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In this work we study a dynamic contact problem between a thermoelastic mixture and a deformable obstacle. The classical normal compliance condition is used for modeling the contact. The variational formulation of this problem is written as a nonlinear coupled system of three parabolic variational equations. An existence and uniqueness result is proved using the Faedo-Galerkin method. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the decay of the discrete energy and the linear convergence of the algorithm are deduced. Finally, some numerical simulations are presented to show the convergence and the behavior of the solution.

Using matrix norm techniques, we give a unified convergence analysis of a general projection method for computing stationary density functions of Frobenius-Perron operators associated with piecewise C 2 and stretching transformations of the unit interval.

The weak and strong ergodic properties of nonautonomous linear ordinary differential equations are considered. It is shown that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to the Perron vectors of the strongly positive transition matrix at infinity (weak ergodicity). If, in addition, the coefficient matrix function is uniformly continuous, then the convergence of the normalized positive solutions to the same strongly positive limiting vector (strong ergodicity) is equivalent to the convergence of the Perron vectors of the coefficient matrices.

We study for the first time the ill-posed backward problem for a contaminated nonlinear predator-prey system whose velocities of migration depend on the total average populations in the considered space domain. We propose a new regularized problem for which we are able to prove its unique solvability in Theorem 1 . Moreover, under some mild assumptions on the true solution, we give useful and rigorous error estimates and convergence rates in both the L 2 - and H 1 -norms in Theorem 2 , Theorem 3 , respectively. Furthermore, numerical simulations are performed to illustrate the accuracy and stability of the regularized solution.

In this study, we consider nonlinear partial differential equations (PDEs) and stochastic PDEs (SPDEs) with Neumann boundary conditions on domains that satisfy the Lions–Sznitmann–Saisho conditions. Using the convergence result based on the penalization approximation for stochastic differential equations with normal reflections on nonsmooth and nonconvex domains, we establish the existence and comparison principle for the viscosity solutions of nonlinear PDEs with the Neumann conditions associated with the optimal control problem. We also obtain for nonlinear SPDEs and backward SPDEs with Neumann condition the representations of the stochastic viscosity solutions.

In this paper, we establish a new estimate (including lower and upper bounds) for an important quantity involved in the convergence analysis of smoothed aggregation algebraic multigrid methods. The new upper bound improves the existing ones. And our upper bound is optimal.

In this study, we mainly consider the order reduction for the solution coefficient vectors of the classical continuous space-time finite element (CCSTFE) method for the two-dimensional (2D) unsteady Stokes equation. First, we construct the CCSTFE model for the 2D unsteady Stokes equation about vorticity–stream functions, and show the existence, stability, and convergence of the CCSTFE solutions. We then apply proper orthogonal decomposition (POD) to develop a reduced-order extrapolation continuous space-time finite element (ROECSTFE) model, as well as discussing the existence, stability, and convergence of the ROECSTFE solutions based on matrix analysis so the theoretical analysis is highly concise, and we use a flowchart to search for the ROECSTFE solutions. Finally, we provide two numerical examples to confirm the correctness of the theoretical results obtained. We also show that the computational results agree well with the theoretical results, thereby demonstrating the effectiveness and feasibility of the proposed ROECSTFE method. Therefore, both the theory and method are totally different from the existing POD reduced order methods.

In this paper, we study the full compressible magnetohydrodynamic (MHD) flows with the viscous stress tensor given by the Maxwell law in R 3 . First, local existence of solutions for general large initial data and global existence of solutions for small initial data are shown. Then we obtain the convergence of solutions for the system with the Maxwell law to the corresponding solution for the classical system with the Newtonian law when the relaxation time goes to 0.

A continuum steady-state model of oxygen transport in the brain is studied. The unique solvability of the corresponding nonlinear boundary-value problem is proven. The convergence of a simple iterative procedure for finding solutions of the boundary-value problem is established. Numerical experiments showing the consistency of the model are conducted.

