We consider the following classical autonomous variational problem

\[ \textrm{minimize\,} \left\{F(v)=\int_a^b f(v(x),v'(x))\ \dx\,:\,v\in AC([a,b]), \;v(a)=\alpha,\; v(b)=\beta\right\},\]

where the Lagrangian f is possibly neither continuous, nor convex, nor coercive.We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existenceor non-existence criteria.

We prove that the critical points of the 3d nonlinear elasticity functionalon shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0to the critical points of the vonKármán functional on S, recently proposed in [Lewicka et al., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)].This result extends the statement in [Müller and Pakzad, Comm. Part. Differ. Equ. 33 (2008) 1018–1032], derived for the case of plates when $S\subset\mathbb{R}^2$.The convergence holds provided the elastic energies of the 3d deformations scale like h4 and the external body forces scale like h3.

Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with measures and functions of bounded variation as controls are considered. Existence and uniqueness of the corresponding predual problems are discussed, as is the solution of the optimality systems by a semismooth Newton method. Numerical examples illustrate the structural differences in the optimal controls in these Banach spaces, compared to those obtained in corresponding Hilbert space settings.

We present a model of the full thermo-mechanical evolution of a shape memory body undergoing a uniaxial tensile stress. The well-posedness of the related quasi-static thermo-inelastic problem is addressed by means of hysteresis operators techniques. As a by-product, details on a time-discretization of the problem are provided.

In the present work we introduce a new family of cell-centered Finite Volume schemes for anisotropic and heterogeneous diffusion operators inspired by the MPFA L method.A very general framework for the convergence study of finite volume methods is provided and then used to establish the convergence of the new method.Fairly general meshes are covered and a computable sufficient criterion for coercivity is provided. In order to guarantee consistency in the presence of heterogeneous diffusivity, we introduce a non-standard test space in $H_0^1$ (Ω) and prove its density. Thorough assessment on a set of anisotropic heterogeneous problems as well as a comparison with classical multi-point Finite Volume methods is provided.

When applied to the linear advection problem in dimension two, the upwind finite volume method is a non consistent scheme in the finite differences sense but a convergent scheme. According to our previous paper [Bouche et al.,SIAM J. Numer. Anal.43 (2005) 578–603], a sufficient condition in order to complete the mathematical analysis of the finite volume scheme consists in obtaining an estimation of order p, less or equal to one, of a quantity that depends only on the mesh and on the advection velocity and that we called geometric corrector. In [Bouche et al., Hermes Science publishing,London, UK (2005) 225–236], we prove that, on the mesh given by Peterson [SIAM J. Numer. Anal.28 (1991) 133–140] and for a subtle alignment of the direction of transport parallel to the vertical boundary, the infinite norm of the geometric corrector only behaves like h 1/2 where h is a characteristic size of the mesh. This paper focuses on the case of an oblique incidence i.e. a transport direction that is not parallel to the boundary, still with the Peterson mesh. Using various mathematical technics, we explicitly compute an upper bound of the geometric corrector and we provide a probabilistic interpretation in terms of Markov processes. This bound is proved to behave like h, so that the order of convergence is one. Then the reduction of the order of convergence occurs only if the direction of advection is aligned with the boundary.

The construction of a well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition is proposed. A suitable parametrix is obtained by using a new unknown and an approximation of the transparency condition. We prove the well-posedness of the equation for any wavenumber. Finally, some numerical comparisons with well-tried method prove the efficiency of the new formulation.

We introduce in this paper some elements for the mathematical and numerical analysis of algebraic turbulence models foroceanic surface mixing layers. In these models the turbulent diffusions are parameterized by means of the gradient Richardson number, that measures the balance between stabilizing buoyancy forces and destabilizing shearing forces. We analyze the existence and linear exponential asymptotic stability of continuous and discrete equilibria states. We also analyze the well-posedness of a simplified model, by application of the linearization principle for non-linear parabolic equations. We finally present some numerical tests for realistic flows in tropical seas that reproduce the formation of mixing layers in time scales of the order of days, in agreement with the physics of the problem. We conclude that the typical mixing layers are transient effects due to the variability of equatorial winds. Also, that these states evolve to steady states in time scales of the order of years, under negative surface energy flux conditions.

