Reduction of PDEs on Unbounded Domains. Application: Unsteady Water-Waves Problem

Summary. For a certain class of partial differential equations in cylindrical domains, we show that all small time-dependent solutions are described by a reduced system of equations on the real line, which contains nonlocal terms. As an application, we investigate the system describing nonlinear water waves travelling on the free surface of an inviscid fluid. Two-dimensional gravity waves are characterized by the parameter λ , the inverse square of the Froude number. For λ close to the critical value λ ₀ =1 , we obtain a reduced system of four nonlocal equations. We show that the terms of lowest order in μ=λ-1 lead to the Korteweg—de Vries equation for the lowest-order approximation of the free surface. Received February 23, 1994; final revision received October 13, 1997; accepted for publication October 16, 1997.     

The Value Function of Singularly Perturbed Control Systems

The limit as ɛ→ 0 of the value function of a singularly perturbed optimal control problem is characterized. Under general conditions it is shown that limit value functions exist and solve in a viscosity sense a Hamilton—Jacobi equation. The Hamiltonian of this equation is generated by an infinite horizon optimization on the fast time scale. In particular, the limit Hamiltonian and the limit Hamilton—Jacobi equation are applicable in cases where the reduction of order, namely setting ɛ = 0 , does not yield an optimal behavior. Accepted 18 November 1999     

Numerical Solutions to the Darboux Problem with Functional Dependence

The paper deals with the Darboux problem for the equation D _{xy z} (x,y) = f(x,y,z( _x,y ) where z( _x,y ) is a function defined by . We construct a general class of difference methods for this problem. We prove the existence and uniqueness of solutions to implicit functional difference equations by means of a comparison method; moreover we give an error estimate. The convergence of explicit difference schemes is proved under a general assumption that given functions satisfy nonlinear estimates of the Perron type. Our results are illustrated by a numerical example.     

A natural extension of the Pythagorean equation to higher dimensions

The Pythagorean equation is extended to higher dimensions via circulant matrices. This form allows for the set of solutions to be expressed in a clean yet non-trivial way. The cubic case, namely the equation x ³+y ³+z ³−3xyz=1, was studied by Ramanujan and displays many interesting properties. The general case highlights the use of circulant matrices and systems of differential equations. The structure of the solutions also allows parametrized solutions of the Fermat equation in degrees 3 and 4 to be given in terms of theta functions.      

Analytic properties of conditional curvatures of convex hypersurfaces and the dirichlet problem for the Monge-Ampère equation

The existence and uniqueness of a surface with given geometric characteristics is one of the important topical problems of global differential geometry. By stating this problem in terms of analysis, we arrive at second-order elliptic and parabolic partial differential equations. In the present paper we consider generalized solutions of the Monge-Ampère equation ||z _ij || = ϕ(x, z, p) in Λ ⁿ , wherez = z(x ₁,...,z _n ) is a convex function,p = (p ₁,...,P _n) = (∂z/∂x ₁,...,ϖz/ϖx _n), andz _ij =ϖ ² z/ϖx _i ϖx _j. We consider the Cayley-Klein model of the space Λ ⁿ and use a method based on fixed point principle for Banach spaces. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 763–768, November, 1998.     

All traveling wave exact solutions of three kinds of nonlinear evolution equations

In this article, we employ the complex method to obtain all meromorphic exact solutions of complex Klein–Gordon (KG) equation, modified Korteweg‐de Vries (mKdV) equation, and the generalized Boussinesq (gB) equation at first, then find all exact solutions of the Equations KG, mKdV, and gB. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions w_2r,2(z) and simply periodic solutions w_1s,2(z),w_2s,1(z) in these equations such that they are not only new but also not degenerated successively by the elliptic function solutions. We have also given some computer simulations to illustrate our main results. Copyright © 2014 John Wiley & Sons, Ltd.     

All meromorphic solutions of an ordinary differential equation and its applications

Dedicated to Professor Yuzan He on the Occasion of his 80th Birthday In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions w_r,2(z) and simply periodic solutions w_s,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Stochastic 2-D Navier—Stokes Equation

   Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution.     

Viscosity Solutions of an Infinite-Dimensional Black—Scholes—Barenblatt Equation

We study an infinite-dimensional Black—Scholes—Barenblatt equation which is a Hamilton—Jacobi—Bellman equation that is related to option pricing in the Musiela model of interest rate dynamics. We prove the existence and uniqueness of viscosity solutions of the Black—Scholes—Barenblatt equation and discuss their stochastic optimal control interpretation. We also show that in some cases the solution can be locally uniformly approximated by solutions of suitable finite-dimensional Hamilton—Jacobi—Bellman equations.     

