Stochastic functional differential equations with infinite delay

The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment ψγ(t)-bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example.     

Conservation laws for some compacton equations using the multiplier approach

This paper is an application of the variational derivative method to the derivation of the conservation laws for partial differential equations. The conservation laws for (1+1) dimensional compacton k(2,2) and compacton k(3,3) equations are studied via multiplier approach. Also the conservation laws for (2+1) dimensional compacton Zk(2,2) equation are established by first computing the multipliers.     

An approximate method via Taylor series for stochastic functional differential equations

The subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals, sufficiently smooth in the sense of Fréchet derivatives. The approximate equations are defined on equidistant partitions of the time interval, and their coefficients are general Taylor expansions of the coefficients of the initial equation. It will be shown that the approximate solutions converge in the Lp-norm and with probability one to the solution of the initial equation, and also that the rate of convergence increases when degrees in Taylor expansions increase, analogously to real analysis.     

Construction of a high order fluid-structure interaction solver

Accuracy is critical if we are to trust simulation predictions. In settings such as fluid-structure interaction, it is all the more important to obtain reliable results to understand, for example, the impact of pathologies on blood flows in the cardiovascular system. In this paper, we propose a computational strategy for simulating fluid structure interaction using high order methods in space and time.First, we present the mathematical and computational core framework, Life, underlying our multi-physics solvers. Life is a versatile library allowing for 1D, 2D and 3D partial differential solves using h/p type Galerkin methods. Then, we briefly describe the handling of high order geometry and the structure solver. Next we outline the high-order space-time approximation of the incompressible Navier-Stokes equations and comment on the algebraic system and the preconditioning strategy. Finally, we present the high-order Arbitrary Lagrangian Eulerian (ALE) framework in which we solve the fluid-structure interaction problem as well as some initial results.     

Global Hopf bifurcation for differential equations with state-dependent delay

We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S¹-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.     

Fourier multipliers and periodic solutions of delay equations in Banach spaces

In this paper we characterize the existence and uniqueness of periodic solutions of inhomogeneous abstract delay equations and establish maximal regularity results for strong solutions. The conditions are obtained in terms of R-boundedness of linear operators determined by the equations and Lp- Fourier multipliers. Periodic mild solutions are also studied and characterized.     

Filling polygonal holes with minimal energy surfaces on Powell-Sabin type triangulations

In this paper we present two different methods for filling in a hole in an explicit 3D surface, defined by a smooth function f in a part of a polygonal domain D⊂R². We obtain the final reconstructed surface over the whole domain D. We do the filling in two different ways: discontinuous and continuous. In the discontinuous case, we fill the hole with a function in a Powell-Sabin spline space that minimizes a linear combination of the usual seminorms in an adequate Sobolev space, and approximates (in the least squares sense) the values of f and those of its normal derivatives at an adequate set of points. In the continuous case, we will first replace f outside the hole by a smoothing bivariate spline s_f, and then we fill the hole also with a Powell-Sabin spline minimizing a linear combination of given seminorms. In both cases, we obtain existence and uniqueness of solutions and we present some graphical examples, and, in the continuous case, we also give a local convergence result.     

Nonlinear elliptic equations with BMO coefficients in Reifenberg domains

We develop a unifying method to obtain the interior and boundary estimates for the weak solution of a nonlinear elliptic partial differential equation of p-Laplacian type with BMO coefficients in a δ-Reifenberg flat domain. Our results greatly improve the known results for such equations.     

Non-simultaneous blow-up and blow-up rates for reaction-diffusion equations

This paper considers blow-up solutions for reaction-diffusion equations, complemented by homogeneous Dirichlet boundary conditions. It is proved that there exist initial data such that one block or two (separated or contiguous) blocks of n components blow up simultaneously while the others remain bounded. As a corollary, a necessary and sufficient condition is obtained such that any blow-up must be the case for at least two components blowing up simultaneously. We also show some other exponent regions, where any blow-up of k(∈{1,2,…,n}) components must be simultaneous. Moreover, the corresponding blow-up rates and sets are discussed. The results extend those in Liu and Li [B.C. Liu, F.J. Li, Non-simultaneous blow-up of n components for nonlinear parabolic systems, J. Math. Anal. Appl. 356 (2009) 215-231].     

Iterative approximation of a zero of accretive operator in Banach space

This paper introduces an iteration scheme for viscosity approximation of a zero of accretive operator in a reflexive Banach space with weakly continuous duality mapping. A new iterative sequence is introduced and strong convergence of the algorithm {x_n} is proved. The results improve and extend the results of Su [Xiaolong Qin, Yongfu Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 320 (2007) 415-424] and some others.     

Convergence of a stochastic particle approximation for fractional scalar conservation laws

We are interested in a probabilistic approximation of the solution to scalar conservation laws with fractional diffusion and nonlinear drift. The probabilistic interpretation of this equation is based on a stochastic differential equation driven by an α-stable Lévy process and involving a nonlinear drift. The approximation is constructed using a system of particles following a time-discretized version of this stochastic differential equation, with nonlinearity replaced by interaction. We prove convergence of the particle approximation to the solution of the conservation law as the number of particles tends to infinity whereas the discretization step tends to 0 in some precise asymptotics.     

