Non‐linear singular integral equations on a finite interval

A class of nonlinear singular integral equations of Cauchy type on a finite interval is transformed to an equivalent class of (discontinuous) boundary value problems for holomorphic functions in the complex unit disk. Using recent results on the solvability of explicit Riemann–Hilbert problems, we prove the existence of solutions to the integral equation with bounded piecewise continuous nonlinearities. We discuss the influence of parameters and additional conditions and demonstrate the approach for a free boundary problem arising from seepage near a channel. Copyright © 2001 John Wiley & Sons, Ltd.     

Empirical Modeling of a Hydraulic Actuator in a Vehicle Control System

The pressure information of a hydraulic actuator plays a key role in modern vehicle control and fault diagnosis. The difficulty in measuring pressure directly naturally motivates such an indirect approach as an observer whose accuracy depends heavily upon the availability of a high fidelity model of a hydraulic actuator. Notwithstanding its success in understanding the dynamics of the first principle model of a hydraulic actuator, it is not suited to controller/observer design due to its complexity. This paper presents an alternative to the first principle modeling methodology: an empirical approach to hydraulic actuator modeling. Linear and nonlinear system identification techniques are applied to obtain low-order models of a hydraulic actuator. Experimental results show how the empirical models reproduce the key features of a hydraulic actuator with sufficient accuracy.     

Nonlinear Modeling and Tracking Control of a Hydraulic Rotary Vane Actuator

Rotary vane actuators as rotational drives provide rotational movements directly because they are constructed as a joint and actuator in one. So it is possible to pass on the disadvantageous transmission kinematics used with the so far usual differential cylinders at the arms of large manipulators. However, the use of hydraulic rotary vane actuators is associated with high internal oil leakage and/or high friction. Therefore, a nonlinear dynamic model for such an actuator, driving a rigid robot arm, as well as its nonlinear control are derived. To achieve tracking control a model based control law is set up using fundamental linear differential equations for the tracking error. The control law is implemented and tested on a testbed, the produced experimental results are presented. The same control algorithm can also be used to realize nonlinear disturbance attenuation for hydraulic rotary vane actuators via tracking control. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatial Velocity and Temperature Measurements in Non‐Isothermal Liquid Flows

A surface tension driven flow in the liquid vicinity of an air bubble on a heated wall is studied experimentally. The liquid flow caused by the temperature gradient along the surface of the bubble is termed thermocapillary convection. The surface tension force and the buoyancy force oppose one another. The measurement technique is the 3D particle tracking velocity and thermometry, 3D PTV/T, using thermochromic liquid crystals and digital image processing. The paper describes the method in some detail and presents quantitative results for different Marangoni numbers. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interaction of faults under slip‐dependent friction. Non‐linear eigenvalue analysis

We analyse the evolution of a system of finite faults by considering the non‐linear eigenvalue problems associated to static and dynamic solutions on unbounded domains. We restrict our investigation to the first eigenvalue (Rayleigh quotient). We point out its physical significance through a stability analysis and we give an efficient numerical algorithm able to compute it together with the corresponding eigenfunction. We consider the anti‐plane shearing on a system of finite faults under a slip‐dependent friction in a linear elastic domain, not necessarily bounded. The static problem is formulated in terms of local minima of the energy functional. We introduce the non‐linear (static) eigenvalue problem and we prove the existence of a first eigenvalue/eigenfunction characterizing the isolated local minima. For the dynamic problem, we discuss the existence of solutions with an exponential growth, to deduce a (dynamic) non‐linear eigenvalue problem. We prove the existence of a first dynamic eigenvalue and we analyse its behaviour with respect to the friction parameter. We deduce a mixed finite element discretization of the non‐linear spectral problem and we give a numerical algorithm to approach the first eigenvalue/eigenfunction. Finally we give some numerical results which include convergence tests, on a single fault and a two‐faults system, and a comparison between the non‐linear spectral results and the time evolution results. Copyright © 2004 John Wiley & Sons, Ltd.     

