Analysis of a fourth-order compact ADI method for a linear hyperbolic equation with three spatial variables

This paper is concerned with a three-level alternating direction implicit (ADI) method for the numerical solution of a 3D hyperbolic equation. Stability criterion of this ADI method is given by using von Neumann method. Meanwhile, it is shown by a discrete energy method that it can achieve fourth-order accuracy in both time and space with respect to H ¹- and L ²-norms only if stable condition is satisfied. It only needs solution of a tri-diagonal system at each time step, which can be solved by multiple applications of one-dimensional tri-diagonal algorithm. Numerical experiments confirming the high accuracy and efficiency of the new algorithm are provided.     

Bifurcation and chaos near sliding homoclinics

We study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface. We assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach we construct a Melnikov-like function M(α) in such a way that if M(α) has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given.     

Optimal dividend payout for classical risk model with risk constraint

In this paper we consider the problem of maximizing the total discounted utility of dividend payments for a Cramér-Lundberg risk model subject to both proportional and fixed transaction costs.We assume that dividend payments are prohibited unless the surplus of insurance company has reached a level b.Given fixed level b,we derive a integro-differential equation satisfied by the value function.By solving this equation we obtain the analytical solutions of the value function and the optimal dividend strategy when claims are exponentially distributed.Finally we show how the threshold b can be determined so that the expected ruin time is not less than some T.Also,numerical examples are presented to illustrate our results.     

Beale-Kato-Majda type condition for Burgers equation

We consider a multidimensional Burgers equation on the torus Td and the whole space Rd. We show that, in case of the torus, there exists a unique global solution in Lebesgue spaces. For a torus we also provide estimates on the large time behaviour of solutions. In the case of Rd we establish the existence of a unique global solution if a Beale-Kato-Majda type condition is satisfied. To prove these results we use the probabilistic arguments which seem to be new.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated with a metric g of class C² and we prove dispersion and Strichartz estimates for the solution of this equation in terms of geodesics associated with g. We introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the Euclidean case holds. To deal with the case of non-trapped long range perturbation of the Euclidean metric, we prove a global velocity moments effect on the solution. In particular, we obtain global in time Strichartz estimates for metrics such that the dispersion estimate is not satisfied.     

Uniqueness theorem for integral equations and its application

This paper is devoted to answering a question asked recently by Y. Li regarding geometrically interesting integral equations. The main result is to give a necessary and sufficient condition on the parameters so that the integral equation with parameters to be discussed in this paper have regular solutions. In the case such condition is satisfied, we will write down the exact solution. As its application of our method, we should show that the non-existence theory of the solutions of prescribed scalar curvature equation on Sn can be generalized to that of prescribed Branson-Paneitz Q-curvature equations on Sn.     

A highly accurate homogeneous scheme for solving the laplace equation on a rectangular parallelepiped with boundary values in C k, 1

In this paper, a homogeneous scheme with 26-point averaging operator for the solution of Dirichlet problem for Laplace??s equation on rectangular parallelepiped is analyzed. It is proved that the order of convergence is O(h ⁴), where h is the mesh step, when the boundary functions are from C ^{3, 1}, and the compatibility condition, which results from the Laplace equation, for the second order derivatives on the adjacent faces is satisfied on the edges. Futhermore, it is proved that the order of convergence is O(h ⁶(|lnh| + 1)), when the boundary functions are from C ^{5, 1}, and the compatibility condition for the fourth order derivatives is satisfied. These estimations can be used to justify different versions of domain decomposition methods.     

Gevrey normal forms of vector fields with one zero eigenvalue

We study normal forms of isolated singularities of vector fields in Rn or Cn. When all eigenvalues of the linear part of the vector field are nonzero, one can eliminate all so-called nonresonant terms from the equation provided some spectral condition (like Siegel) is satisfied. In this paper, we discuss the case where there is one zero eigenvalue (in that case Siegel's condition is not satisfied), and show that the formal normalizing transformations are either convergent or divergent of at most Gevrey type. In some cases, we show the summability of the normalizing transformations, which leads to the existence of analytic normal forms in complex sectors around the singularity.     

