Stability estimates for a class of semi-linear ill-posed problems

We study a final value problem for a nonlinear parabolic equation with positive self-adjoint unbounded operator coefficients. The problem is ill-posed. The regularized equation is given by a modified quasi-reversibility method. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained.     

Preservation of stability under discretization of systems of ordinary differential equations

We study the stability preservation problem while passing from ordinary differential to difference equations. Using the method of Lyapunov functions, we determine the conditions under which the asymptotic stability of the zero solutions to systems of differential equations implies that the zero solutions to the corresponding difference systems are asymptotically stable as well. We prove a theorem on the stability of perturbed systems, estimate the duration of transition processes for some class of systems of nonlinear difference equations, and study the conditions of the stability of complex systems in nonlinear approximation.     

Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation

In this article, we present an inverse problem for the nonlinear 1D Kuramoto–Sivashinsky (KS) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability for this inverse problem is our main result and it relies on the Bukhge?m–Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.     

A Mixed Problem for a Hyperbolic System on the Plane with Delay in the Boundary Conditions

We examine well-posedness of the boundary value problem in a half-strip for a first-order linear hyperbolic system with delay (lumped and distributed) in the boundary conditions. In the case of the negative real parts of the eigenvalues of the corresponding spectral problem we prove a time uniform estimate for a solution to the homogeneous problem which enables us to justify the linearization principle for analysis of stability of stationary solutions to the nonlinear problem.     

Carleman estimate for parabolic equation with memory effects and exact controllability

A nonlinear parabolic problem with memory effect is considered. We first establish a Carleman type estimate for a linear problem which is the adjoint of a suitable linearization of the nonlinear problem. As a consequence of this estimate we obtain an observability estimate. Then we establish the exact controllability of the linearized system with distributed control over a subdomain. Finally, we arrive at the controllability of the nonlinear system via a classical fixed point theorem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simultaneous Identification of Two Time Independent Coefficients in a Nonlinear Phase Field System

In this article, we study a simultaneous reconstruction of two time independent parameters in a nonlinear phase field system by final overdetermination data. To this end, the given problem is transformed into an optimization problem by using the optimal control framework; then the existence of the minimizer for the control functional is established. Further, we deduce the necessary condition for the minimizer of the control functional. Finally we derive the stability estimate for two coefficients with the upper bound given by some Sobolev norms of the final measurement of the solutions.     

Analysis of a decoupled time‐stepping algorithm for reduced MHD system modeling magneto‐convection

In this article, we analyze the stability and error estimate of a decoupled algorithm for a magneto‐convection problem. Magneto‐convection is assumed to be modeled by a coupled system of reduced magneto‐hydrodynamic (RMHD) equations and convection‐diffusion equation. The proposed algorithm applies the second‐order backward difference formula in time and finite element in space. To obtain a noniterative decouple algorithm from the fully discrete nonlinear system, we use a second‐order extrapolation in time to the nonlinear terms such that their skew symmetry properties are preserved. We prove the stability of the algorithm and derive error estimates without assuming any stability conditions. The algorithm is unconditionally stable and requires the solution of one RMHD problem and one convection‐diffusion equation per time step. Numerical test is presented that illustrates the accuracy and efficiency of the algorithm.     

Logarithmic convergence rates for the identification of a nonlinear Robin coefficient

We consider the identification of a nonlinear corrosion profile from single voltage boundary data and show injectivity of the parameter-to-output map. We demonstrate that Tikhonov regularization can be applied in order to solve the inverse problem in a stable manner despite the presence of noisy data. In combination with a logarithmic stability estimate for the underlying Cauchy problem, rates for the convergence of the regularized solutions are proven using a source condition that does not involve the Fréchet derivative of the parameter-to-output map. We present sufficient conditions for the existence of a source function and illustrate our approach by means of numerical examples.     

Error estimation in nonlinear optimization

Methods are developed and analyzed for estimating the distance to a local minimizer of a nonlinear programming problem. One estimate, based on the solution of a constrained convex quadratic program, can be used when strict complementary slackness and the second-order sufficient optimality conditions hold. A second estimate, based on the solution of an unconstrained nonconvex, nonsmooth optimization problem, is valid even when strict complementary slackness is violated. Both estimates are valid in a neighborhood of a local minimizer. An active set algorithm is developed for computing a stationary point of the nonsmooth error estimator. Each iteration of the algorithm requires the solution of a symmetric, positive semidefinite linear system, followed by a line search. Convergence is achieved in a finite number of iterations. The error bounds are based on stability properties for nonlinear programs. The theory is illustrated by some numerical examples.     

完全非线性伪抛物组的非均匀网格差分格式

完全非线性伪抛物组的非均匀网格差分格式韩永前，袁光伟，周毓麟（北京应用物理与计算数学研究所）ＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＳＷＩＴＨＮＯＮＵＮＩＦＯＲＭＭＥＳＨＥＳＦＯＲＦＵＬＬＹＮＯＮＬＩＮＥＡＲＰＳＥＵＤＯ－ＰＡＲＡＢＯＬＩＣＳＹＳＴＥＭＳ￥Ｈ...     