Non‐linear systems in plankton population dynamics and nitrogen cycles

In this paper we propose a new perspective of population dynamics of plankton, by considering some effects of global ecological cycles, in which a mixed population of plankton is embedded. The propagation of plankton is extremely influenced by various material cycles, such as Nitrogen cycles. Taking this global effect into consideration, we will construct a mathematical model of non‐linear system. Our model is a non‐linear, non‐equilibrium system based on a stochastic model realizing population dynamics of a mixed population of two species of plankton which is placed in a global nitrogen cycle. We show, in this article, that our model gives a new mathematical foundation of phenomena such as water blooms and the predominance of one type of plankton against the other. We calculate the probability of the occurrence of the water bloom of a mixed population and that is where one type of plankton predominates. We show, as a characteristic feature of our model, that the function of predominance has some discontinuity and that there exists a threshold point among the initial values, with respect to the type of plankton that predominates the other. In other words, there is a sort of phase transition in dynamic changes of plankton population, as a result of global ecological cycles. Copyright © 2000 John Wiley & Sons, Ltd.     

Non‐linear stability in the Bénard problem for a double‐diffusive mixture in a porous medium

The linear and non‐linear stability of a horizontal layer of a binary fluid mixture in a porous medium heated and salted from below is studied, in the Oberbeck–Boussinesq–Darcy scheme, through the Lyapunov direct method. This is an interesting geophysical case because the salt gradient is stabilizing while heating from below provides a destabilizing effect. The competing effects make an instability analysis difficult. Unconditional non‐linear exponential stability is found in the case where the normalized porosity ? is equal to one. For other values of ? a conditional stability theorem is proved. In both cases we demonstrate the optimum result that the linear and non‐linear critical stability parameters are the same whenever the Principle of Exchange of Stabilities holds. Copyright © 2001 John Wiley & Sons, Ltd.     

Non‐linear stability for convection with quadratic temperature dependent viscosity

In this paper, we study the non‐linear stability of convection for a Newtonian fluid heated from below, where the viscosity of the fluid depends upon temperature. We are able to show that for Rayleigh numbers below a certain critical value, Ra_c, the rest state of the fluid and the steady temperature distribution remains non‐linearly stable, using the calculations of Diaz and Straughan (Continuum Mech. Thermodyn. 2004; 16 :347–352). The central contribution of this paper lies in a simpler proof of non‐linear stability, than the ones in the current literature, by use of a suitable maximum principle argument. Copyright © 2006 John Wiley & Sons, Ltd.     

Non‐linear bone remodelling: an existence and uniqueness result

Several models have been proposed in order to be able to predict the correct density and stress distributions on long bones. Most of them are of an empiric and/or experimental nature. A few are related to optimization techniques, in the framework of elasticity theory. In this work, we establish an existence and uniqueness result for a modified Cowin–Hegedus model. This models is of great importance not only because it is one of the most sophisticated including a great number of the existing models, but also because it is set in the framework of adaptive elasticity theory giving a sound basis for the development of robust numerical methods. Copyright © 2000 John Wiley & Sons, Ltd.     

Control problem for a non‐linear thermoelasticity system

The control problem for a three‐dimensional non‐linear thermoelasticity system is considered. The system may represent, among others, the dynamical model of shape memory materials. As controls we take distributed heat sources and body forces. The goal functional refers to the desired evolution of displacement, strain and temperature. The continuity and differentiability of solutions with respect to controls is studied. The existence of optimal controls is proved and the necessary optimality conditions are formulated. The existence of adjoint state variables is proved under additional regularity of data. Copyright © 2004 John Wiley & Sons, Ltd.     

Non‐linear stability and convection for laminar flows in a porous medium with Brinkman law

The non‐linear stability of plane parallel shear flows in an incompressible homogeneous fluid heated from below and saturating a porous medium is studied by the Lyapunov direct method.In the Oberbeck–Boussinesq–Brinkman (OBB) scheme, if the inertial terms are negligible, as it is widely assumed in literature, we find global non‐linear exponential stability (GNES) independent of the Reynolds number R. However, if these terms are retained, we find a restriction on R (depending on the inertial convective coefficient) both for a homogeneous fluid and a mixture heated and salted from below. In the case of a mixture, when the normalized porosity ε is equal to one, the laminar flows are GNES for small R and for heat Rayleigh numbers less than the critical Rayleigh numbers obtained for the motionless state. Copyright © 2003 John Wiley & Sons, Ltd.     

