Erential Equations Research Articles

Modeling of the Flow-Reversal Process in a Density Oscillator

Modeling of the Flow-Reversal Process in a Density Oscillator Takeshi Kano and Shuichi Kinoshita Graduate School of Frontier Biosciences, Osaka University, Suita, Osaka 565-0871, Japan (Received 21 August 2007) The density oscillator is known to be a simple model system that exhibits self-sustained oscillation. In a previous study, we have shown that the ow-reversal process is clearly viscosity-dependent and have proposed a model that describes the viscosity-dependent process. In the present paper, we re ne the previous model and describe the dynamical behaviors of the density oscillator by using simple di erential equations with non-dimensionalized formulations. The re ned model is generally in good agreement with the experimental results for uids with various viscosities. PACS numbers: 05.45.-a, 82.40.Bj, 05.90.+m

Identification of the Local Speed Function in a Levy Model for Option Pricing

We propose a non-parametric stable calibration method based on Tikhonov regularization for the local speed function in a local Levy model. The jump term in this model introduces an integral operator into the classic Black-Scholes partial di erential equation such that the associated model calibration to observed option prices can be treated as a parameter identication problem for a partial integro-di erential equation. This problem is shown to be ill-posed and thus requires regularization. It is proven that nonlinear Tikhonov regularization is a stable and convergent method for this problem. Furthermore, convergence rate results are established under an abstract source condition. Finally the theoretical results are underpinned by numerical illustrations including a real-world example.

Derivatives and Eulerian Numbers

The aim of the present note is to give a closed form formula for the nth deriva-tive of a function which satis es Riccati’s di erential equation with constantcoecients. The paper is organized as follows. In section 2 we recall the de -nition and basic properties of Eulerian numbers. We introduce and prove thementioned formula in section 3. Some examples where the formula can be ap-plied are included in section 4. In one of them, using a formula derived byHamilton, we arrive at an elegant integral expression for Eulerian numbers.

Correct Specification and Identification of Nonparametric Transformation Models

This paper derives necessary and sufficient conditions for nonparametric transformation models to be (i) correctly speci fied, and (ii) identi fied. Our correct speci fication conditions come in a form of partial diff erential equations; when satis fied by the true distribution, they ensure that the observables are indeed generated from a nonparametric transformation model. Our nonparametric identifi cation result is global; we derive it under conditions that are substantially weaker than full independence. In particular, we show that a completeness assumption combined with independence with respect to one of the regressors suffices for the model to be identi fied.

Projection of diffusions on submanifolds: Application to mean force computation

AbstractWe consider the problem of sampling a Boltzmann‐Gibbs probability distribution when this distribution is restricted (in some suitable sense) on a submanifold Σ of ℝn implicitly defined by N constraints q1(x) = ⃛ = qN(x) = 0 (N < n). This problem arises, for example, in systems subject to hard constraints or in the context of free energy calculations. We prove that the constrained stochastic differential equations (i.e., diffusions) proposed in [7, 13] are ergodic with respect to this restricted distribution. We also construct numerical schemes for the integration of the constrained diffusions. Finally, we show how these schemes can be used to compute the gradient of the free energy associated with the constraints. © 2007 Wiley Periodicals, Inc.

On asymptotic behavior of positive solutions of $\bm{x" = e^{\alpha \lambda t} x^{1 + \alpha \/}}$ with $\bm{\alpha < -1\/}$

In this paper we shall show asymptotic behavior of all positive solutions of the second order nonlinear di¤erential equation written in the title. It will complete this task to obtain an analytical expression or an asymptotic form of every solution valid in a neighborhood of an end of its domain.

Characterization of the linear partial di¤erential equations that admit solution operators on Gevrey classes

Characterization of the linear partial di¤erential equations that admit solution operators on Gevrey classes

Second-order hyperbolic s.p.d.e.’s driven by homogeneous Gaussian noise on a hyperplane

We study a class of hyperbolic stochastic partial di®erential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spatially homogeneous within the hyperplane. Two natural notions of solutions are function-valued solutions and random field solutions. For the linear form of the equations, we identify the necessary and sufficient condition on the spectral measure of the spatial covariance for existence of each type of solution, and it turns out that the conditions di®er. In spatial dimensions 2 and 3, under the condition for existence of a random field solution to the linear form of the equation, we prove existence and uniqueness of a random field solution to non-linear forms of the equation.

Experimental and numerical study of the Marangoni convection due to localized laser heating

Abstract We study experimentally and theoretically convective flows, which are in-duced in a horizontal liquid layer by a concentrated heat source: the split co-herent beam of laser radiation. The layer surface is deformable. Dependingon controlling factors, laboratory experiments demonstrate a variety of flowstructures and surface configurations. The flow primarily has a single vortexpattern, but in a certain range of governing parameters, secondary nonsta-tionary vortices are superimposed. Among the surface configurations thereare the concave meniscus, the convex one, and the concavo-convex one. Thenumerical simulation is performed on a mathematical model, which involvesnonlinear partial di¤erential equations for two-dimensional amplitude func-tions associated with distributions of temperature, vorticity, and surface de-formation. In the long-wave approximation, the model describes the contri-bution of thermocapillary convection to the heat transfer as well as thedegree of the interface deformation. The proposed model generalizes the ex-isting one by taking into account the heating inhomogeneity.