A recent article of the first author presented a constructive proof for the Weierstrass approximation theorem for a wide class of weighted spaces of continuous functions defined on [ 0 , ∞ ) , using Chlodovsky's generalization of the Bernstein polynomial operators. Here, we investigate necessary and sufficient conditions related to the uniform boundedness of a sequence of such operators and to their convergence properties when the weight is a classical Freud weight of the form w ( x ) = e − x α , α ≥ 1 .

We prove that even irregular convergence of semigroups of operators implies similar convergence of mild solutions of the related semi-linear equations with Lipschitz continuous nonlinearity. This result is then applied to three models originating from mathematical biology: shadow systems, diffusions on thin layers, and dynamics of neurotransmitters, 21 pages, no figures. Minor revision: Some typos fixed

This study presents an efficient, accurate, effective and unconditionally stable time stepping scheme for the Darcy-Brinkman equations in double-diffusive convection. The stabilization within the proposed method uses the idea of stabilizing the curvature for velocity, temperature and concentration equations. Accuracy in time is proven and the convergence results for the fully discrete solutions of problem variables are given. Several numerical examples including a convergence study are provided that support the derived theoretical results and demonstrate the efficiency and the accuracy of the method. (C) 2018 Elsevier Inc. All rights reserved.

In this paper, we mainly utilize proper orthogonal decomposition (POD) to reduce the order for the coefficient vector of the classical Crank–Nicolson finite spectral element (CCNFSE) method of the two-dimensional (2D) non-stationary Boussinesq equations about vorticity-stream functions so that the reduced-order method maintains all the advantages of the CCNFSE method. Toward this end, we first establish a CCNFSE format with the second-order time accuracy for the two-dimensional (2D) non-stationary Boussinesq equations about vorticity-stream functions and analyze the existence, stability, and convergence of the CCNFSE solutions. And then, by POD, we establish a reduced-order extrapolated Crank–Nicolson finite spectral element (ROECNFSE) method and analyze the existence, stability, and convergence of the ROECNFSE solutions as well as offer the flowchart for seeking ROECNFSE solutions. Finally, we use two sets of numerical experiments to validate that the numerical computational consequences are accorded with the theoretical ones such that the effectiveness and feasibility of the ROECNFSE method are further verified. Both theory and method in this paper is new and completely different from the existing reduced-order methods.

This work is concerned with various weighted four dimensional matrix summability methods in modular function spaces associating with generalized difference operator involving ( p , q ) -gamma function. Following very recent results of Kadak (2017) [20] , we first introduce a new type of difference operator on double function sequences and, also define modularly weighted A-statistical convergence and modularly statistically N ˜ A -summability by means of weighted four dimensional regular matrix A. We also present some important inclusion relations between newly proposed summability methods. Moreover, based on the concept of modularly statistically N ˜ A -summability, we prove a Korovkin type approximation theorem for functions of two variables in modular spaces. Finally, in order to show that our proposed method and its applications to approximation results are stronger than the existing methods, we display an illustrative example using bivariate ( p , q ) -Bernstein Kantorovich operators.

In this paper, we study on the local and semi-local convergence and dynamic of an iterative optimal two-step Steffensen's method. We study local convergence by using Lipschitz conditions and continuity of differentiable function F. We also obtain semi-local convergence by using majorizing sequences. By this approach, we can obtain the convergence of the method by minimum of auxiliary sequences and few initial conditions. Moreover, we analyze the stability of the method via Julia sets and parameter planes that is a novelty for Steffensen-type methods. We extend this method for obtaining the sign function of a matrix. Numerical examples are given to confirm our theoretical results.

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise., Comment: 13 pages

We study the local convergence to equilibria, as time goes to infinity, of trajectories of semilinear wave systems with friction damping on the boundary and subject to nonlinear potential energy. Estimates for the speed of convergence are obtained in terms of the behavior of the nonlinear feedback close to the origin. As an example of application, we show that the trajectories of a sine-Gordon system with nonlinear boundary damping, approach equilibria at least polynomially.

