On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

On local convergence of a symmetric semi-discrete scheme for an abstract analogue of the Kirchhoff equation

The present work considers a nonlinear abstract hyperbolic equation with a self-adjoint positive definite operator, which represents a generalization of the Kirchhoff string equation. A symmetric three-layer semi-discrete scheme is constructed for an approximate solution of a Cauchy problem for this equation. Value of the gradient in the nonlinear term of the scheme is taken at the middle point. It makes possible to find an approximate solution at each time step by inverting the linear operator. Local convergence of the constructed scheme is proved. Numerical calculations for different model problems are carried out using this scheme.     

Numerical implementation of iteration processes applied in the study of nonlinear boundary value problems of the theory of elasticity and filtration

Nonlinear elastic problems for hardening media are solved by applying the universal iteration process proposed by A.I. Koshelev in his works on the regularity of solutions to quasilinear elliptic and parabolic systems. This requires numerically solving a linear elliptic system at each step of the iteration procedure. The method is numerically implemented in the MATLAB environment by using its PDE Toolbox. A modification of the finite-element procedure is suggested in order to solve a linear algebraic system at each iteration step. The computer model is tested on simple examples. The same nonlinear problems are also solved by the method of elastic solutions, which consists in replacing the Laplace operator in the universal iteration process by the Lamé operator of linear elasticity. As is known, this iteration process converges to a weak solution of the nonlinear problem, provided that the displacements are fixed on the boundary. The method is tested on examples with stresses on the boundary. The third part of the paper is devoted to the nonlinear filtration problem. General properties of the iteration process for nonlinear parabolic systems have been studied by A.I. Koshelev and V.M. Chistyakov. The numerical implementation is based on slightly modified PDE Toolbox procedures. The program is tested on simple examples.     

Two-Stage Method of Construction of Regularizing Algorithms for Nonlinear Ill-Posed Problems

For an equation with a nonlinear differentiable operator acting in a Hilbert space, we study a two-stage method of construction of a regularizing algorithm. First, we use the Lavrentiev regularization scheme. Then we apply to the regularized equation either Newton’s method or nonlinear analogs of α-processes: the minimum error method, the minimum residual method, and the steepest descent method. For these processes, we establish the linear convergence rate and the Fejér property of iterations. Two cases are considered: when the operator of the problem is monotone and when the operator is finite-dimensional and its derivative has nonnegative spectrum. For the two-stage method with a monotone operator, we give an error bound, which has optimal order on the class of sourcewise representable solutions. In the second case, the error of the method is estimated by means of the residual. The proposed methods and their modified analogs are implemented numerically for three-dimensional inverse problems of gravimetry and magnetometry. The results of the numerical experiment are discussed.     

Estimating the Accuracy of a Method of Auxiliary Boundary Conditions in Solving an Inverse Boundary Value Problem for a Nonlinear Equation

An inverse boundary value problem for a nonlinear parabolic equation is considered. Two-sided estimates for the norms of values of a nonlinear operator in terms of those of a corresponding linear operator are obtained.On this basis, two-sided estimates for the modulus of continuity of a nonlinear inverse problem in terms of that of a corresponding linear problem are obtained. A method of auxiliary boundary conditions is used to construct stable approximate solutions to the nonlinear inverse problem. An accurate (to an order) error estimate for the method of auxiliary boundary conditions is obtained on a uniform regularization class.     

一类非线性抛物方程Cauchy问题解的存在性条件

研究了带源项的非线性抛物方程Cauchy问题解存在的必要条件以及解所应具有的性质,通过把文献中的线性算子推广到形式较一般的带源项的非线性抛物算子,利用其中处理线性问题的方法来处理非线性问题．     

A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some appropriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.     

A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation

We study a nonlocal mixed problem for a nonlinear pseudoparabolic equation, which can, for example, model the heat conduction involving a certain thermodynamic temperature and a conductive temperature. We prove the existence, uniqueness and continuous dependence of a strong solution of the posed problem. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplicator. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.     

