Showing papers in "Applications of Mathematics in 2011"
TL;DR: In this paper, the authors proved the convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman Filter, where the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables.
Abstract: Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and Lp bounds on the ensemble then give Lp convergence.
83 citations
TL;DR: In this paper, the Coulomb friction law and non-penetration condition are assumed to hold on the whole crack surface, and the existence theorem of the solution is proved by using Goursat-Kolosov-Muskhelishvili stress functions.
Abstract: We study a model of interfacial crack between two bonded dissimilar linearized elastic media. The Coulomb friction law and non-penetration condition are assumed to hold on the whole crack surface. We define a weak formulation of the problem in the primal form and get the equivalent primal-dual formulation. Then we state the existence theorem of the solution. Further, by means of Goursat-Kolosov-Muskhelishvili stress functions we derive convergent expansions of the solution near the crack tip.
30 citations
TL;DR: In this article, the authors consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate and give suitable boundary conditions on the traction at the inflow/outflow part of boundary.
Abstract: We consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate. Suitable boundary conditions on the traction at the inflow/outflow part of boundary are given. As an advantage of this, the mean value of the pressure over the domain is no more a free parameter which would have to be prescribed otherwise. We prove the existence and uniqueness of weak solutions (the latter for small data) and discuss particular applications of the results.
26 citations
TL;DR: A new algorithm for solving large scale bound constrained minimization problems is proposed, based on an accurate identification technique of the active set, which is superlinear without requiring the strict complementarity assumption.
Abstract: A new algorithm for solving large scale bound constrained minimization problems is proposed. The algorithm is based on an accurate identification technique of the active set proposed by Facchinei, Fischer and Kanzow in 1998. A further division of the active set yields the global convergence of the new algorithm. In particular, the convergence rate is superlinear without requiring the strict complementarity assumption. Numerical tests demonstrate the efficiency and performance of the present strategy and its comparison with some existing active set strategies.
23 citations
TL;DR: In this paper, a generalization of the Zlamal minimum angle condition to simplicial partitions in any space dimension is discussed. But the generalization is not discussed in this paper.
Abstract: The famous Zlamal’s minimum angle condition has been widely used for construction of a regular family of triangulations (containing nondegenerating triangles) as well as in convergence proofs for the finite element method in 2d. In this paper we present and discuss its generalization to simplicial partitions in any space dimension.
22 citations
TL;DR: In this article, an L 2-estimation of the finite element error for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh is proved.
Abstract: An L 2-estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space.
16 citations
TL;DR: In this article, the authors prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions.
Abstract: Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.
13 citations
TL;DR: In this article, the authors consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain and show the existence of a weak solution for arbitrarily large data for the pressure law.
Abstract: We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain. We show the existence of a weak solution for arbitrarily large data for the pressure law p(ϱ, ϑ) ∼ ϱγ + ϱϑ if γ > 1 and p(ϱ, ϑ) ∼ ϱ lnα(1 + ϱ) + ϱϑ if γ = 1, α > 0, depending on the model for the heat flux.
12 citations
TL;DR: In this article, the authors considered a fractional impulsive boundary value problem on infinite intervals and obtained the existence, uniqueness and computational method of unbounded positive solutions, which they used to solve the problem.
Abstract: In this paper, we consider a fractional impulsive boundary value problem on infinite intervals. We obtain the existence, uniqueness and computational method of unbounded positive solutions.
11 citations
TL;DR: In this article, a class of discrete logistic single-species systems with time-varying pure-delays and feedback control is studied, and sufficient and necessary conditions for the permanence and extinction of species are obtained.
Abstract: A class of nonautonomous discrete logistic single-species systems with time-varying pure-delays and feedback control is studied. By introducing a new research method, almost sufficient and necessary conditions for the permanence and extinction of species are obtained. Particularly, when the system degenerates into a periodic system, sufficient and necessary conditions on the permanence and extinction of species are obtained. Moreover, a very important fact is found in our results, that is, the feedback control and delays are harmless for the permanence and extinction of species for discrete single-species systems. This shows that in a discrete single-species system introducing the feedback control to factitiously control the permanence and extinction of species is useless.
10 citations
TL;DR: In this paper, a mathematical model for flow and transport processes of diluted solutions in domains separated by a leaky semipermeable membrane is proposed, which takes into account the property of the membrane to partly reject the solute, the accumulation of rejected solute at the membrane, and the influence of the solution concentration on the volume flow, known as osmotic effect.
Abstract: In this paper, we propose a mathematical model for flow and transport processes of diluted solutions in domains separated by a leaky semipermeable membrane. We formulate transmission conditions for the flow and the solute concentration across the membrane which take into account the property of the membrane to partly reject the solute, the accumulation of rejected solute at the membrane, and the influence of the solute concentration on the volume flow, known as osmotic effect.
TL;DR: In this article, a mathematical analysis of poroacoustic traveling wave phenomena is presented, assuming that the fluid phase satisfies the perfect gas law and that the drag offered by the porous matrix is described by Darcy's law, exact traveling wave solutions (TWS)s, asymptotic/approximate expressions, are derived and examined.
