A Hilbert Space Approach to Fractional Differential Equations

Journal of Dynamics and Differential Equations - We study fractional differential equations of Riemann–Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of...     

A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term

We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering integro-differential equations, neutral differential equations and general delay differential equations within a unified framework. We show that reasonable differential equations lead to causal solution operators.     

Geometric Approach to Nonvariational Singular Elliptic Equations

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms, as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter (γ - 1), for 0 < γ < 1, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases.We establish existence as well as sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and we obtain fine geometric-measure properties of the free boundary ${\mathfrak{F} = \partial{u > 0}}$ . In particular, we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region {u > 0} and the ${\mathcal{H}^{n-1}}$ almost-everywhere weak differentiability property of ${\mathfrak{F}}$ .     

A New Rational Approach to Jet Noise Reduction

This paper offers a new rational approach to jet noise suppression methodology. The tools required to implement such an approach are discussed. These involve the use of global velocimetry to measure those Lighthill turbulent stress tensor terms thought most important to noise generation. The framework of the dynamic systems model as it applies to the development of a noise suppression model is discussed, as well as the latest results on the development of an actuator for control input. The dynamic systems model offers a unique opportunity to study the influence of a concept noise reduction scheme on noise reduction through appropriate algorithms that relate turbulent jet processes to noise. Received 24 January 1997 and accepted 30 May 1997     

A General Approach for Front-Propagation in Functional Reaction-Diffusion Equations

We investigate the persistence of front propagation for functional reaction-diffusion equations

Stability Results for Second-Order Evolution Equations with Memory and Switching Time-Delay

It is well-known that wave-type equations with memory, under appropriate assumptions on the memory kernel, are uniformly exponentially stable. On the other hand, time delay effects may destroy this behavior. Here, we consider the stabilization problem for second-order evolution equations with memory and intermittent delay feedback. We show that, under suitable assumptions involving the delay feedback coefficient and the memory kernel, asymptotic or exponential stability are still preserved. In particular, asymptotic stability is guaranteed if the delay feedback coefficient belongs to \(L^1(0, +{\infty })\) and the time intervals where the delay feedback is off are sufficiently large.     

Lagrangian Approach to Evolution Equations: Symmetries and Conservation Laws

We show that one can apply a Lagrangian approach to certain evolution equations by considering them together with their associated equations. Consequently, one can employ Noether's theorem and derive conservation laws from symmetries of coupled systems of evolution equations. We discuss in detail the linear and non-linear heat equations as well as the Burgers equation and obtain new non-local conservation laws for the non-linear heat and the Burgers equations by extending their symmetries to the associated equations. We also provide Lagrangians for non-linear Schrödinger and Korteweg—de Vries type systems.     

一种求解双曲方程新的有限差分算法

本文提出一种适于求解一阶双向系统的新的差分格式。它的建立方法是：将所要求解的方程与解的空间导数所满足的微分方程同时离散化，然后再通过插值函数构成封闭的离散变量代数方程。在线性情况下的误差分析表明：该格式的幅值与位相误差均小于常用的一、二阶差分格式；当其应用于非线性气动方程求解时，基本上可以消除数值扩散与振荡这两种非正常现象。     

A Geometrical Approach to the Study¶of Unbounded Solutions¶of Quasilinear Parabolic Equations

In this article, we are interested in the existence and uniqueness of solutions for quasilinear parabolic equations set in the whole space ? ^N . We consider, in particular, cases when there is no restriction on the growth or the behavior of these solutions at infinity. Our model equation is the mean-curvature equation for graphs for which Ecker and Huisken have shown the existence of smooth solutions for any locally Lipschitz continuous initial data. We use a geometrical approach which consists in seeing the evolution of the graph of a solution as a geometric motion which is then studied by the so-called “level-set approach”. After determining the right class of quasilinear parabolic PDEs which can be taken into account by this approach, we show how the uniqueness for the original PDE is related to “fattening phenomena” in the level-set approach. Existence of solutions is proved using a local L ^∞ bound obtained by using in an essential way the level-set approach. Finally we apply these results to convex initial data and prove existence and comparison results in full generality, i.e., without restriction on their growth at infinity.     

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.     

A New Approach to Asymptotic Diagonalization of Linear Differential Systems

We study the asymptotic diagonalization of a system consisting of an -matrix plus a finite number of -perturbations on an interval I ₀=[t ₀, ), where p, m _i[1, ). Using linear skew-product flows and spectral theory, we show that if the unperturbed system has full spectrum over its omega-limit set, then the entire system is asymptotically diagonalizable almost everywhere.     