The sufficient conditions of the almost sure exponential stability of the exact solution for the stochastic pantograph differential equation are considered, with a Khasminskii-type condition. The almost sure exponential stability of the numerical solutions by the Euler–Maruyama method and the backward Euler–Maruyama method is also discussed, based on the discrete semimartingale convergence theorem. We present the sufficient conditions for the stability of the Euler–Maruyama method, with one extra condition when compared with the exact solution. We show that the backward Euler–Maruyama method can share almost the same conditions for the almost sure exponential stability as the exact solution.

In this paper we study two optimal design problems associated to fractional Sobolev spaces Ws,p(Ω). Then we find a relationship between these two problems and finally we investigate the convergence when s↑1. Fil: Fernandez Bonder, Julian. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Spedaletti, Juan Francisco. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi". Universidad Nacional de San Luis. Facultad de Ciencias Físico, Matemáticas y Naturales. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi"; Argentina

We study strong (pathwise) approximation of Cox–Ingersoll–Ross processes. We propose a Milstein-type scheme that is suitably truncated close to zero, where the diffusion coefficient fails to be locally Lipschitz continuous. For this scheme we prove positive convergence rates for the full parameter range including the accessible boundary regime. The error criterion is given by the maximal L p -distance of the solution and its approximation on a compact interval. In the particular case of a squared Bessel process of dimension δ > 0 the convergence rate is given by min ⁡ ( 1 , δ ) / ( 2 p ) .

In this paper, we study the minimal Keller-Segel model with a logistic source and obtain quantitative and qualitative descriptions of the competition between logistic damping and other ingredient, especially, chemotactic aggregation to guarantee boundedness and convergence. More specifically, we establish how precisely strong a logistic source can prevent blow-up, and then we obtain an explicit relationship between logistic damping and other ingredient, especially, chemotactic aggregation so that convergences are ensured and their respective convergence rates are explicitly calculated out. Known results in the literature are completed and refined. Furthermore, our findings provide clues on how to produce blowup solutions for KS chemotaxis models with logistic sources., Comment: 34 pages, a second revised version; detected typos were amended

Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.

Maximal estimates for Schrodinger means and convergence almost everywhere of sequences of Schrodinger means are studied.

We investigate the asymptotic behavior of solutions to a nonhomogeneous second order difference inclusion of monotone type and prove new results on the existence and approximation of zeroes of the operator. In particular, we introduce new weighted averages of the solutions and prove their convergence to a zero of the operator under different control conditions on the parameters.

In this paper, the state reconstruction of the wave equation with general viscosity and non-collocated observation and control are given from both the theoretical aspect and the numerical aspect. The forward-backward observers-based algorithm is utilized to solve this problem. The formula for calculating the initial value is derived. Moreover, the iterative sequence is also built for any given guess value and it is showed that it strongly converges to initial value. However, because the exponential stabilities of the error systems between the original systems and the forward∖backward observers play important roles in the involved algorithm, the exponential stability of some related system is firstly discussed. Furthermore, combining the theory of port-Hamiltonian system and the finite difference, the semi-discretization scheme of the finite difference with order reduction is given and the uniform exponential stability of the semi-discretization system is verified by the method paralleling to the continuous system. Finally, the convergence analysis of the finite difference scheme is given and the convergence of the solution of the mixed finite element scheme induced by the finite difference scheme with order reduction to the solution of the continuous counterpart is also presented.

We consider the 3D magnetohydrodynamic-alpha model driven by a multiplicative noise. The model arises in the modelling of turbulent flows in magneto fluids. We prove the existence and uniqueness of the probabilistic strong solution. We also propose and analyze a numerical scheme for the approximation of the unique strong solution. We prove the convergence in mean square of the scheme, which is a linear evolution equation with additive noise.

For scalar conservation laws posed on bounded domains in R d , the convergence of quasilinear parabolic viscous approximations to the entropy solution in the sense of Bardos-Leroux-Nedelec is established.