Abstract: A mathematical analysis of poroacoustic traveling wave phenomena is presented. Assuming that the fluid phase satisfies the perfect gas law and that the drag offered by the porous matrix is described by Darcy’s law, exact traveling wave solutions (TWS)s, as well as asymptotic/approximate expressions, are derived and examined. In particular, stability issues are addressed, shock and acceleration waves are shown to arise, and special/limiting cases are noted. Lastly, connections to other fields are pointed out and possible extensions of this work are briefly discussed.
TL;DR: In this paper, the authors examine worst-case analysis from the standpoint of classical Decision Theory and elucidate how this analysis is expressed in the framework ofWald's famous Maximin paradigm for decision-making under strict uncertainty.
Abstract: We examine worst-case analysis from the standpoint of classical Decision Theory. We elucidate how this analysis is expressed in the framework ofWald’s famous Maximin paradigm for decision-making under strict uncertainty. We illustrate the subtlety required in modeling this paradigm by showing that information-gap’s robustness model is in fact a Maximin model in disguise.
TL;DR: In this article, the effect of the solid phase during liquid/solid phase change by numerical simulation of a benchmark test based on the wellknown and debated experiment of melting of a pure gallium slab counducted by Gau & Viskanta in 1986 is analyzed.
Abstract: We analyse the effect of the mechanical response of the solid phase during liquid/solid phase change by numerical simulation of a benchmark test based on the wellknown and debated experiment of melting of a pure gallium slab counducted by Gau & Viskanta in 1986. The adopted mathematical model includes the description of the melt flow and of the solid phase deformations. Surprisingly the conclusion reached is that, even in this case of pure material, the contribution of the solid phase to the balance of the momentum of the system influences significantly the numerical solution and is necessary in order to get a better match with the experimental observations. Here an up-to-date list of the most meaningful mathematical models and numerical simulations of this test is discussed and the need is shown of an accurate revision of the numerical simulations of melting/solidification processes of pure materials (e.g. artificial crystal growth) produced in the last thirty years and not accounting for the solid phase mechanics.
TL;DR: In this paper, the authors establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some densely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces.
Abstract: In this paper we establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some densely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces. We rely on a fixed point theorem for the sum of completely continuous and contraction operators.
TL;DR: In this paper, the Navier-Stokes system is studied on a family of domains with rough boundaries formed by oscillating riblets and it is shown that the limit velocity field satisfies boundary conditions of a mixed type depending on the characteristic direction of the riblet.
Abstract: The Navier-Stokes system is studied on a family of domains with rough boundaries formed by oscillating riblets. Assuming the complete slip boundary conditions we identify the limit system, in particular, we show that the limit velocity field satisfies boundary conditions of a mixed type depending on the characteristic direction of the riblets.
TL;DR: In this article, the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions was studied and the Leray-Schauder fixed point theorem was proved.
Abstract: We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions
$$\left\{ \begin{gathered} u^{(4)} (t) = f(t,u(t),u'(t),u''(t),u'''(t)),a.e.t \in [0,1], \hfill \\ u(0) = a,u'(0) = b,u(1) = c,u''(1) = d, \hfill \\ \end{gathered} \right. $$
where the nonlinear term f(t, u0, u1, u2, u3) is a strong Caratheodory function. By constructing suitable height functions of the nonlinear term f(t, u0, u1, u2, u3) on bounded sets and applying the Leray-Schauder fixed point theorem, we prove that the equation has a solution provided that the integration of some height function has an appropriate value.
TL;DR: By using the coincidence degree theory, this paper studied a type of p-Laplacian neutral Rayleigh functional differential equation with deviating argument to establish new results on the existence of T-periodic solutions.
Abstract: By using the coincidence degree theory, we study a type of p-Laplacian neutral Rayleigh functional differential equation with deviating argument to establish new results on the existence of T-periodic solutions
TL;DR: In this paper, the uniqueness problem for solutions to the second order ODE of the form x + f(x, t) = 0, subject to appropriate initial conditions, under the sole assumption that f is non-decreasing with respect to x, for each t fixed.
Abstract: We are concerned with the uniqueness problem for solutions to the second order ODE of the form x″+f(x, t) = 0, subject to appropriate initial conditions, under the sole assumption that f is non-decreasing with respect to x, for each t fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable.
TL;DR: In this paper, the authors apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential equation, and prove the existence of the worst scenario through the convergence of a sequence of approximate worst scenarios.
Abstract: We apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential equation In contrast to the one-dimensional problem investigated by P Harasim in Appl Math 53 (2008), No 6, 583–598, the two-dimensional problem requires stronger assumptions restricting the admissible set to ensure the monotonicity of the nonlinear operator in the examined state problem, and, as a result, to show the existence and uniqueness of the state solution The existence of the worst scenario is proved through the convergence of a sequence of approximate worst scenarios Furthermore, it is shown that the Galerkin approximation of the state solution can be calculated by means of the Kachanov method as the limit of a sequence of solutions to linearized problems
TL;DR: In this paper, a new analytical method for solving an initial value problem for a time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered.
