THE HIGH ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING PARABOLIC EQUATIONS 3-DIMENSION

In this paper, an explicit three_level symmetrical differencing scheme with parameters for solving parabolic partial differential equation of three_dimension will be considered. The stability condition and local truncation error for the scheme are r<1/2 and O( Δ t 2+ Δ x 4+ Δ y 4+ Δ z 4) ,respectively.

A POSTERIORI ERROR ESTIMATE FOR BOUNDARY CONTROL PROBLEMS GOVERNED BY THE PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we discuss the a posteriori error estimate of the finite element approximation for the boundary control problems governed by the parabolic partial differential equations. Three different a posteriori error estimators are provided for the parabolic boundary control problems with the observations of the distributed state, the boundary state and the final state. It is proven that these estimators are reliable bounds of the finite element approximation errors, which can be used as the indicators of the mesh refinement in adaptive finite element methods.

OSCILLATION THEOREM TO SYSTEMS OF IMPULSIVE NEUTRAL DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we study the oscillation of solutions to the systems of impulsive neutral delay parabolic partial differential equations. Under two different boundary conditions, we obtain some sufficient conditions for oscillation of all solutions to the systems.

OSCILLATION OF SYSTEMS OF IMPULSIVE DELAY PARABOLIC EQUATIONS ABOUT BOUNDARY VALUE PROBLEMS

In this paper, oscillation of solutions to a class of impulsive delay parabolic partial differential equations system with higher order Laplace operator is studied. Under two different boundary value conditions, we establish some sufficient criteria with respect to the oscillations of such systems, employing first-order impulsive delay differential inequalities. The results fully reflect the influence action of impulsive and delay in oscillation.

IMPROVEMENT ON STABILITY AND CONVERGENCE OF A. D. I. SCHEMES

Alternating direction implicit (A.D.I.) schemes have been proved valuable in the approximation of the solutions of parabolic partial differential equations in multi-dimensional space. Consider equations in the form partial derivative u/partial derivative t - partial derivative/partial derivative x(a(x,y,t) partial derivative u/partial derivative x) - partial derivative/partial derivative y(b(x,y,t) partial derivative u partial derivative y) = f Two A.D.I. schemes, Peaceman-Rachford scheme and Douglas scheme will be studied. In the literature, stability and convergence have been analysed with Fourier Method, which cannot be extended beyond the model problem with constant coefficients. Additionally, L-2 energy method has been introduced to analyse the case of non-constant coefficients, however, the conclusions are too weak and incomplete because of the so-called 'equivalence between L-2 norm and H-1 semi-norm'. In this paper, we try to improve these conclusions by H-1 energy estimating method. The principal results are that both of the two A.D.I. schemes are absolutely stable and converge to the exact solution with error estimations O(Delta t(2) + h(2)) in discrete H-1 norm. This implies essential improvement of existing conclusions.

A Fitted Numerov Method for Singularly Perturbed Parabolic Partial Differential Equation with a Small Negative Shift Arising in Control Theory

In this paper,a fitted Numerov method is constructed for a class of singularly perturbed one-dimensional parabolic partial differential equations with a small negative shift in the temporal variable.Similar boundary value problems are associated with a furnace used to process a metal sheet in control theory.Here,the study focuses on the effect of shift on the boundary layer behavior of the solution via finite difference approach.When the shift parameter is smaller than the perturbation parameter,the shifted term is expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed.The proposed finite difference scheme is unconditionally stable.When the shift parameter is larger than the perturbation parameter,a special type of mesh is used for the temporal variable so that the shift lies on the nodal points and an exponentially fitted scheme is developed.This scheme is also unconditionally stable.The applicability of the proposed methods is demonstrated by means of two examples.

On the Rayleigh-Plateau instability

In this paper,we study the surface instability of a cylindrical pore in the absence of stress.This instability is called the Rayleigh-Plateau instabilty.We consider the model developed by Spencer et al.[18],Kirill et al.[10]and Boutat et al.[2]in the case without stress.We obtain a nonlinear parabolic PDE of order four.We show the local existence and uniqueness of the solution of this problem by using Faedo-Galerkin method.The main results are the global existence of the solution and the convergence to the mean value of the initial data for long time.Numerical tests are also presented in this study.