In this paper, we study the Cauchy problem for the hyperbolic p-system with time-gradually-degenerate damping term − 1 ( 1 + t ) λ u for 0 ⩽ λ 1 , and show that the damped p-system has a couple of global solutions uniquely, and such solutions tend time-asymptotically to the shifted nonlinear diffusion waves, which are the solutions of the corresponding nonlinear parabolic equation governed by the Darcy's law. We further derive the convergence rates when the initial perturbations are in L 2 . The approach adopted is the technical time-weighted energy method.

The classical concept of synchronization is usually related to the locking of the basic frequencies and instantaneous phases of regular oscillations, and this question is addressed by studying specific kinds of coupled systems. This work presents a different point of view. We do not study the convergence of coupled systems to a synchronized behaviour, but try to answer the following question: in a population of coupled differential systems, when each cell (subsystem) exhibits a periodic behaviour, is the whole trajectory periodic? We define generalized spatial automata, with reference to continuous spatial automata, by means of coupling maps and associated measures on the set of cells: the main idea is the fact that a cell interprets its own environment via the states of the whole population and according to its own state. A natural partition of periods is such that cells belong to the same class if their trajectories share a common period. We demonstrate that in a general case where cells belong to an a priori unstructured set, and their trajectories evolve in possibly distinct Banach spaces, the set of classes of periods is generally countable. In particular, when the set of cells is endowed with a Borelian structure, all the cells necessarily share a common period.

We study a nonlocal biharmonic MEMS equation with the fringing field. It arises in the Micro-Electro Mechanical System (MEMS) devices. First, we establish the local solution and extend it globally in time. Next, we discuss the dynamical properties of omega limit set. Especially, we show that the omega limit set is included in the set of the stationary solution. Finally, we consider the convergence rate to the stationary solution.

We consider a Lotka-Volterra system with both local and nonlocal intraspecific and interspecific competitions, where nonlocal competitions depend on both spatial and temporal effects in a general form. Firstly, global stability of two constant semi-trivial equilibria and global convergence of the coexistence equilibrium are derived by using the functional and energy method, which implies that strengths of nonlocal intraspecific competitions have great effects on these global dynamics but the nonlocal interspecific competitions not and extends global results of Gourley and Ruan (2003) [11] . Secondly, global attracting region of each constant semi-trivial equilibrium is limited by its environment capacity regardless of the distinction of local and nonlocal intraspecific competitions. Thirdly, in the weak competition case, the coexistence equilibrium becomes Turing unstable when the kernels are chosen as generally distributed delay functions in temporal and the nonlocal intraspecific competitions are suitably strong. Additionally, spatially homogeneous and inhomogeneous periodic solutions are found numerically.

This paper address the existence and uniqueness, in suitable functional spaces, of the solution of a non linear system of equations originating from the implicit time discretization of the coupled unsteady Navier-Stokes and Allen-Cahn equations. This result is based on the convergence analysis of an original stabilized fixed point algorithm, from which we also get a maximum principle. Numerical experiments are performed to illustrate the influence of the implicit formulation in comparison to a simpler approach based on a semi-explicit time scheme.

The viscous and inviscid aggregation equation with Newtonian potential models a number of different physical systems and has close analogs in 2D incompressible fluid mechanics. We consider a slight generalization of these equations in the whole space establishing well-posedness of the viscous and inviscid equations, spatial decay of the viscous solutions, and the convergence of viscous solutions to the inviscid solution as the viscosity goes to zero.

There are generalizations and complementations of results from our joint papers with Filipow ( [15] and [16] ). We study relationships between ideal equal convergence and various kinds of ideal convergences of sequences of real functions. Furthermore we consider ideal version of the bounding number on sets from coideals.