The dynamics of dendritic growth of a crystal in an undercooled melt isdetermined by macroscopic diffusion-convection of heat and by capillary forcesacting on the nanometer scale of the solid-liquid interface width.Its modelling is useful for instance in processing techniques based on casting.The phase-field method is widely used to study evolution of such microstructural phase transformations ona continuum level; it couples the energy equation to a phenomenological Allen-Cahn/Ginzburg-Landauequation modelling the dynamics of an order parameter determining the solid and liquid phases,including also stochastic fluctuations to obtain the qualitatively correctresult of dendritic side branching.This work presents a method to determine stochastic phase-field models from atomisticformulations by coarse-graining molecular dynamics. It has three steps: (1) a precisequantitative atomistic definition of the phase-field variable, based on the localpotential energy;(2) derivation of its coarse-graineddynamics model, from microscopic Smoluchowski molecular dynamics (that is Brownian or over damped Langevin dynamics);and (3) numerical computation of the coarse-grained model functions. The coarse-grained model approximates Gibbs ensemble averages of the atomistic phase-field, bychoosing coarse-grained drift and diffusion functions that minimize the approximation error of observables in thisensemble average.

This paper deals with two-species convolution diffusion-competition models of Lotka–Volterra type with delays which describe more accurate information than the Laplacian diffusion-competition models. We first investigate the existence of travelling wave solutions of a class of nonlocal convolution diffusion systems with weak quasimonotonicity or weak exponential quasimonotonicity by a cross-iteration technique and Schauder’s fixed point theorem. When the results are applied to the convolution diffusion-competition models with delays, we establish the existence of travelling wave solutions as well as asymptotic behaviour.

It is shown that a simple deduction engine can be developed for a propositional logic that follows the normal rules of classical logic in symbolic form, but the description of what is known about a proposition uses two numeric state variables that conveniently describe unknown and inconsistent, as well as true and false. Partly true and partly false can be included in deductions. The multi-valued logic is easily understood as the state variables relate directly to true and false. The deduction engine provides a convenient standard method for handling multiple or complicated logical relations. It is particularly convenient when the deduction can start with different propositions being given initial values of true or false. It extends Horn clause based deduction for propositional logic to arbitrary clauses. The logic system used has potential applications in many areas. A comparison with propositional logic makes the paper self-contained.

This paper deals with the problem of discrete and distributed time-delay exponential stability for deterministic and uncertain stochastic high-order neural networks. The concept of a parameter weak coupling linear matrix inequality set (PWCLMIS) is developed. New results are derived in terms of PWCLMIS. Large mixed time delays can be obtained by using this approach. Furthermore, these results are more general than some previous existence results. Two numerical examples are given to show the merit of the approach.

A general analytic approach is proposed for nonlinear eigenvalue problems governed by nonlinear differential equations with variable coefficients. This approach is based on the homotopy analysis method for strongly nonlinear problems. As an example, a beam with arbitrary variable cross section acted on by a compressive axial load is used to show its validity and effectiveness. This approach provides us with great freedom to transfer the original nonlinear buckling equation with variable coefficients into an infinite number of linear differential equations with constant coefficients that are much easier to solve. More importantly, it provides us with a convenient way to guarantee the convergence of solution series. As an example, the beam displacement and the critical buckling load can be obtained for arbitrary variable cross sections. The influence of nonuniformity of moment of inertia is investigated in detail and the optimal distributions of moment of inertia are studied. It is found that the critical buckling load of a beam with the optimal distribution of moment of inertia can be approximately 21–22% larger than that of a uniform beam with the same average moment of inertia. Mathematically, this approach is rather general and thus can be used to solve many other linear/nonlinear differential equations with variable coefficients.

We consider families of general two-point quadrature formulae, using the extension of Montgomery’s identity via Taylor’s formula. The formulae obtained are used to present a number of inequalities for functions whose derivatives are from Lp spaces and Bullen-type inequalities.

The Sherman–Morrison formula is one scheme for computing the approximate inverse preconditioner of a large linear system of equations. However, parallelizing a preconditioning approach is not straightforward as it is necessary to include a sequential process in the matrix factorization. In this paper, we propose a formula that improves the performance of the Sherman–Morrison preconditioner by partially parallelizing the matrix factorization. This study shows that our parallel technique implemented on a PC cluster system of eight processing elements significantly reduces the computational time for the matrix factorization compared with the time taken by a single processor. Our study has also verified that the Sherman–Morrison preconditioner performs better than ILU or MR preconditioners.

In this paper, we consider an insurance company whose surplus (reserve) is modeled by a jump diffusion risk process. The insurance company can invest part of its surplus in n risky assets and purchase proportional reinsurance for claims. Our main goal is to find an optimal investment and proportional reinsurance policy which minimizes the ruin probability. We apply stochastic control theory to solve this problem. We obtain the closed form expression for the minimal ruin probability, optimal investment and proportional reinsurance policy. We find that the minimal ruin probability satisfies the Lundberg equality. We also investigate the effects of the diffusion volatility parameter, the market price of risk and the correlation coefficient on the minimal ruin probability, optimal investment and proportional reinsurance policy through numerical calculations.

In this study, Liénard equations in their general form are treated using the Adomian decomposition method. The special structure of the Liénard equation is exploited to obtain a numerically efficient algorithm suitable for solution by a computer program.