ODE-Based Multistep Schemes for Backward Stochastic Differential Equations

In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of reference ordinary differential equations(ODEs),which can be directly discretized by many standard ODE solvers,yielding the corresponding numerical schemes for BSDEs.In particular,by applying strong stability preserving(SSP)time discretizations to the reference ODEs,we can propose new SSP multistep schemes for BSDEs.Theoretical analyses are rigorously performed to prove the consistency,stability and convergency of the proposed SSP multistep schemes.Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.

EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS FOR A NONLINEAR PARABOLIC EQUATION RELATED TO IMAGE ANALYSIS

In this paper we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a nonlinear parabolic partial differential equation, which is related to the Malik-Perona model in image analysis.

Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem

A mesh-independent,robust,and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented.We first consider a Lavrentiev regularization of the state-constrained optimization problem.Then,a multigrid scheme is designed for the numerical solution of the regularized optimality system.Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration.Results of numerical experiments and theoretical twogrid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.

On the speed of convergence of Picard iterations of backward stochastic differential equations

It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearities converge at least exponentially fast to the solution.In this paper we prove that this convergence is in fact at least square-root factorially fast.We show for one example that no higher convergence speed is possible in general.Moreover,if the nonlinearity is zindependent,then the convergence is even factorially fast.Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations.

A New Computational Approach for Solving Optimal Control of Linear PDEs Problem

In this paper, we present a new computational approach for solving an internal optimal control problem, which is governed by a linear parabolic partial differential equation. Our approach is to approximate the PDE problem by a nonhomogeneous ordinary differential equation system in higher dimension. Then, the homogeneous part of ODES is solved using semigroup theory. In the next step, the convergence of this approach is verified by means of Toeplitz matrix. In the rest of the paper, the optimal control problem is solved by utilizing the solution of homogeneous part. Finally, a numerical example is given.

WEAK APPROXIMATION OF OBLIQUELY REFLECTED DIFFUSIONS IN TIME-DEPENDENT DOMAINS

In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in Rd, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to 1/2 and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains.

An L_2-theory for a class of SPDEs driven by Lévy processes

In this paper we present an L2-theory for a class of stochastic partial differential equations driven by Levy processes. The coefficients of the equations are random functions depending on time and space variables, and no smoothness assumption of the coefficients is assumed.

Attractors for a Three-Dimensional Thermo-Mechanical Model of Shape Memory Alloys

In this note, we consider a Fremond model of shape memory alloys. Let us imagine a piece of a shape memory alloy which is fixed on one part of its boundary, and assume that forcing terms, e.g., heat sources and external stress on the remaining part of its boundary, converge to some time-independent functions, in appropriate senses, as time goes to infinity. Under the above assumption, we shall discuss the asymptotic stability for the dynamical system from the viewpoint of the global attractor. More precisely, we generalize the paper dealing with the one-dimensional case. First, we show the existence of the global attractor for the limiting autonomous dynamical system; then we characterize the asymptotic stability for the non-autonomous case by the limiting global attractor.

ANALYSIS OF MULTI-INDEX MONTE CARLO ESTIMATORS FOR A ZAKAI SPDE

In this article, we propose a space-time Multi-Index Monte Carlo （MIMC） estimator for a one-dimensional parabolic stochastic partial differential equation （SPDE） of Zakai type. We compare the complexity with the Multilevel Monte Carlo （MLMC） method of Giles and Reisinger （2012）, and find, by means of Fourier analysis, that the MIMC method： （i） has suboptimal complexity of 0（ε^-21 ｜ogε｜） for a root mean square error （RMSE） z if the same spatial discretisation as in the MLMC method is used; （ii） has a better complexity of 0（ε^-21 ｜ogε｜） if a carefully adapted discretisation is used; （iii） has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.