We prove maximum and comparison principles for the discrete fractional derivatives in the integers. Regularity results when the space is a mesh of length h, and approximation theorems to the continuous fractional derivatives are shown. When the functions are good enough (Holder continuous), these approximation procedures give a measure of the order of approximation. These results also allow us to prove the coincidence, for Holder continuous functions, of the Marchaud and Grunwald–Letnikov derivatives in every point and the speed of convergence to the Grunwald–Letnikov derivative. The discrete fractional derivative will be also described as a Neumann–Dirichlet operator defined by a semi-discrete extension problem. Some operators related to the Harmonic Analysis associated to the discrete derivative will be also considered, in particular their behavior in the Lebesgue spaces l p ( Z ) .

In this paper, we establish approximate representations of solutions to stochastic Volterra integral equations (SVIEs, for short), by virtue of solutions to a family of stochastic differential equations. As an application, we present two algorithms for solving SVIEs: stochastic theta method and splitting method, and prove the global 1/2 order convergence rates in L 2 norm.

In the present paper we introduce a new method to compute approximate analytical solutions for nonlinear differential equations, called the Least Squares Homotopy Perturbation Method. The method is based on the well-known Homotopy Perturbation Method and its main feature is an accelerated convergence compared to the regular Homotopy Perturbation Method. The comparison with previous results emphasizes the high accuracy of the method.

This paper is concerned with the strong approximation of a semi-linear stochastic wave equation with strong damping, driven by additive noise. Based on a spatial discretization performed by a spectral Galerkin method, we introduce a kind of accelerated exponential time integrator involving linear functionals of the noise. Under appropriate assumptions, we provide error bounds for the proposed full-discrete scheme. It is shown that the scheme achieves higher strong order in time direction than the order of temporal regularity of the underlying problem, which allows for higher convergence rate than usual time-stepping schemes. For the space-time white noise case in two or three spatial dimensions, the scheme still exhibits a good convergence performance. Another striking finding is that, even for the velocity with low regularity the scheme always promises first order strong convergence in time. Numerical examples are finally reported to confirm our theoretical findings., Comment: We are now preparing a paper on the weak approximation of such problem

In this article, we are concerned with averaging principle for stochastic hyperbolic–parabolic equations driven by Poisson random measures with slow and fast time-scales. We first establish the existence and uniqueness of weak solutions of the stochastic hyperbolic–parabolic equations. Then, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic wave equation is an average with respect to the stationary measure of the fast varying process. Finally, we derive the rate of strong convergence for the slow component towards the solution of the averaged equation.

Property UC, introduced in 2009, plays a crucial role in several existence and convergence results for best proximity points and fixed points. In this paper, we present two approximation theoretic characterizations of uniform convexity and as consequences of these results, we characterize the uniform convexity in terms of property UC. Characterizations of strong convexity in terms of property UC are also presented.

This paper deals with an accelerated iterative procedure for a coupled system of partial differential equations arising from a catalytic converter model. Based on the main theorem, a finite difference based numerical method is developed. The monotone property as well as the convergence analysis and the error estimate of the proposed finite difference scheme are proved theoretically. The efficiency of the proposed scheme is illustrated by providing a comparative numerical study with the existing method. The proposed iterative procedure reduces the number of iterations at least by forty percent for certain benchmarks available in the literature.

In this article, we consider the time viscosity-splitting method for the Boussinesq problem. By using this technique, the considered problem is decoupled into two subproblems, and each subproblem is solved more easily than the original one. In the first step, two linear elliptic problems with semi-implicit schemes for the convection terms are solved. The advantage of using such numerical technique is that a linear system with the constant coefficient matrices is obtained and then the computation becomes easy. In the second step, a system consists of a Stokes problem and a linear elliptic problem is solved. The main results of this article include that (1) Establish the stability of numerical solutions in the time viscosity-splitting method; (2) Provide the convergence results of strongly second order for the velocity and temperature and strongly first order for the pressure. Finally, some numerical results are provided to display the performances of the developed numerical algorithms.