Blowup in the Nonlinear Schrödinger Equation Near Critical Dimension

This article contains an analysis of the cubic nonlinear Schrödinger equation and solutions that become singular in finite time. Numerical simulations show that in three dimensions the blowup is self-similar and symmetric. In two dimensions, the blowup still appears to be symmetric but is no longer self-similar. In the case that the dimension, d, is greater than and exponentially close to 2 in terms of a small parameter associated to the norm of the blow-up solution, a locally unique, monotonically decreasing in modulus, self-similar solution that satisfies the boundary and global conditions associated with the blow-up solution is constructed in Kopell and Landman [1995, SIAM J. Appl., Math.55, 1297-1323]. In this article, it is shown that this locally unique solution also exists for d > 2 and algebraically close to 2 in the same small parameter. The central idea of the proof involves constructing a pair of manifolds of solutions (to the nonautonomous ordinary differential equation satisfied by the self-similar solutions) that satisfy the conditions at r = 0 and the asymptotic conditions respectively and then showing that these intersect transversally. A key step involves tracking one of the manifolds over a midrange in which the ordinary differential equation has a turning point and hence obtaining good control over the solutions on the manifold.     

A Hamiltonian with periodic orbits having several delays

In 1974 Kaplan and Yorke introduced a certain delay equation with an arbitrary number of delays, x^′(t)=-f(x(t-1))-f(x(t-2))-?-f(x(t-n)), conjecturing that it has periodic solutions when f is an odd homeomorphism of the reals which is differentiable at the origin and infinity. By coupling the delay equation to a vector field on Rⁿ⁺¹ they were able to prove, when certain conditions on the derivative of f at 0 and ∞ are satisfied, one delay and two delay versions. We find that the closed orbits of the coupled vector field occur at the points of intersection of two hyperplane fields which are invariant under the flow and invariant under deformation to a linear vector field. By analyzing properties of the linear vector field we are able to give an elementary construction of periodic solutions to the general delay equation when conditions naturally extending those of Kaplan and Yorke are satisfied.     

On an algebraic version of the Knizhnik–Zamolodchikov equation

A difference equation analogue of the Knizhnik?CZamolodchikov equation is exhibited by developing a theory of the generating function H(z) of Hurwitz polyzeta functions to parallel that of the polylogarithms. By emulating the role of the KZ equation as a connection on a suitable bundle, a difference equation version of the notion of connection is developed for which H(z) is a flat section. Solving a family of difference equations satisfied by the Hurwitz polyzetas leads to the normalized multiple Bernoulli polynomials (NMBPs) as the counterpart to the Hurwitz polyzeta functions, at tuples of non-positive integers. A generating function for these polynomials satisfies a similar difference equation to that of H(z), but in contrast to the fact that said polynomials have rational coefficients, the algebraic independence of the usual Hurwitz zeta functions is proven, and the Hurwitz polyzeta functions are shown to satisfy no algebraic relations other than those arising from the shuffle relations. The values of the NMBPs at z=1 provide a regularization of the multiple zeta values at tuples of negative integers, which is shown to agree with the regularization given in Akiyama et al. (Acta Arith. 98:107?C116, 2001). Various elementary properties of these values are proven.     

A polynomial time algorithm to solve the single-item capacitated lot sizing problem with minimum order quantities and concave costs

This paper deals with the single-item capacitated lot sizing problem with concave production and storage costs, and minimum order quantity (CLSP-MOQ). In this problem, a demand must be satisfied at each period t over a planning horizon of T periods. This demand can be satisfied from the stock or by a production at the same period. When a production is made at period t, the produced quantity must be greater to than a minimum order quantity (L) and lesser than the production capacity (U). To solve this problem optimally, a polynomial time algorithm in O(T⁵) is proposed and it is computationally tested on various instances.     

Cyclotomic p-adic multi-zeta values in depth two

In this paper we compute the values of the p-adic multiple polylogarithms of depth two at roots of unity. Our method is to solve the fundamental differential equation satisfied by the crystalline frobenius morphism using rigid analytic methods. The main result could be thought of as a computation in the p-adic theory of higher cyclotomy. We expect the result to be useful in proving non-vanishing results since it gives quite explicit formulas.     

