i.e., the property that, for all arbitrary small time horizon T and for all initial condition in a sufficiently small neighborhood of K, there exists a solution to the controlled stochastic differential equation which reaches K before T.     

Newton's method for nonlinear ordinary and partial differential equations

A unified approach is presented for proving the local, uniform and quadratic convergence of the approximate solutions and a-posteriori error bounds obtained by Newton's method for systems of nonlinear ordinary or partial differential equations satisfying an inverse-positive property. An important step is to show that, at each iteration, the linearized problem is inverse-positive. Many classes of problems are shown to satisfy this property. The convergence proofs depend crucially on an error bound derived previously by Rosen and the author for quasilinear elliptic, parabolic and hyperbolic problems.     

Estimates of solutions of two-point problems for systems of first-order ordinary differential equations and the method of lines

In the paper estimates are established on the solution of systems of first-order differential equations subject to two-point conditions of the form which enable us, in particular, to obtain an estimate of the order of uniform convergence of the method of lines for solving periodic boundary-value problems for second-order nonlinear parabolic partial differential equations of form For problems with boundary conditions containing the derivatives where the functions satisfy, in a small neighborhood of the solutionu(t,x) being examined of Eq. (3), the inequalities, for which the approximation is only of the first order relative to the net steph, the uniform convergence of the approximate solutions to the exact one is established to the second order relative toh.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 277–296, 1979.     

Continuous and discrete time Zhang dynamics for time-varying <Emphasis Type="Italic">4</Emphasis>th root finding

Recently, a special class of neural dynamics has been proposed by Zhang et al. for online solution of time-varying and/or static nonlinear equations. Different from eliminating a square-based positive error-function associated with gradient-based dynamics (GD), the design method of Zhang dynamics (ZD) is based on the elimination of an indefinite (unbounded) error-function. In this paper, for the purpose of online solution of time-varying 4th root, both continuous-time ZD (CTZD) and discrete-time ZD (DTZD) models are developed and investigated. In addition, power-sigmoid activation function is exploited in Zhang dynamics, which makes ZD models possess the property of superior convergence and better accuracy. To summarize generalization for possible widespread application, such approach is further extended to general time-varying nonlinear equations solving. Computer-simulation results demonstrate the efficacy of the ZD models for finding online time-varying 4th root and solving general time-varying equations.     

The presentation of explicit analytical solutions of a class of nonlinear evolution equations

In this paper, we introduce a function set Ω_m. There is a conjecture that an arbitrary explicit travelling-wave analytical solution of a real constant coefficient nonlinear evolution equation is necessarily a linear (or nonlinear) combination of the product of some elements in Ω_m. A widespread applicable approach for solving a class of nonlinear evolution equations is established. The new analytical solutions to two kinds of nonlinear evolution equations are described with the aid of the guess.     

Contractivity of θ-method for semi-discrete systems

We consider nonlinear semi-discrete problems that derive by reaction diffusion systems of partial differential equations, when finite difference methods or Faedo Galerkin methods are used for spatial discretization. The aim of this article is to give sufficient conditions for the contractivity of the θ-method, in a norm generated by a positive diagonal matrix G. We show that the numerical contractivity property is obtained if some matrices, constructed by means of the Jacobian matrix of nonlinear term, are M-matrices. © 1996 John Wiley & Sons, Inc.     

The Painlevé Property and Hirota's Method

The connection between the Painlevé property for partial differential equations, proposed by Weiss, Tabor, and Carnevale, and Hirota's method for calculating N-soliton solutions is investigated for a variety of equations including the nonlinear Schrödinger and mKdV equations. Those equations which do not possess the Painlevé property are easily seen not to have self-truncating Hirota expansions. The Bäcklund transformations derived from the Painlevé analysis and those determined by Hirota's method are shown to be directly related. This provides a simple route for demonstrating the connection between the singular manifolds used in the Painlevé analysis and the eigenfunctions of the AKNS inverse scattering transform.     

Maximum error estimates for a compact difference scheme of the coupled nonlinear Schrödinger–Boussinesq equations

The coupled nonlinear Schrödinger–Boussinesq (SBq) equations describe the nonlinear development of modulational instabilities associated with Langmuir field amplitude coupled to intense electromagnetic wave in dispersive media such as plasmas. In this paper, we present a conservative compact difference scheme for the coupled SBq equations and analyze the conservative property and the existence of the scheme. Then we prove that the scheme is convergent with convergence order O(τ² + h⁴) in L_∞‐norm and is stable in L_∞‐norm. Numerical results verify the theoretical analysis.     

A Local Functional Calculus and Related Results on the Single-Valued Extension Property

We study the local functional calculus of an operator T having the single-valued extension property. We consider a vector f(T, v) for an analytic function f on a neighborhood of the local spectrum of a vector v with respect to T and show that the local spectrum of v and the local spectrum of f(T, v)are equal with the possible exception of points of the local spectrum of v that are zeros of f, that is, we show that s_T \sigma_{T} (v) is equal to s_T \sigma_{T} (f(T,v)) union the set of zeros of f on s_T \sigma_{T} (v). This local functional calculus extends the Riesz functional calculus for operators. For an analytic function f on a neighborhood of s \sigma (T), we use the above mentioned proposition to obtain proofs of the results that if T has the single-valued extension property, then f(T) also has the single-valued extension property, and conversely if f is not constant on each connected component of a neighborhood of s \sigma (T) and f(T) has the singlevalued extension property, then T also does.