General three-point quadrature formulas for the approximate evaluation of an integral of a function f over [0,1], through the values f(x), f(1/2), f(1−x), f′(0) and f′(1), are derived via the extended Euler formula. Such quadratures are sometimes called “corrected” or “quadratures with end corrections” and have a higher accuracy than the adjoint classical formulas, which only include the values f(x), f(1/2)and f(1−x) . The Gauss three-point, corrected Simpson, corrected dual Simpson, corrected Maclaurin and corrected Gauss two-point formulas are recaptured as special cases. Finally, sharp estimates of error are given for this type of quadrature formula.

The problem of stagnation point flow with heat transfer of an electrically conducting fluid impinging normally on a permeable axisymmetric surface in the presence of a uniform transverse magnetic field is analysed. The governing nonlinear differential equations and their associated boundary conditions are reduced to dimensionless form using suitable similarity transformations. Comparison with previously published work shows good agreement. Effects of the injection–suction parameter, magnetic parameter and Prandtl number on the flow and thermal fields are presented. The investigations show that the wall shear stress and heat transfer rate from the surface increase with increased applied magnetic field. An increase in the velocity and thermal boundary layer thicknesses is observed with an increase in the wall injection, while the velocity and thermal boundary layers become thinner when increasing the wall suction and applied magnetic field.

The creeping flow of an incompressible viscous liquid past a porous approximately spherical shell is considered. The flow in the free fluid region outside the shell and in the cavity region of the shell is governed by the Navier–Stokes equations. The flow within the porous annular region of the shell is governed by Brinkman’s model. The boundary conditions used at the interface are continuity of the velocity, continuity of the pressure and Ochoa-Tapia and Whitaker’s stress jump condition. An exact solution for the problem and an expression for the drag on the porous approximately spherical shell are obtained. The drag is evaluated numerically for several values of the parameters governing the flow.

A synthesis is presented of two recent studies on modelling the nonlinear neuro-mechanical hearing processes in mosquitoes and in mammals. In each case, a hierarchy of models is considered in attempts to understand data that shows nonlinear amplification and compression of incoming sound signals. The insect’s hearing is tuned to the vicinity of a single input frequency. Nonlinear response occurs via an arrangement of many dual capacity neuro-mechanical units called scolopidia within the Johnston’s organ. It is shown how the observed data can be captured by a simple nonlinear oscillator model that is derived from homogenization of a more complex model involving a radial array of scolopidia. The physiology of the mammalian cochlea is much more complex, with hearing occurring via a travelling wave along a tapered, compartmentalized tube. Waves travel a frequency-dependent distance along the tube, at which point they are amplified and “heard”. Local models are reviewed for the pickup mechanism, within the outer hair cells of the organ of Corti. The current debate in the literature is elucidated, on the relative importance of two possible nonlinear mechanisms: active hair bundles and somatic motility. It is argued that the best experimental agreement can be found when the nonlinear terms include longitudinal coupling, the physiological basis of which is described. A discussion section summarizes the lessons learnt from both studies and attempts to shed light on the more general question of what constitutes a good mathematical model of a complex physiological process.

We analyze a nonlinear discrete scheme depending on second-order finite differences. Thisis the two-dimensional analog of a scheme which in one dimension approximates afree-discontinuity energy proposed by Blake and Zisserman as a higher-order correction ofthe Mumford and Shah functional. In two dimension we give a compactness result showingthat the continuous problem approximating this difference scheme is still defined onspecial functions with bounded hessian, and we give an upper and a lower bound in terms ofthe Blake and Zisserman energy. We prove a sharp bound by exhibiting thediscrete-to-continuous Γ-limit for a special class of functions, showingthe appearance new ‘shear’ terms in the energy, which are a genuinely two-dimensionaleffect.

In this paper, using direct and inverse images for fractional stochastic tangent sets, weestablish the deterministic necessary and sufficient conditions which control that thesolution of a given stochastic differential equation driven by the fractional Brownianmotion evolves in some particular sets K. As a consequence, a comparisontheorem is obtained.

We study a two-phase pipe flow model with relaxation terms in the momentum and energy equations, driving the model towards dynamic and thermal equilibrium. These equilibrium states are characterized by the velocities and temperatures being equal in each phase. For each of these relaxation processes, we consider the limits of zero and infinite relaxation times. By expanding on previously established results, we derive a formulation of the mixture sound velocity for the thermally relaxed model. This allows us to directly prove a subcharacteristic condition; each level of equilibrium assumption imposed reduces the propagation velocity of pressure waves. Furthermore, we show that each relaxation procedure reduces the mixture sound velocity with a factor that is independent of whether the other relaxation procedure has already been performed. Numerical simulations indicate that thermal relaxation in the two-fluid model has negligible impact on mass transport dynamics. However, the velocity difference of sonic propagation in the thermally relaxed and unrelaxed two-fluid models may significantly affect practical simulations.