Existence and stability of solutions for a class of partial neutral functional differential equations

In this work, we consider a class of nonlinear partial neutral functional di¤erential equations with a nondensely defined Hille-Yosida operator. We first prove the local existence, uniqueness and regularity of solutions. Second, we study the global existence and stability. In the end, we extend in the autonomous case, results of Hale ([21], [23]) concerning dissipativeness and existence of a global attractor to our situation.

A simple fast method in finding the analytical solutions to a class of nonlinear partial differential equations

By introducing,a new transformation and selecting appropriate trial functions ,nonlinear partial diff erential equations can be converted to algebraic equations,and their related coe f ficients can be easily determined by making use of the method of undetermined coefficients. Finally,the analytical solutions to a class of nonlinear partial di fferential equations are successfully derived. One can easily see that this meth od used herein is particularly simple.

Positive solutions for first order nonlinear functional boundary value problems on infinite intervals

1. IntroductionBoundaryvalueproblemsonin nite intervalsappearinmanyproblemsofpracti-cal interest, for example in linear elasticity problems, nonlinear uid ow problemsand foundation engineering (see e.g. [1,10,16] and the references therein). Thisis the reason why these problems have been studied quite extensively in the lit-erature, especially the ones involving second order di erential equations. Secondorder boundary value problems on in nite intervals are treated with various meth-ods, such as xed point theorems (see e.g. [1,5,6,10,14-16,23]), upper and lowersolutions method (see e.g. [2,8,9]), diagonalization method (see e.g. [1,18,20]) andothers. An interesting overview on in nite interval problems, including real worldexamples, history and various methods of solvability, can be found in the recentbook of Agarwal and O’ Regan [1].A rather less extensive study has been done on rst order boundary value prob-lems on in nite intervals. One of the major ways to deal with such problems is touse numerical methods, see for example [10,16]. In short, the basic idea in this caseis to build a nite interval problem such that its solution approximates the solutionof the in nite problemquite well on the nite interval. The diculty of this methodlies in setting the nite interval boundary value problem in such a way that theapproximation is accurate. Another way to deal with boundary value problems onin nite intervals is to use xed point theorems. See for example [7,9]. Using thisapproach one will have to reformulate the boundary value problem to an operatorequation and to use an appropriate compactness criterion on in nite intervals forthe corresponding operator (see Lemma 2.2 below).In the recent years a growing interest has arisen for positive solutions of bound-ary value problems. See for example [10,11,17,18,23]. Also, nowadays, functionalboundary value problems are extensively investigated, usually via xed point the-orems (see [6,12,21,22] and the references therein).

Kinetics Behavior of G1-to-S Cell Cycle Phase Transition Model

Recently, the study of dynamics for many kinds of cellular molecules is a topic, because we can notunderstand an entire picture of cell, regardless of rapid increase of information for it individually.Knowing the dynamics, we can nd many insights about mechanics of what happens in cell, cell cycle,apoptosis, development, and transformation.A cell cycle is a critical phenomenon for transformation, apoptosis, and development, which forma complex and highly integrated network [3]. Although many of the regulatory molecules have beenidenti ed, dynamics of the complicated network is too complex to be understood by intuition alone.Therefore, by using a simulation based on a mathematical model of cell cycle G1-to-S phasetransition which consists of di erential equations, we shall understand the key feature which controlsthecellcycle. Untilnow, manymathematical modelsofregulationofcell cycleG1-to-Sphasetransitionhave been previously proposed and simulated [1, 2, 4]. The most detailed model of models is by Agudaand Tang [1]. We simulated cell cycle G1-to-S phase transition based on this model, and analyze thedynamics of cell cycle.

Multilinear singular integrals

We survey the thoery of multilinear singular integral operators with modulation symmetry. The basic example for this theory is the bilinear Hilbert transform and its multilinear variants. We outline a proof of boundedness of Carleson's operator which shows the close connection of this operator to multilinear singular inte- grals. We discuss particular multilinear singular integrals which historically arose in the study of eigenfunctions of Schrodinger operators. This survey article arose from a series of three expository lectures given at the 6th International Conference on Harmonic Analysis and Partial Di! erential Equations at El Escorial 2000. I would like to thank the organizers for organizing this stimulating and successful conference. The article is divided into three chapters following the three lectures at El Escorial. The first section gives an introduction into the subject of multilinear singular integrals with modulation symmetries as it has evolved over the past five years. The second section presents a proof of boundedness of the Carleson operator, the essential ingredient in the proof of Carleson's theorem on almost everywhere convergence of Fourier series. We follow closely the work (29), with additional comments as pre- sented during the lecture. While Carleson's operator is not a multilinear operator itself, it serves as a good model for the type of arguments used in the theory: thus the theory extends to other objects than multilinear operators. The third section is again purely expository and gives an overview of how the discussed theory of multilinear singular integrals plays a role in eigenfunction expansions of one dimensional Schrodinger operators. In fact, some of the most prominent objects in the theory such as the bilinear Hilbert transform and Carleson's operator enter directly into these multilinear expansions.