A power penalty approach to a Nonlinear Complementarity Problem

We propose a novel power penalty approach to a Nonlinear Complementarity Problem (NCP) in which the NCP is approximated by a nonlinear equation containing a power penalty term. We show that the solution to the penalty equation converges to that of the NCP at an exponential rate when the function involved is continuous and ξ-monotone. A higher convergence rate is also obtained when the function becomes Lipschitz continuous. Numerical results are presented to confirm the theoretical findings.     

Convergence of a fixed point iteration method for the OSV model

In image processing, image denoising and texture extraction are important problems in which many new methods recently have been developed. One of the most important models is the OSV model [S. Osher, A. Solé, L. Vese, Image decomposition and restoration using total variation minimization and the H^-1 norm, Multiscale Model. Simul. A SIAM Interdisciplinary J. 1(3) (2003) 349-370] which is constructed by the total variation and H^-1 norm. This paper proves the existence of the minimizer of the functional from the OSV model and analyzes the convergence of an iterative method for solving the problems. Our iteration method is constructed by a fixed point iteration on the fourth order partial differential equation from the computation of the associated Euler-Lagrange equation, and the limit of our iterations satisfies the minimizer of the functional from the OSV model. In numerical experiments, we compare the numerical results of our works with those of the ROF model [L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992) 259-268].     

Statistical convergence and ideal convergence for sequences of functions

Let I⊂P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M,μ), we obtain a statistical version of the Egorov theorem (when μ(X)<∞). We show that, in its assertion, equi-statistical convergence on a big set cannot be replaced by uniform statistical convergence. Also, we consider statistical convergence in measure and I-convergence in measure, with some consequences of the Riesz theorem. We prove that outer and inner statistical convergences in measure (for sequences of measurable functions) are equivalent if the measure is finite.     

Existence of weak solutions to the three-dimensional steady compressible Navier-Stokes equations

We prove the existence of a spatially periodic weak solution to the steady compressible isentropic Navier-Stokes equations in R³ for any specific heat ratio γ>1. The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density, and the method of weak convergence.     

Delay-dependent stability of symmetric schemes in Boundary Value Methods for DDEs

We consider the symmetric schemes in Boundary Value Methods (BVMs) applied to delay differential equations y^′(t)=ay(t)+by(t-τ) with real coefficients a and b. If the numerical solution tends to zero whenever the exact solution does, the symmetric scheme with (k₁+m,k₂)-boundary conditions is called τ_k1,k2(0)-stable. Three families of symmetric schemes, namely the Extended Trapezoidal Rules of first (ETRs) and second (ETR₂s) kind, and the Top Order Methods (TOMs), are considered in this paper.By using the boundary locus technology, the delay-dependent stability region of the symmetric schemes are analyzed and their boundaries are found. Then by using a necessary and sufficient condition, the considered symmetric schemes are proved to be τ_ν,ν-1(0)-stable.     

On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations

The present paper is concerned with asymptotic behaviours of the solutions to the micropolar fluid motion equations in R². Upper and lower bounds are derived for the L² decay rates of higher order derivatives of solutions to the micropolar fluid flows. The findings are mainly based on the basic estimates of the linearized micropolar fluid motion equations and generalized Gronwall type argument.     

An analytic approximate method for solving stochastic integrodifferential equations

In this paper we compare the solution of a general stochastic integrodifferential equation of the Ito type, with the solutions of a sequence of appropriate equations of the same type, whose coefficients are Taylor series of the coefficients of the original equation. The approximate solutions are defined on a partition of the time-interval. The rate of the closeness between the original and approximate solutions is measured in the sense of the Lp-norm, so that it decreases if the degrees of these Taylor series increase, analogously to real analysis. The convergence with probability one is also proved.     

Chromatic characterization of biclique covers

A biclique B of a simple graph G is the edge-set of a complete bipartite subgraph of G. A biclique cover of G is a collection of bicliques covering the edge-set of G. Given a graph G, we will study the following problem: find the minimum number of bicliques which cover the edge-set of G. This problem will be called the minimum biclique cover problem (MBC). First, we will define the families of independent and dependent sets of the edge-set E(G) of G: F⊆E(G) will be called independent if there exists a biclique B⊆E(G) such that F⊆B, and will be called dependent otherwise. From our study of minimal dependent sets we will derive a 0-1 linear programming formulation of the following problem: find the maximum weighted biclique in a graph. This formulation may have an exponential number of constraints with respect to the number of nodes of G but we will prove that the continuous relaxation of this integer program can be solved in polynomial time. Finally we will also study continuous relaxation methods for the problem (MBC). This research was motivated by an open problem of Fishburn and Hammer.