Numerical analysis of evolution problems in nonlinear small strains elastoviscoplasticity

Summary The present paper deals with the mathematical and numerical analysis of evolution problems in nonlinear small strains viscoelasticity of Burger's type. After a brief review of the mechanical model, the viscoelastic problem to be solved is written as an abstract evolution problem. The associated operator is proved to be maximal monotone, thus implying existence and uniqueness of solutions. This problem is then solved numerically by a backward Euler discretization in time, a finite element approximation in space and by using a preconditioned conjugate gradient algorithm for solving the resulting nonlinear algebraic systems. Numerical results are finally presented to illustrate the solution procedure.     

D-admissible hybrid control of a class of singular systems

This paper mainly studies the problem of designing a hybrid state feedback D-admissible controller for a class of linear and nonlinear singular systems. Based on the relationship between singular discrete systems and singular delta operator systems, several necessary and sufficient conditions for a linear singular delta operator system to be D-admissible (i.e. regular, causal and all finite poles lie in a prescribed circular region) with different representations are derived. Then, the existence conditions and explicit expressions of a desirable D-admissible controller are given by means of matrix inequalities and strict linear matrix inequalities, respectively. We further extend the obtained results to singular delta operator systems with Lipschitz nonlinear perturbations, and the design methods of hybrid controller are presented for the nonlinear case as well. Finally, numerical examples as well as simulations are provided to illustrate the effectiveness of the theoretical outcomes obtained in the paper.     

Variational Primitive of a Differential Form

In this paper, a Dirichlet-to-Neumann operator related to the Cauchy problem for the gradient operator with data on a part of the boundary is defined. To this end, a nonlinear relaxation of this problem, which is a mixed boundary problem of Zaremba type for the p-Laplace equation, is considered.     

Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient

In this paper we consider an initial boundary value problem for a parabolic inclusion whose multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz function, and whose elliptic operator may be a general quasilinear operator of Leray-Lions type. Recently, extremality results have been obtained in case that the governing multivalued term is of special structure such as, multifunctions given by the usual subdifferential of convex functions or subgradients of so-called dc-functions. The main goal of this paper is to prove the existence of extremal solutions within a sector of appropriately defined upper and lower solutions for quasilinear parabolic inclusions with general Clarke's gradient. The main tools used in the proof are abstract results on nonlinear evolution equations, regularization, comparison, truncation, and special test function techniques as well as tools from nonsmooth analysis.     

A note on convection with nonlinear heat flux

The problem of thermal convection is investigated when the heat flux is a nonlinear function of the temperature gradient. A complete analysis of the linear instability problem is given. The nonlinear stability problem is studied in a case which is believed to be physically relevant and the stability threshold is compared directly to that found by linear instability theory.      

On mild solutions of gradient systems in Hilbert spaces

We consider the Cauchy problem for an infinite-dimensional Ornstein-Uhlenbeck equation perturbed by gradient of a potential. We prove some results on existence and uniqueness of mild solutions of the problem. We also provide stochastic representation of mild solutions in terms of linear backward stochastic differential equations determined by the Ornstein-Uhlenbeck operator and the potential.     

An inexact Newton method for nonlinear two-point boundary-value problems

The method of quasilinearization for nonlinear two-point boundary-value problems is an application of Newton's method to a nonlinear differential operator equation. Since the linear boundary-value problem to be solved at each iteration must be discretized, it is natural to consider quasilinearization in the framework of an inexact Newton method. More importantly, each linear problem is only a local model of the nonlinear problem, and so it is inefficient to try to solve the linear problems to full accuracy. Conditions on size of the relative residual of the linear differential equation can then be specified to guarantee rapid local convergence to the solution of the nonlinear continuous problem. If initial-value techniques are used to solve the linear boundary-value problems, then an integration step selection scheme is proposed so that the residual criteria are satisfied by the approximate solutions. Numerical results are presented that demonstrate substantial computational savings by this type of economizing on the intermediate problems.This work was supported in part by DOE Contract DE-AS05-82-ER13016 and NSF Grant RII-89-17691 and was part of the author's doctoral thesis at Rice University. It is a pleasure to thank the author's thesis advisors, Professor R. A. Tapia and Professor J. E. Dennis, Jr.     

