Asymptotic analysis of steady solutions of the KdVB equation with application to resonant sloshing

The forced Korteweg-de Vries equation with Burgers’ damping (fKdVB) on a periodic domain, which arises as a model for water waves in a shallow tank with forcing near resonance, is considered. A method for construction of asymptotic solutions is presented, valid in cases where dispersion and damping are small. Through variation of a detuning parameter, families of resonant solutions are obtained providing detailed insight into the resonant response character of the system and allowing for direct comparison with the experimental results of Chester and Bones (1968).     

Uniform asymptotic expansions of solutions of the Mathieu equation and the modified Mathieu equation

By the method of the model equation, uniform asymptotic expansions of the Floquet solutions of the Mathieu equation and two linearly independent solutions of the modified Mathieu equation are obtained for any real values of the separation parameter contained in these equations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 60–91, 1976.     

Asymptotic solutions and stability analysis for generalized non-homogeneous Mathieu equation

The asymptotic solutions and transition curves for the generalized form of the non-homogeneous Mathieu differential equation are investigated in this paper. This type of governing differential equation of motion arises from the dynamic behavior of a pendulum undergoing a butterfly-type end support motion. The strained parameter technique is used to obtain periodic asymptotic solutions. The transition curves for some special cases are presented and their corresponding periodic solutions with the periods of 2π and 4π are evaluated. The stability analyses of those transition curves in the ε–δ plane are carried out, analytically, using the multiple scales method. The numerical simulations for some typical points in the ε–δ plane are performed and the dynamic characteristics of the resulting phase plane trajectories are discussed.     

Calculation of solutions to the Mathieu equation and of related quantities

For the Mathieu equation, we consider finding eigenvalues with a given index (on the basis of oscillation theorems for the relevant difference equations), the stability of solutions to the difference equations, correct definition and calculation of eigenvalues and Mathieu functions with noninteger numbers, correct definition and calculation of the Mathieu characteristic exponent, and the calculation of values of solutions to the Mathieu equation for large arguments. Numerical algorithms are proposed for the problems listed above.     

The viscous Cahn-Hilliard equation with inertial term

We consider a hyperbolic relaxation of the viscous Cahn-Hilliard equation. This equation describes the early stages of spinodal decomposition in certain glasses. We establish the existence of families of exponential attractors and inertial manifolds which are continuous at any parameter of viscosity ?≥0. Continuity properties of the global attractors are also examined.     

Construction of active adaptation networks with application to inertial navigation systems

The principles of the theory of active identification are applied to construct networks for performance criterion evaluation. The topic is considered in application to adaptive control in inertial navigation. Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 2, pp. 110–115, 1994.     

Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term χtt, χ being the order parameter, which is linearly coupled with an evolution equation for the (relative) temperature ?. The latter can be of hyperbolic type if the Cattaneo-Maxwell heat conduction law is assumed. The state variables and the chemical potential are subject to the homogeneous Neumann boundary conditions. We first provide conditions which ensure the well-posedness of the initial and boundary value problem. Then, we prove that the corresponding dynamical system is dissipative and possesses a global attractor. Moreover, assuming that the nonlinear potential is real analytic, we establish that each trajectory converges to a single steady state by using a suitable version of the ?ojasiewicz-Simon inequality. We also obtain an estimate of the decay rate to equilibrium.     

Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point,II — Its relevance to the Mathieu equation and the Legendre equation

We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrödinger equation. In Part II, using a WKB-theoretic transformation to the algebraic Mathieu equation constructed in Part I, we calculate the alien derivative of its Borel transformed WKB solutions at each fixed singular point relevant to the simple poles through the analysis of Borel transformed WKB solutions of the Legendre equations. In the course of the calculation of the alien derivative we make full use of microdifferential operators whose symbols are given by the infinite series that appear in the coefficients of the algebraic Mathieu equation and the Legendre equation.     

Periodic solutions of a resonant third-order equation

We study the existence of periodic solutions for a third-order equation of resonant type. Under suitable conditions we prove the existence of at least one periodic solution of the problem applying Mawhin coincidence degree theory.     

A parabolic equation with nonlocal diffusion without a smooth inertial manifold

A family of parabolic integro-differential equations with nonlocal diffusion on the circle which have no smooth inertial manifold is presented.     

New estimating equation approaches with application in lifetime data analysis

Estimating equation approaches have been widely used in statistics inference. Important examples of estimating equations are the likelihood equations. Since its introduction by Sir R. A. Fisher almost a century ago, maximum likelihood estimation (MLE) is still the most popular estimation method used for fitting probability distribution to data, including fitting lifetime distributions with censored data. However, MLE may produce substantial bias and even fail to obtain valid confidence intervals when data size is not large enough or there is censoring data. In this paper, based on nonlinear combinations of order statistics, we propose new estimation equation approaches for a class of probability distributions, which are particularly effective for skewed distributions with small sample sizes and censored data. The proposed approaches may possess a number of attractive properties such as consistency, sufficiency and uniqueness. Asymptotic normality of these new estimators is derived. The construction of new estimation equations and their numerical performance under different censored schemes are detailed via Weibull distribution and generalized exponential distribution.     

Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point,I — Its WKB-theoretic transformation to the Mathieu equation

We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrödinger equation. Our emphasis is put on the analysis of the singularity structure of its Borel transformed WKB solutions near fixed singular points relevant to the two simple poles contained in the potential of the equation. In Part I, we focus our attention on the construction and analytic properties of a WKB-theoretic transformation that transforms an M2P1T equation to an algebraic Mathieu equation. That transformation plays an important role in Part II ([7]) when we discuss the singularity structure of Borel transformed WKB solutions of an M2P1T equation.     

Uncertain partial differential equation with application to heat conduction

This paper first presents a tool of uncertain partial differential equation, which is a type of partial differential equations driven by Liu processes. As an application of uncertain partial differential equation, uncertain heat equation whose noise of heat source is described by Liu process is investigated. Moreover, the analytic solution of uncertain heat equation is derived and the inverse uncertainty distribution of solution is explored. This paper also presents a paradox of stochastic heat equation.     

Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions

This paper is concerned with the asymptotic behavior of solution to the Cahn-Hilliard equation

(0.1)     

Convergence of semigroups with an application to the carleman equation

We present a simple proof of a convergence result for nonlinear semigroups which can be applied to obtain the fluid dynamical limit of Carleman's equation. In addition, we explicitly identify a closed exponential generator of the limit semigroup. Our results are closely related to those of T. G. Kurtz and H. P. McKean.