Equivalency for Disconjugate Operators

The solutions of the dfferential equation L_nx + a₀(t)x = 0, where L_n is a disconjugate operator and a₀ is of one sign, are studied according to their behavior as t → ∞. We prove equivalence theorems between solutions of this equation and its adjoint in terms of property A and property B. These theorems generalize the results known for n = 3 and for odd order binomial equations. Some applications are given too.     

A fast matrix–vector multiplication method for solving the radiosity equation

A “fast matrix–vector multiplication method” is proposed for iteratively solving discretizations of the radiosity equation (I — К)u = E. The method is illustrated by applying it to a discretization based on the centroid collocation method. A convergence analysis is given for this discretization, yielding a discretized linear system (I — K _n )u _n = E _n. The main contribution of the paper is the presentation of a fast method for evaluating multiplications K_n v, avoiding the need to evaluate K_n explicitly and using fewer than O(n ²) operations. A detailed numerical example concludes the paper, and it illustrates that there is a large speedup when compared to a direct approach to discretization and solution of the radiosity equation. The paper is restricted to the surface S being unoccluded, a restriction to be removed in a later paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Decomposition Theorem for the Solutions to the Interface Problems of Quasi-Linear Elliptic Equations

Interface problems for second order quasi-linear elliptic partial differential equations in a two-dimensional space are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a finite sum of functions of the form cr^a log^m rφ(θ), where the coefficients c depend on the H^1-norm of the solution, the C^(0,δ) -norm of the solution, and the equation only; and the other one of which is a regular one, the norm of which is also estimated.     

Uniqueness theorem of solutions for stochastic differential equation in the plane

LetM={M _z, z ∈ R ₊ ² } be a continuous square integrable martingale andA={A _z, z ∈ R ₊ ² be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane:dX _z=α(z, X_z)dM_z+β(z, X_z)dA_z, z∈R ₊ ² , X_z=Z_z, z∈∂R ₊ ² , whereR ₊ ² =[0, +∞)×[0,+∞) and ∂R ₊ ² is its boundary,Z is a continuous stochastic process on ∂R ₊ ² . We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2]. Supported by the National Science Foundation and the Postdoctoral Science Foundation of China     

Geometrical conditions for oscillation of second-order half-linear differential equations

This paper deals with the second-order half-linear differential equation

Concerning the Sierpinski-Schinzel system of Diophantine equations

It is proved that the equation (x ²−1)(y ²−1)=(z ²−1)², |x|≠|y|, |z|≠1, is not solvable in integersx,y,z under the conditionx−y=kz, wherek is a positive integer different from 2. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 181–187, August, 1999.     

Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation

We show that under the Euler integral transformation with the kernel (x−z)^−α, some solutions of the Fuchs equations (the original pair for the Painlevé VI equation) pass into solutions of a system of the same form with the parameters changed according to the Okamoto transformation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 355–364, March, 2006.     

Oscillation and asymptotic behaviour of a class of higher-order non-linear difference equations

The dynamical characteristics of scalar difference equations of the form?x _{n +1}=f ₁(x _{n −τ1})+f ₂(x _{n −τ2}), n=0,1,2,...,?are investigated. A necessary and sufficient condition is obtained for all positive solutions to be oscillatory about a unique positive equilibrium point and sufficient criteria for the global attractivity of the equilibrium are established. Also, the stability and periodicity of more general equations are studied via comparison with the corresponding properties of an associated first-order non-linear equation. Received: September 5, 2001; in final form: March 7, 2002?Published online: April 14, 2003     

The Moment Problem Associated with the q -Laguerre Polynomials

   Abstract. We consider the indeterminate Stieltjes moment problem associated with the q -Laguerre polynomials. A transformation of the set of solutions, which has all the classical solutions as fixed points, is established and we present a method to construct, for instance, continuous singular solutions. The connection with the moment problem associated with the Stieltjes—Wigert polynomials is studied; we show how to come from q -Laguerre solutions to Stieltjes—Wigert solutions by letting the parameter α —> ∞ , and we explain how to lift a Stieltjes—Wigert solution to a q -Laguerre solution at the level of Pick functions. Based on two generating functions, expressions for the four entire functions from the Nevanlinna parametrization are obtained.