The averaging principle is an important method to extract effective macroscopic dynamic from complex systems with slow component and fast component. This paper concerns the L p -strong convergence of the averaging principle for two-time-scales stochastic partial differential equations (SPDEs) driven by Wiener processes and Poisson jumps. To achieve this, a key step is to show the existence for an invariant measure with exponentially ergodic property for the fast equation, where the dissipative conditions are needed. Furthermore, it is shown that under suitable assumptions the slow component L p -strongly converges to the solution of the averaged equation. The rate of the convergence is also obtained as a byproduct.

This paper deals with the well-posedness of families of finite dimensional vector optimization problems ordered by components (Pareto problem). For this problem, two Hadamard well-posedness concepts are introduced. These concepts involve the existence and uniqueness of efficient/weak efficient solutions, and also the continuous behavior of these solutions with respect to perturbations of the data. The perturbations in the last property are formulated through a variational convergence notion of the objective functions and by considering approximate solutions of the perturbed problems. Necessary and sufficient conditions for the well-posedness of Pareto optimization problems are obtained in general, and also under convexity and quasiconvexity assumptions. To do this, asymptotic mathematical tools are employed. Finally, it is proved that the convex Pareto optimization problems are “essentially” well-posed in the sense of category theory.

In this paper, we consider a generalized differential equation with feedback and time-varying delays and prove the existence of a unique pseudo compact almost automorphic solution that is exponentially attractive. We show our result by applying the Banach fixed point theorem and the properties of pseudo compact almost automorphic function. We apply our results and obtain new criteria for the existence and convergence dynamics of pseudo compact almost automorphic solutions for some models that involves sum of nonlinear terms that satisfy a general condition. These models are very general and include sums of unimodals terms with monotones, such as Nicholson, Mackey-Glass, Lasota-Wazeska and others. The results obtained are new and they recover, extend and improve recent works.

In this paper, the sufficient and necessary conditions for complete convergence and the Kolmogorov strong law of large numbers for weighted sums of m-extended negatively dependent random variables are presented. Some applications of the main results are also provided, including the weak and strong consistency of the least squares estimator in multiple linear regression models, strong consistency of conditional Value-at-risk estimator, and the asymptotics of the quasi-renewal counting process. Finally, some numerical simulations are carried out to confirm the theoretical results.

This paper develops a splitting-up method to solution of backward stochastic parabolic partial differential equations. Both splitting equations are simpler for the numerical computations. Two different approximation schemes are given, and their convergence results are proved. The first-order convergence rate is proved in the linear case.

For admissible rearrangements of the Vilenkin–Chrestenson–Paley system, perfect sets of uniqueness of positive measure are constructed. As far as can be seen, these are the first examples of uniqueness sets of positive measure for a complete in L 2 orthogonal system. Also a mixed type (convergence + uniqueness) problem for summable functions and Fourier–Vilenkin–Chrestenson series is considered and several recovering theorems are proved.

A modified alternating direction implicit scheme for the time integration of linear isotropic Maxwell equations with strictly positive conductivity on cuboids is constructed. A key feature of the proposed scheme is its uniform exponential stability, being achieved by coupling the Maxwell system with an additional damped PDE and adding artificial damping to the scheme. The implicit steps in the resulting time integrator further decouple into essentially one-dimensional elliptic problems, requiring only linear complexity. The convergence of the scheme to the solution of the original Maxwell system is analyzed in the abstract time-discrete setting, providing an error bound in a space related to H − 1 .

In this paper we present a Newton-type two-step method to approximate a solution of a nonlinear equation in Banach spaces. We establish a semilocal convergence theorem under Newton-Kantorovich-type conditions, both the convergence order and the efficiency index of the developed method are found. The iterative procedure presented here is a simple modification of the Newton-Kantorovich method, however, with the same number of function and derivative evaluations at each iteration, it is improved in two important aspects: Firstly, the convergence order is increased from 2 for the Newton-Kantorovich method to 1 + 2 ≈ 2.414 for the new method. Secondly, the efficiency index (convergence order per function evaluation) is improved from 2 ≈ 1.414 to ( 1 + 2 ) 1 / 2 ≈ 1.553 . We illustrate some of our results with a numerical example.