A higher‐order PDE‐based image registration approach

This paper addresses the problem of image registration with higher‐order partial differential equation (PDE) methods. From the study of existing affine‐linear and non‐linear methods, a new framework is proposed that unifies common image registration methods within a generic formulation. Currently image registration strategies are classified into either affine‐linear or non‐linear methods subject to the underlying transformations. The new approach combines both strategies to obtain proper approximations which are invariant under global geometrical distortion (shearing), anisotropic resolution (scale changes), as well as rotation and translation. To achieve this favourable property, a modified gradient flow approach is proposed which uses an operator with a kernel consisting of affine‐linear transformations. An approximation with finite differences leads to a large singular linear system. The pseudo‐inverse solution of this system can be computed efficiently by augmenting the singular system to a regular system. Numerical experiments show the improvements compared to unmodified gradient flow approaches. Copyright © 2005 John Wiley & Sons, Ltd.     

A SPARSE SUBSPACE TRUNCATED NEWTON METHOD FOR LARGE-SCALE BOUND CONSTRAINED NONLINEAR OPTIMIZATION

In this paper we report a sparse truncated Newton algorithm for handling large-scale simple bound nonlinear constrained minimixation problem. The truncated Newton method is used to update the variables with indices outside of the active set, while the projected gradient method is used to update the active variables. At each iterative level, the search direction consists of three parts, one of which is a subspace truncated Newton direction, the other two are subspace gradient and modified gradient directions. The subspace truncated Newton direction is obtained by solving a sparse system of linear equations. The global convergence and quadratic convergence rate of the algorithm are proved and some numerical tests are given.     

On convergence of the modified Newton’s method under Hölder continuous Fréchet derivative

In this paper, the upper and lower estimates of the radius of the convergence ball of the modified Newton’s method in Banach space are provided under the hypotheses that the Fréchet derivative of the nonlinear operator are center Hölder continuous for the initial point and the solution of the operator. The error analysis is given which matches the convergence order of the modified Newton’s method. The uniqueness ball of solution is also established. Numerical examples for validating the results are also provided, including a two point boundary value problem.     

Buckling and postcritical behaviour of the elastic infinite plate strip resting on linear elastic foundation

In this paper the von Kármán model for thin, elastic, infinite plate strip resting on a linear elastic foundation of Winkler type is studied. The infinite plate strip is simply-supported and subjected to evenly distributed compressive loads. The critical values of bifurcation parameters and buckling modes for given frequency of longitudinal waves are found on the basis of investigation of linearized problem. The mathematical nonlinear model is reduced to operator equation with Fredholm type operator of index 0 depending on parameters defined in corresponding Hölder spaces. The Lyapunov-Schmidt reduction and the Crandall-Rabinowitz bifurcation theorem (gradient case) are used to examine the postcritical behaviour of the plate. It is proved that there exists maximal frequency of longitudinal waves depending on the compressive load and the stiffness modulus of foundation.     

An efficient preconditioning scheme for iterative numerical solutions of partial differential equations.

Numerical solutions of time dependent and or nonlinear partial differential equations often require several solutions of a sparse linear system. If this system is factorized it may not fit into the computer core; if it is solved by an iterative process like the conjugate gradient algorithm it takes too much computing time. We show that if the small elements of the factorized matrix are deleted then the resulting operator is an excellent preconditioning operator for the conjugate gradient algorithm. Tests on two problems show that 90% of the main storage space can be saved without increasing the computing time as compared with a direct factorization method.