Non‐isolated quasi‐degrees

We show that non‐isolated from below 2‐c.e. Q ‐degrees are dense in the structure of c.e. Q ‐degrees. We construct a 2‐c.e. Q ‐degree, which can't be isolated from below not only by c.e. Q ‐degrees, but by any Q ‐degree. We also prove that below any c.e. Q ‐degree there is a 2‐c.e. Q ‐degree, which is non‐isolated from below and from above (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non‐commutative ‐divergence functional

We introduce the non‐commutative f‐divergence functional for an operator convex function f, where and are continuous fields of Hilbert space operators and study its properties. We establish some relations between the perspective of an operator convex function f and the non‐commutative f‐divergence functional. In particular, an operator extension of Csiszár's result regarding f‐divergence functional is presented. As some applications, we establish a refinement of the Choi–Davis–Jensen operator inequality, obtain some unitarily invariant norm inequalities and give some results related to the Kullback–Leibler distance.     

Non‐effective Quantifier Elimination

Genera connections between quantifier elimination and decidability for first order theories are studied and exemplified.     

Non‐Cayley‐isomorphic self‐complementary circulant graphs

Non‐CI self‐complementary circulant graphs of prime‐squared order are constructed and enumerated. It is shown that for prime p, there exists a self‐complementary circulant graph of order p² not Cayley isomorphic to its complement if and only if p ≡ 1 (mod 8). Such graphs are also enumerated. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 128–141, 2000     

Non‐smooth forced oscillations of a delaminated sandwich beam

Forced oscillations of delaminated sandwich structures are dominated by impacts. During motion the gap between the two adjacent parts of the structure opens and closes periodically. Each contact gives rise to an impact, which leads to energy loss. A 4 DOF beam model with concentrated masses allows discussion of principle on the influence of internal dissipation due to the impacts on the non‐linear system's response and the evolution of impacts near resonance points. Experimental investigations confirm the numerical results.     

Velocity averaging,kinetic formulations,and regularizing effects in quasi‐linear PDEs

We prove in this paper new velocity‐averaging results for second‐order multidimensional equations of the general form ??(?_x, v)f(x, v) = g(x, v) where ??(?_x, v) := a (v) · ?_x ? ? ${}_{\text{x}}^{\text{?}}$ · b (v)?_x. These results quantify the Sobolev regularity of the averages, ∫_v f(x, v)?(v)dv, in terms of the nondegeneracy of the set {v: |??(iξ, v)| ≤ δ} and the mere integrability of the data, (f, g) ∈ (L, L). Velocity averaging is then used to study the regularizing effect in quasi‐linear second‐order equations, ??(?_x, ρ)ρ = S(ρ), which use their underlying kinetic formulations, ??(?_{x, v})χ_ρ = g_S. In particular, we improve previous regularity statements for nonlinear conservation laws, and we derive completely new regularity results for convection‐diffusion and elliptic equations driven by degenerate, nonisotropic diffusion. © 2007 Wiley Periodicals, Inc.     

Non‐selfsimilar global solutions to a two‐dimensional system of conservation laws

This paper considers the two‐dimensional Riemann problem for a system of conservation laws that models the polymer flooding in an oil reservoir. The initial data are two different constant states separated by a smooth curve. By virtue of a nonlinear coordinate transformation, this problem is converted into another simple one. We then analyze rigorously the expressions of elementary waves. Based on these preparations, we obtain respectively four kinds of non‐selfsimilar global solutions and their corresponding criteria. It is shown that the intermediate state between two elementary waves is no longer a constant state and that the expression of the rarefaction wave is obtained by constructing an inverse function. These are distinctive features of the non‐selfsimilar global solutions. Copyright © 2017 John Wiley & Sons, Ltd.