A Variational Approach for the 2-Dimensional Semi-Geostrophic Shallow Water Equations

Existence of weak solutions to the 3-D semi-geostrophic equations with rigid boundaries was proved by Benamou and Brenier [3], using Monge transport theory. This paper extends the results to a free surface boundary condition, which is more physically appropriate. This extension is at present for the 2-D shallow water case only. In addition, we establish stronger time regularity than was possible in [3]. Accepted October 9, 2000?Published online February 14, 2001     

A New Approach to the Fundamental Theorem of Surface Theory

The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a _αβ ) of order two and a field of symmetric matrices (b _αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of , then there exists an immersion such that these fields are the first and second fundamental forms of the surface , and this surface is unique up to proper isometries in . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a _αβ and b _αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation

A New Approach to Non-Isothermal Models for Nematic Liquid Crystals

We introduce a new class of non-isothermal models describing the evolution of nematic liquid crystals and prove their consistency with the fundamental laws of classical thermodynamics. The resulting system of equations captures all essential features of physically relevant models; in particular, the effect of stretching of the director field is taken into account. In addition, the associated initial-boundary value problem admits global-in-time weak solutions without any essential restrictions on the size of the initial data.     

Complexity Versus Complex Systems: A New Approach to Scientific Discovery

Extraction of quantitative features from observations via suitable measuring devices M means that the words of science are coded as numbers, and the syntaxis is a set of mathematical rules. Once general premises are available all consequences can be worked in a purely deductive way. This characteristic of science displays two orders of drawbacks, namely, undecidability of deductive procedures, and intractability of computer modelings of complex situations. The way out of such a crisis consists in an adaptive strategy, that is, in a frequent readjustment of M suggested by the observed events. As a consequence, M provides different data streams (words) for the same observed events, as it is tuned to different resolutions. The adaptive strategy here introduced should by no means be confused with the adaptivity of a learning machine, which—inputted by a data stream—readjusts itself over a class of theoretical explanations in order to select the optimal one, thus providing knowledge conditional on the assigned input. On the contrary, physics aims at extracting regular patterns out of things, by a trial and error procedure which includes not only modifications of the explanations for fixed data sets, but also exploring different data sets via modified M's. This M-adjustment is a pre-linguistic endeavour, not expressible by a formal language. Such an essential characteristic of the physical program means that physics can not be performed by a machine.     

A New Approach to Front Propagation Problems: Theory and Applications

In this paper we present a new definition for the global in time propagation (motion) of fronts (hypersurfaces, boundaries) with a prescribed normal velocity, past the first time they develop singularities. We show that if this propagation satisfies a geometric maximum principle (inclusion‐avoidance)‐type property, then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures. This new approach, which is more geometric and, as it turns out, equivalent to the level‐set method, is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction‐diffusion equations. We finally present a number of new asymptotic results. Among them are the asymptotics of (i) reaction‐diffusion equations with rapidly oscillating coefficients, (ii) fully nonlinear nonlocal (integral differential) equations and (iii) stochastic Ising models with long-range anisotropic interactions and general spin flip dynamics. (Accepted July 8, 1996)     

A New Class of Equations for Rotationally Constrained Flows

The incompressible Navier–Stokes equation is considered in the limit of rapid rotation (small Ekman number). The analysis is limited to horizontal scales small enough so that both horizontal and vertical velocities are comparable, but the horizontal velocity components are still in geostrophic balance. Asymptotic analysis leads to a pair of nonlinear equations for the vertical velocity and vertical vorticity coupled by vertical stretching. Statistically stationary states are maintained against viscous dissipation by boundary forcing or energy injection at larger scales. For thermal forcing direct numerical simulation of the reduced equations reveals the presence of intense vortical structures spanning the layer depth, in excellent agreement with simulations of the Boussinesq equations for rotating convection by Julien et al. (1996). Received 30 May 1997 and accepted 4 January 1998     

Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

This paper is on the so called inverse problem of ordinary differential equations, i.e. the problem of determining the differential system satisfying a set of given properties. More precisely we characterize under very general assumptions the ordinary differential equations in \(\mathbb {R}^N\) which have a given set of either \(M\) partial integrals, or \(M first integral, or \(M partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of \(N-1\) independent first integrals. We give two relevant applications of the solutions of these inverse problem to constrained Lagrangian and Hamiltonian systems respectively. Additionally we provide the general solution of the inverse problem in dynamics.     

边界元法边界层处理的一种新方法—拟多连域法

提出了处理边界元法边界层问题的新方法。这种方法在求得边界未知量后，在距边界较远的域中构造一工作边界，将实际边界与工作边界看作一多连域问题求得工作边界的未知量。再将工作边界与边界层边界视为一多连域问题求解，可得到满足精度要求的边界附近点的位移与应力。这种方法理论简洁、计算方便、有效、精度高，对于需求边界上多点的未知量问题很具优越性