Numerical approximation of Quasi‐Newtonian flows by ALE‐FEM

The aim of this work is to investigate the numerical approximation of a nonlinear, time‐dependent quasi‐Newtonian flow problem formulated in the framework of Arbitrary Lagrangian Eulerian method. We present some stability results and convergence analysis of finite element solutions for semidiscrete and fully discretized problems, respectively. Numerical results supporting the derived error estimate are also presented. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Direct problem for a nonlinear evolutionary system

In this article, we consider the first initial boundary-value problem for an evolutionary system describing nonlinear interactions of electromagnetic and elastic waves. The system under study consists of three coupled differential equations, one of them is a hyperbolic equation (an analogue of the Lamé equations) and the other two equations form a parabolic system (an analogue of the diffusion Maxwell system). Existence and uniqueness results are established. We also prove the stability estimate of a weak solution.     

一类带非线性边界条件的拟线性抛物方程的第二初边值问题

利用先验估计的方法讨论具非线性第二边界条件的快扩散方程解的存在性、唯一性、稳定性和渐近性.主要结果是:1)存在唯的整体广义解,解连续依赖于初值;2)存在T₀,t<T₀时解是无穷次可微的正则解;3)当t充分大时,解一致收敛到零.     

Upper Estimate of the Interval of Existence of Solutions of a Nonlinear Timoshenko Equation

Solutions to the initial-boundary value problem for a nonlinear Timoshenko equation are considered. Conditions on the initial data and nonlinear term are given so that solutions to the problem under consideration do not exist for all t > 0. An upper estimate of the t-interval of the existence of solutions is obtained. An estimate of the growth rate of the solutions is given.     

Linear and nonlinear response of pipeline suspension bridge due to moving fluid

Basing on the nonlinear dynamic model of flexible pipeline suspended by spatial system of cables, described in Ref. [1], the linear and nonlinear vibrations are investigated in order to estimate the nonlinear effects. The model is based on substructure technique and formulated including features specific to analyzed structure, for example large displacements and time dependent parameters appearing in equations of motion due to fluid flowing inside the pipeline. Due to the fact that modelling problem for the analyzed structure is one's own complicated, a simple case when the conveying fluid is idealized simply as a ballast moving inside the pipe is considered. This paper presents a short numerical analysis of linear and nonlinear, static and dynamic response of exemplary structure for three different cases: during filling the pipe with fluid, when the pipeline is completely filled and during emptying the pipe. Moreover, for the linear problem, the influence of a speed of the fluid on the stability of the pipeline suspension bridge is investigated. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation of rated influence in parabolic systems.L 2-approach

By using observations of solutions of the first initial boundary-value problem for a parabolic quasilinear equation with fast random oscillations, we estimate the nonlinear term of the equation. In the metric of the space L₂, we study large deviations of a nonparametric estimate of nonlinear influence.     

Minimization and error estimates for a class of the nonlinear Schrödinger eigenvalue problems

It is shown that the nonlinear eigenvalue problem can be transformed into a constrained functional problem. The corresponding minimal function is a weak solution of this nonlinear problem. In this paper, one type of the energy functional for a class of the nonlinear Schrödinger eigenvalue problems is proposed, the existence of the minimizing solution is proved and the error estimate is given out.     

Contrast structures in multidimensional singularly perturbed reaction-diffusion-advection problems

We consider a nonlinear boundary value problem for an equation of reaction-diffusion-advection type whose solutions have internal transition layers (contrast structures). We prove the existence of such solutions, construct their asymptotics, and estimate its accuracy. The analysis is carried out with the use of the asymptotic method of differential inequalities and permits one to prove the Lyapunov stability of the contrast structures.     

A Posteriori Error Estimates for Axisymmetric and Nonlinear Problems

We propose and examine the primal and dual finite element method for solving an axially symmetric elliptic problem with mixed boundary conditions. We derive an a posteriori error estimate and generalize the method used for a nonlinear elliptic problem. Finally, an a posteriori error estimate for a nonlinear parabolic problem based on the concept of hierarchical finite element basis functions is introduced.     

Existence and global exponential stability of pseudo almost periodic solution for SICNNs with mixed delays

Shunting Inhibitory Artificial Neural Networks are biologically inspired networks in which the synaptic interactions are mediated via a nonlinear mechanism called shunting inhibition, which allows neurons to operate as adaptive nonlinear filters. This paper considers the problem of existence and exponential stability of the pseudo almost periodic solution for shunting inhibitory cellular neural networks with mixed delays. The Banach fixed point theorem and the variant of a certain integral inequality with explicit estimate are used to establish the results. The results of this paper are new and they complement previously known results.