Abstract: The time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered. A new analytical method for solving an initial value problem for this system is the main object of the paper. This method consists in the following: the initial value problem is written in terms of Fourier images with respect to lateral space variables, then the resulting problem is reduced to an operator integral equation. After that the operator integral equation is solved by the method of successive approximations. Finally, a solution of the original initial value problem is found by the inverse Fourier transform.
TL;DR: In this paper, the authors give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation and the convergence of the solution of the evolution equation to its stationary state.
Abstract: We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation
$$u_{tt} + 2u_t - a_{ij} (u_t ,
abla u)\partial _i \partial _j u = f$$
corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation
$$ - a_{ij} (0,
abla v)\partial _i \partial _j v = h$$
. We then give conditions for the convergence, as t → ∞, of the solution of the evolution equation to its stationary state.
TL;DR: In this article, a polynomial interpolation method is proposed for finite element derivative recovery by using the polynomially interpolated interpolation technique, and the recovered derivatives possess superconvergence on the recovery domain and ultraconvergence at the interior mesh points for finite elements approximations to elliptic boundary problems.
Abstract: A new finite element derivative recovery technique is proposed by using the polynomial interpolation method. We show that the recovered derivatives possess superconvergence on the recovery domain and ultraconvergence at the interior mesh points for finite element approximations to elliptic boundary problems. Compared with the well-known Z-Z patch recovery technique, the advantage of our method is that it gives an explicit recovery formula and possesses the ultraconvergence for the odd-order finite elements. Finally, some numerical examples are presented to illustrate the theoretical analysis.
TL;DR: In this article, the authors discuss the globalization of modified Levenberg-Marquardt methods for nonsmooth equations and their applications to nonlinear complementarity problems and show that the present methods are promising.
Abstract: In this paper, we discuss the globalization of some kind of modified Levenberg-Marquardt methods for nonsmooth equations and their applications to nonlinear complementarity problems. In these modified Levenberg-Marquardt methods, only an approximate solution of a linear system at each iteration is required. Under some mild assumptions, the global convergence is shown. Finally, numerical results show that the present methods are promising.
TL;DR: In this article, the authors study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate, and establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain.
Abstract: We study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate. For large class of models we establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain. The proof relies on the reconstruction of the globally integrable pressure, available due to considered Navier’s slip boundary conditions, and on the so-called L
∞-truncation method, used to obtain the strong convergence of the velocity gradient. The important point of the approach consists in the choice of an appropriate form of the balance of energy.
Journal Article•
TL;DR: In this article, the authors established existence and uniqueness theorems for solutions of second order nonlinear differential equations on a finite interval subject to linear impulse conditions and periodic boundary conditions.
Abstract: In this study, we establish existence and uniqueness theorems for solutions of second order nonlinear differential equations on a finite interval subject to linear impulse conditions and periodic boundary conditions. The results obtained yield periodic solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.
TL;DR: In this paper, the authors deal with the dimensional reduction from 2D to 1D in magnetoelastic interactions and identify the limit problem by using the so-called energy method.
Abstract: The paper deals with the dimensional reduction from 2D to 1D in magnetoelastic interactions. We adopt a simplified, but nontrivial model described by the Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We identify the limit problem by using the so-called energy method.
TL;DR: Based on the coincidence degree theory of Mawhin, this article obtained a new general existence result for the following higher-order multi-point boundary value problem at resonance, where two of the boundary value conditions are responsible for resonance.
Abstract: Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance $\begin{gathered}
x^{(n)} (t) = f(t,x(t),x'(t),...,x^{(n - 1)} (t)),t \in (0,1), \hfill \\
x(0) = \sum\limits_{i = 1}^m {a_i x(\xi _i ),x'(0) = ... = x^{(n - 2)} (0) = 0,x^{(n - 1)} (1) = } \sum\limits_{j = 1}^l {\beta _j x^{(n - 1)} (\eta _j )} , \hfill \\
\end{gathered}
$
where f: [0, 1] × ℝ
n
→ ℝ is a Caratheodory function, 0 < ξ
1 < ξ
2 < … < ξ
m
< 1, α
i
∈ ℝ, i = 1, 2, …, m, m ≥ 2 and 0 < η
1 < … < η
l
< 1, β
j
∈ ℝ, j = 1, …, l, l ≥ 1. In this paper, two of the boundary value conditions are responsible for resonance.
TL;DR: In this article, a non-interior continuation method for the solution of nonlinear complementarity problem with P0-function (P0-NCP) is proposed, which is based on a smoothing symmetric perturbed minimum function (SSPM-function).
Abstract: In this paper, we consider a new non-interior continuation method for the solution of nonlinear complementarity problem with P0-function (P0-NCP). The proposed algorithm is based on a smoothing symmetric perturbed minimum function (SSPM-function), and one only needs to solve one system of linear equations and to perform only one Armijo-type line search at each iteration. The method is proved to possess global and local convergence under weaker conditions. Preliminary numerical results indicate that the algorithm is effective.