Maximum principle for optimal control of non-well posed elliptic differential equations

In this paper, we shall study two optimal control problems (P1) and (P2) governed by some semilinear elliptic di%erential equations which can admit more than one solution. We shall call such systems non-well posed systems. We obtain the Pontryagin maximum principle for problems (P1) and (P2). In the 6rst problem (P1), the cost functional may be non-smooth and the set of controls is convex while in the second problem (P2), the cost functional is smooth and the set of controls is non-convex. In both problem (P1) and problem (P2), we consider some state constraint of integral type. Due to some practical interests, many authors studied optimal control for elliptic di%erential equations. Here, we cite [1–10,12–15]. However, most of these works, except for [7,10,13], deal with state equations which are governed by monotone operators and admit a unique solution corresponding to each control. In such problems, the Pontryagin maximum principle was obtained by studying the variations of the state with respect to some perturbations of the control (cf. [1,2,12,15]). In the present work, the state equations are governed by non-monotone operators and can admit more than one solution. We cannot obtain the variations of the state with respect to the control with the same methods in [1,2,12,15]. When the state equations admit more than one solution, we even do not know how to make the sensitivity analysis of the state with respect to the control.

Oscillation of solutions of first-order neutral differential equations

First order neutral di¤erential equations are studied and sufficient conditions are derived for every solution to be oscillatory. Our approach is to reduce the oscillation of neutral differential equations to the nonexistence of eventually positive solutions of non-neutral differential inequalities. Our results extend and improve several known results in the literature.

Sensitivity Analysis of a Nonlinear Obstacle Plate Problem

PRAXIS/PCEX/C/MAT/38/96 \Variational Models and Optimization, and Sapiens Proj99, No. 34471 \Nonlinear Partial Di erential Equations and Interfaces Problems of the Ministry of Science and Technology of Portugal

Chaotic dynamics for perturbations of infinite-dimensional Hamiltonian systems

where w(t; z)∈R is the transverse de9ection of the axis of the beam; w(t; 0)=w(t; 1)= wzz(t; 0)=wzz(t; 1)=0, is an external load, ? 0 is a ratio indicating the extensional rigidity and is the damping. The #rst result on the existence of a chaotic dynamics for system (H ) has been given by Holmes and Marsden in [10] for a speci#c periodic forcing perturbation of the type P(t; w(t; z))=f(z) cos(!t). They use the theory of invariant manifolds and of non-linear semigroups in order to extend the classical Melnikov approach for planar ordinary di=erential equations to system (H ).

The control of boundary value problems for quasilinear impulsive integro-differential equations

The theory of impulsive di'erential equations is emerging as an important area of investigation since it is richer in problems in comparison with the corresponding theory of di'erential equations and many of the mathematical problems encountered in studying the impulsive di'erential equations cannot be treated with the usual techniques of ordinary di'erential equations [7,9,11]. Moreover, impulsive di'erential systems represent a natural framework for mathematical modelling of several processes in natural science. That is why in recent years the study of such systems has been very intensive. However, the theory of integro-di'erential equations with impulse actions on surfaces has not yet been su9ciently elaborated when compared to that of impulsive di'erential equations and integro-di'erential equations. There are also several problems on the controllability problem for impulsive systems which are connected with the results of the theory of integral and integro-di'erential equations and have not been well investigated yet [5,6,8,10].

Modeling and analysis of a marine bacteriophage infection with latency period

In a previous paper [4] the authors proposed a simple model to describe the epidemics induced by bacteriophages in marine bacteria populations like cyanobacteria and heterotrophic bacteria where the environment is the thermoclinic layer of the sea within which bacteriophages and bacteria are assumed to be homogeneously distributed. The main model simpli cation was in modeling the latent period of infected bacteria in order to describe the model with three nonlinear ordinary di erential equations. However, modeling of the latent period by suitable delay terms looks to be biologically reasonable and mathematically challenging, the ndings of which can be interesting to compare with the outcomes of our previous model [4]. Hence, in the following we rst recall the biological justi cation for the model and then introduce the model itself comparing it with other models on the same topic. The experimental evidence of the bacteriophage infection of marine bacteria can be found, for example, in the papers by Sieburth [14], Moebus [11], Bergh et al. [3]; Proctor and Fuhrman [12]. It is