Numerically solving the Boltzmann kinetic equations with the small Knudsen number ischallenging due to the stiff nonlinear collision terms. A class of asymptotic-preservingschemes was introduced in [F. Filbet and S. Jin,J. Comput. Phys. 229 (2010)7625–7648] to handle this kind of problems. The idea is to penalize the stiff collisionterm by a BGK type operator. This method, however, encounters its own difficulty whenapplied to the quantum Boltzmann equation. To define the quantum Maxwellian (Bose-Einsteinor Fermi-Dirac distribution) at each time step and every mesh point, one has to invert anonlinear equation that connects the macroscopic quantity fugacity with density andinternal energy. Setting a good initial guess for the iterative method is troublesome inmost cases because of the complexity of the quantum functions (Bose-Einstein orFermi-Dirac function). In this paper, we propose to penalize the quantum collision term bya ‘classical’ BGK operator instead of the quantum one. This is based on the observationthat the classical Maxwellian, with the temperature replaced by the internal energy, hasthe same first five moments as the quantum Maxwellian. The scheme so designed avoids theaforementioned difficulty, and one can show that the density distribution is still driventoward the quantum equilibrium. Numerical results are presented to illustrate theefficiency of the new scheme in both the hydrodynamic and kinetic regimes. We also developa spectral method for the quantum collision operator.

In this article, we provide a priori error estimates for the spectral andpseudospectral Fourier (also called planewave) discretizations of theperiodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectraldiscretization of the periodic Kohn-Shammodel, within the local density approximation (LDA). These modelsallow to compute approximations of the electronic ground state energy and densityof molecular systems in the condensed phase. The TFW model is strictlyconvex with respect to the electronic density, and allows for acomprehensive analysis. This is not the case for the Kohn-Sham LDAmodel, for which the uniqueness of the ground state electronic densityis not guaranteed. We prove that, for any local minimizer $\Phi^0$ of the Kohn-Sham LDA model, and under a coercivity assumption ensuring the local uniqueness of this minimizer up to unitary transform, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of $\Phi^0$ for large enough energy cut-offs, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.

This paper presents the first analytical solutions for the three-dimensional motion of two idealized mobiles controlled by a particular guidance law designed to avoid a collision with minimal path deviation. The mobiles can be regarded as particles, and guidance can be interpreted as complex forces of interaction between the particles. The motion is then a generalized form of two-body Newtonian dynamics. If the mobiles have equal speeds, the relative motion is determined through various transformations of the differential equations. Solvability relies on congruence and symmetries of the paths, which is exploited to reduce the original twelve first-order differential equations to three first-order equations for the relative motion. The resulting state space is partitioned into five invariant subsets, with various symmetries and stabilities. One of these sets describes planar motion, where simple explicit solutions are given. In nonplanar motion, the solution is formally reduced to quadrature. A numerical calculation gives the separation at the closest point of approach, which provides control over minimum separation. The results should be of interest because of their application, which includes, most importantly, the prevention of midair collisions between aircraft, but also potential application to land, water and space vehicles. The solutions should be of interest to mathematical specialists in dynamical systems, because of some novel constants of the motion, novel symmetries, and the associated reducibility of the equations.

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.

We study the problem of flatness of two-input driftless control systems. Although acharacterization of flat systems of that class is known, the problems of describing allflat outputs and of calculating them is open and we solve it in the paper. We show thatall x-flat outputs are parameterized by an arbitrary function of threecanonically defined variables. We also construct a system of 1st order PDE’s whosesolutions give all x-flat outputs of two-input driftless systems. Weillustrate our results by describing all x-flat outputs of models of anonholonomic car and the n-trailer system.

We consider an insurance company whose surplus is governed by a jump diffusion risk process. The insurance company can purchase proportional reinsurance for claims and invest its surplus in a risk-free asset and a risky asset whose return follows a jump diffusion process. Our main goal is to find an optimal investment and proportional reinsurance policy which maximizes the expected exponential utility of the terminal wealth. By solving the corresponding Hamilton–Jacobi–Bellman equation, closed-form solutions for the value function as well as the optimal investment and proportional reinsurance policy are obtained. We also discuss the effects of parameters on the optimal investment and proportional reinsurance policy by numerical calculations.

In this article we derive a macroscopic model for the time evolution of root density, starting from a discrete mesh of roots, using homogenization techniques. In the microscopic model each root grows vertically according to an ordinary differential equation. The roots growth rates depend on the spatial distribution of nutrient in the soil, which also evolves in time, leading to a fully coupled non-linear problem. We derive an effective partial differential equation for the root tip surface and for the nutrient density.

The aim of this paper is to analyze a low order finite element methodfor a stiffened plate. The plate is modeled by Reissner-Mindlinequations and the stiffener by Timoshenko beams equations. Theresulting problem is shown to be well posed. In the case of concentricstiffeners it decouples into two problems, one for the in-plane plate deformation and the other for the bending of the plate. The analysisand discretization of the first one is straightforward. The second oneis shown to have a solution bounded above and below independently of thethickness of the plate. A discretization based on DL3 finite elementscombined with ad-hoc elements for the stiffener is proposed.Optimal order error estimates are proved for displacements, rotationsand shear stresses for the plate and the stiffener. Numerical tests arereported in order to assess the performance of the method. Thesenumerical computations demonstrate that the error estimates areindependent of the thickness, providing a numerical evidence that themethod is locking-free.


A number of approaches for discretizing partial differential equations with random dataare based on generalized polynomial chaos expansions of random variables. These constitutegeneralizations of the polynomial chaos expansions introduced by Norbert Wiener toexpansions in polynomials orthogonal with respect to non-Gaussian probability measures. Wepresent conditions on such measures which imply mean-square convergence of generalizedpolynomial chaos expansions to the correct limit and complement these with illustrativeexamples.