An Auxiliary Equation for the Bellman Equation in a One-Dimensional Ergodic Control

In this paper we consider the Bellman equation in a one-dimensional ergodic control. Our aim is to show the existence and the uniqueness of its solution under general assumptions. For this purpose we introduce an auxiliary equation whose solution gives the invariant measure of the diffusion corresponding to an optimal control. Using this solution, we construct a solution to the Bellman equation. Our method of using this auxiliary equation has two advantages in the one-dimensional case. First, we can solve the Bellman equation under general assumptions. Second, this auxiliary equation gives an optimal Markov control explicitly in many examples. \keywords{Bellman equation, Auxiliary equation, Ergodic control.} \amsclass{49L20, 35G20, 93E20.} Accepted 11 September 2000. Online publication 16 January 2001.     

On a Solution of the Quaternion Matrix Equation X - A X B = C and Its Application

This paper first studies the solution of a complex matrix equation X - AXB = C, obtains an explicit solution of the equation by means of characteristic polynomial, and then studies the quaternion matrix equation X - A X B = C, characterizes the existence of a solution to the matrix equation, and derives closed-form solutions of the matrix equation in explicit forms by means of real representations of quaternion matrices. This paper also gives an application to the complex matrix equation X - AXB =C.     

Integrability properties of a generalized reduction of the KdV6 equation

We consider the integrability properties of a generalized version of a similarity reduction of the so-called KdV6 equation, an equation that has recently generated much interest. We give a linear problem for this generalized reduction and show that it satisfies the requirements of the Ablowitz-Ramani-Segur algorithm. In addition we give a Bäcklund transformation to a related equation, giving also an auto-Bäcklund transformation for this last. Our results mirror those for the Korteweg-de Vries equation itself, which has a similarity reduction to an ordinary differential equation which is related by a Bäcklund transformation to the second Painlevé equation, this last having an auto-Bäcklund transformation.     

一阶迭代微分方程的解析解

讨论了一阶迭代微分方程解析解的存在性,通过构造一个辅助方程的幂级数解给出该方程的解析解.     

Solutions of nonlinear differential equations

We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in the algebra of new generalized functions. The solution of such an equation is a new generalized function. In this article we formulate necessary and sufficient conditions for when the solution of the given equation in the algebra of new generalized functions is associated with an ordinary function. Moreover, a class of all possible associated functions is described.     

Solution of the equivalence problem for the Painlevé IV equation

We solve the equivalence problem for the Painlevé IV equation, formulating the necessary and sufficient conditions in terms of the invariants of point transformations for an arbitrary second-order differential equation to be equivalent to the Painlevé IV equation. We separately consider three pairwise nonequivalent cases: both equation parameters are zero, a = b = 0; only one parameter is zero, b = 0; and the parameter b ?? 0. In all cases, we give an explicit point substitution transforming an equation satisfying the described test into the Painlevé IV equation and also give expressions for the equation parameters in terms of invariants.     

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.     

Coefficient characterization of linear differential equations with maximal symmetries

A characterization of the general linear equation in standard form admitting a maximal symmetry algebra is obtained in terms of a simple set of conditions relating the coefficients of the equation. As a consequence, it is shown that in its general form such an equation can be expressed in terms of only two arbitrary functions, and its connection with the Laguerre–Forsyth form is clarified. The characterizing conditions are also used to derive an infinite family of semi-invariants, each corresponding to an arbitrary order of the linear equation. Finally a simplifying ansatz is established, which allows an easier determination of the infinitesimal generators of the induced pseudo group of equivalence transformations, for all the three most common canonical forms of the equation.     

五阶势MKdV方程的一个不变性

§1Introduction Asweknow,Backlundtransformation[1-3]isaverypowerfulwayforfindingnonline evolutionequations.Inrecentdecades,Painlevéanalysis[4]hasbecomeaverypopu methodtoobtainBacklundtransformation.Inreference[5],fordevelopingthetheory Painlevéanalysis,AndrewPickeringintroduceanewexpansionvariableZwhichsatisf thefollowingRicattiSystem:Zx=1-AZ-BZ2,Zt=-C+(AC+Cx)Z-(D-BC)Z2.(1.Astheapplicationofthenewexpansionvaraible,thepotentialfifth-orderMKd equation(PMKdV5)-vxt+(vxxxxx-10k2v2…     

Lipschitzian semigroups and abstract functional differential equations

The paper is devoted to studying an abstract functional differential equation by a nonlinear semigroup approach. We first prove in details the equivalence of the well posedness of an abstract functional differential equation and an associated abstract Cauchy problem in the sense of strong solutions. Secondly, a sufficient condition is derived for well posedness of the abstract functional differential equation. Thirdly, we present principles of linearized stability for the abstract functional differential equation. Finally, the results obtained are applied to a reaction-diffusion equation with delays.     

Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing 2-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy type arguments combined with the 2-Wasserstein distance. This together with the quantitative error estimate between the pressureless Euler system and the aggregation equation in 2-Wasserstein distance in [Commun. Math. Phys, 365, (2019), 329–361] establishes the quantitative bounds on the error between the kinetic equation and the aggregation equation.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

Equations with Random Gaussian Operators with Unbounded Mean Value

We consider an equation in a Hilbert space with a random operator represented as a sum of a deterministic, closed, densely defined operator and a Gaussian strong random operator. We represent a solution of an equation with random right-hand side in terms of stochastic derivatives of solutions of an equation with deterministic right-hand side. We consider applications of this representation to the anticipating Cauchy problem for a stochastic partial differential equation.     

Formal analysis of the Cauchy problem for a system associated with the (2+1)-dimensional Krichever-Novikov equation

The singularity manifold equation of the Kadomtsev-Petviashvili equation, the so-called Krichever-Novikov equation, has an exact linearization to an overdetermined system of partial differential equations in three independent variables. We study in detail the Cauchy problem for this system as an example for the use of the formal theory of differential equations. A general existence and uniqueness theorem is established. Formal theory is then contrasted with Janet-Riquier theory in the formulation of Reid. Finally, the implications of the results for the Krichever-Novikov equation are outlined.     

Exact master equations describing reduced dynamics of the Wigner function

Master equations of different types describe the evolution (reduced dynamics) of a subsystem of a larger system generated by the dynamic of the latter system. Since, in some cases, the (exact) master equations are relatively complicated, there exist numerous approximations for such equations, which are also called master equations. In the paper, we develop an exact master equation describing the reduced dynamics of the Wigner function for quantum systems obtained by a quantization of a Hamiltonian system with a quadratic Hamilton function. First, we consider an exact master equation for first integrals of ordinary differential equations in infinite-dimensional locally convex spaces. After this, we apply the results obtained to develop an exact master equation corresponding to a Liouville-type equation (which is the equation for first integrals of the (system of) Hamilton equation(s)); the latter master equation is called the master Liouville equation; it is a linear first-order differential equation with respect to a function of real variables taking values in a space of functions on the phase space. If the Hamilton equation generating the Liouville equation is linear, then the vector fields that define the first-order linear differential operators in the master Liouville equations are also linear, which in turn implies that for a Gaussian reference state the Fourier transform of a solution of the master Liouville equation also satisfies a linear differential equation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 203–219, 2005.     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

Periodic solutions of retarded functional differential equations

The classical criterion for the existence of a periodic solution of a forced, linear, retarded functional differential equation (rfde) references the solution of an adjoint equation which is an advanced functional differential equation. This paper introduces a new criterion which does away with the (anti-causal) advanced functional differential equation.     

Integrating Scalar Ordinary Differential Equations with Symmetry Revisited

The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems. This formulation immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group. To determine the noninvariant solutions the problem splits into three cases. If the dimension of the symmetry groups is less than the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions can be found by integrating a quotient Pfaffian system on a quotient space, and integrating an equation of fundamental Lie type associated with the symmetry group. If the dimension of the symmetry group is equal to the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions are obtained either by solving an equation of fundamental Lie type associated with the symmetry group, or the solutions are invariant under a one-parameter subgroup. If the dimension of the symmetry group is greater than the order of the equation, then there exists an open dense set of initial conditions where the solutions can either be determined by solving an equation of fundamental Lie type for a solvable Lie group, or are invariant. In each case the initial conditions, the quotient Pfaffian system, and the equation of Lie type are all determined algebraically. Examples of scalar ordinary differential equations and a Pfaffian system are given.     

On the intermediate integral for Monge-Ampère equations

Goursat showed that in the presence of an intermediate integral, the problem of solving a second-order Monge-Ampère equation can be reduced to solving a first-order equation, in the sense that the generic solution of the first-order equation will also be a solution of the original equation. An attempt by Hermann to give a rigorous proof of this fact contains an error; we show that there exists an essentially unique counterexample to Hermann's assertion and state and prove a correct theorem.

     

Nonlinear wave propagation through a stratified atmosphere

We examine the propagation of sound waves through a stratified atmosphere. The method of multiple scales is employed to obtain an asymptotic equation which describes the evolution of sound waves in an atmosphere with spatially dependant density and entropy fields. The evolution equation is an inviscid Burger-like equation which contains quadratic and cubic nonlinearities, and a curvature term all of which are functions of the space variables. A model equation is derived when the modulations of the signal in a direction transverse to the direction of propagation become significant.