Diffusion equation for multivalued stochastic differential equations

Deterministic oscillations with bilinear hysteresis are governed by a multivalued differential equation of the type ξ′ + kξ?b(ξ) + g, where k is maximal monotonic and b is Lipschitzian. An existence and uniqueness result is proven for corresponding stochastic equation. The diffusion equation satisfied by the laws of ξ(t) is established. In the particular case k = 0, this equation is equivalent to the Fokker-Planck equation.     

The Reduced Ostrovsky Equation: Integrability and Breaking

The reduced Ostrovsky equation is a modification of the Korteweg‐de Vries equation, in which the usual linear dispersive term with a third‐order derivative is replaced by a linear nonlocal integral term, which represents the effect of background rotation. This equation is integrable provided a certain curvature constraint is satisfied. We demonstrate, through theoretical analysis and numerical simulations, that when this curvature constraint is not satisfied at the initial time, then wave breaking inevitably occurs.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

Properties of equation reformulation of the Karush–Kuhn–Tucker condition for nonlinear second order cone optimization problems

We give an equation reformulation of the Karush–Kuhn–Tucker (KKT) condition for the second order cone optimization problem. The equation is strongly semismooth and its Clarke subdifferential at the KKT point is proved to be nonsingular under the constraint nondegeneracy condition and a strong second order sufficient optimality condition. This property is used in an implicit function theorem of semismooth functions to analyze the convergence properties of a local sequential quadratic programming type (for short, SQP-type) method by Kato and Fukushima (Optim Lett 1:129–144, 2007). Moreover, we prove that, a local solution x* to the second order cone optimization problem is a strict minimizer of the Han penalty merit function when the constraint nondegeneracy condition and the strong second order optimality condition are satisfied at x*.     

Identifying the radiative coefficient of an evolutional type heat conduction equation by optimization method

This paper studies an evolutional type inverse problem of identifying the radiative coefficient of heat conduction equation when the over-specified data is given. Problems of this type have important applications in several fields of applied science. Being different from other ordinary inverse coefficient problems, the unknown coefficient in this paper depends on both the space variable x and the time t. Based on the optimal control framework, the inverse problem is transformed into an optimization problem and a new cost functional is constructed in the paper. The existence, uniqueness and stability of the minimizer of the cost functional are proved, and the necessary conditions which must be satisfied by the minimizer are also given. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations.     

Stationary analysis of the shortest queue first service policy

We analyze the so-called shortest queue first (SQF) queueing discipline whereby a unique server addresses queues in parallel by serving at any time, the queue with the smallest workload. Considering a stationary system composed of two parallel queues and assuming Poisson arrivals and general service time distributions, we first establish the functional equations satisfied by the Laplace transforms of the workloads in each queue. We further specialize these equations to the so-called “symmetric case,” with same arrival rates and identical exponential service time distributions at each queue; we then obtain a functional equation $$\begin{aligned} M(z) = q(z) \cdot M \circ h(z) + L(z) \end{aligned}$$ for unknown function M, where given functions \(q, L\) , and \(h\) are related to one branch of a cubic polynomial equation. We study the analyticity domain of function M and express it by a series expansion involving all iterates of function \(h\) . This allows us to determine empty queue probabilities along with the tail of the workload distribution in each queue. This tail appears to be identical to that of the head-of-line preemptive priority system, which is the key feature desired for the SQF discipline.     

Implicit function theorem without a priori assumptions about normality

The equation F(x, σ) = 0,x ∈ K, in which σ is a parameter and x is an unknown taking values in a given convex cone in a Banach space X, is considered. This equation is examined in a neighborhood of a given solution (x _*, σ_*) for which the Robinson regularity condition may be violated. Under the assumption that the 2-regularity condition (defined in the paper), which is much weaker than the Robinson regularity condition, is satisfied, an implicit function theorem is obtained for this equation. This result is a generalization of the known implicit function theorems even for the case when the cone K coincides with the entire space X.