An almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction-diffusion problem

Summary. This paper is concerned with a high order convergent discretization for the semilinear reaction-diffusion problem: , for , subject to , where . We assume that on , which guarantees uniqueness of a solution to the problem. Asymptotic properties of this solution are discussed. We consider a polynomial-based three-point difference scheme on a simple piecewise equidistant mesh of Shishkin type. Existence and local uniqueness of a solution to the scheme are analysed. We prove that the scheme is almost fourth order accurate in the discrete maximum norm, uniformly in the perturbation parameter . We present numerical results in support of this result. Received February 25, 1994     

A second-order difference scheme for a parameterized singular perturbation problem

In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.     

Iterative approximation of a common solution of a split equilibrium problem,a variational inequality problem and a fixed point problem

In this paper, we introduce an iterative method to approximate a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem for a nonexpansive mapping in real Hilbert spaces. We prove that the sequences generated by the iterative scheme converge strongly to a common solution of the split equilibrium problem, the variational inequality problem and the fixed point problem for a nonexpansive mapping. The results presented in this paper extend and generalize many previously known results in this research area.     

Error expansion for a first‐order upwind difference scheme

A singularly perturbed convection‐diffusion problem is considered. The problem is discretized using a simple firstorder upwinded difference scheme on general meshes. We derive an expansion of the error of the scheme that allows us to give error bounds in the discrete maximum norm, uniformly with respect to the perturbation parameter, for both a defect correction method and the Richardson extrapolation technique. This generalizes and simplifies results obtained in earlier publications [2], [8].     

Coefficient Stability of the Solution of a Difference Scheme Approximating a Mixed Problem for a Semilinear Parabolic Equation

We study the coefficient stability of a difference scheme approximating a mixed problem for a one-dimensional semilinear parabolic equation. We obtain sufficient conditions on the input data under which the solutions of the differential and difference problems are bounded. We also obtain estimates of perturbations of the solution of a linearized difference scheme with respect to perturbations of the coefficients; these estimates agree with the estimates for the differential problem.     

Applying a Power Penalty Method to Numerically Pricing American Bond Options

In this paper, we aim to develop a numerical scheme to price American options on a zero-coupon bond based on a power penalty approach. This pricing problem is formulated as a variational inequality problem (VI) or a complementarity problem (CP). We apply a fitted finite volume discretization in space along with an implicit scheme in time, to the variational inequality problem, and obtain a discretized linear complementarity problem (LCP). We then develop a power penalty approach to solve the LCP by solving a system of nonlinear equations. The unique solvability and convergence of the penalized problem are established. Finally, we carry out numerical experiments to examine the convergence of the power penalty method and to testify the efficiency and effectiveness of our numerical scheme.     

Recovering the Initial Distribution for a Time-Fractional Diffusion Equation

We consider the inverse problem of reconstructing the initial condition of a one-dimensional time-fractional diffusion equation from measurements collected at a single interior location over a finite time-interval. The method relies on the eigenfunction expansion of the forward solution in conjunction with a Tikhonov regularization scheme to control the instability inherent in the problem. We show that the inverse problem has a unique solution provided exact data is given, and prove stability results regarding the regularized solution. Numerical realization of the method and illustrations using a finite-element discretization are given at the end of this paper.     

Pointwise error estimates for a generalized Oseen problem and an application to an optimal control problem

We derive pointwise error estimates for a generalized Oseen when it is approximated by a low order Taylor‐Hood finite element scheme in two dimensions. The analysis is based on estimates for regularized Green's functions associated with a generalized Oseen problem on weighted Sobolev spaces and weighted interpolation results. We apply the maximum norm results to obtain convergence in an optimal control problem governed by a generalized Oseen equation and present a numerical example that allows us to show the behavior of the error.     

Uniform second-order pointwise convergence of a central difference approximation for a quasilinear convection-diffusion problem

A singularly perturbed quasilinear two-point boundary value problem with an exponential boundary layer is considered. The problem is discretized using the standard central difference scheme on generalized Shishkin-type meshes. We give a uniform second-order error estimate in a discrete L_∞ norm. Numerical experiments support the theoretical results.     

Limit Galerkin-Petrov schemes for a nonlinear convection-diffusion equation

In the present paper, we suggest a version of the nonconformal finite-element method (a perturbed Galerkin method) for approximating a quasilinear convection-diffusion equation in divergence form. A grid scheme is constructed with the use of an approach based on the Galerkin-Petrov approximation to the mixed statement of the original problem. The separated coordinate approximation of the solution components for the mixed problem permits one to take into account the direction of convective transport and preserve the main properties of the spatial operator of the original problem. We prove the stability of the line method scheme and a two-layer weighted scheme for the original problem.     

An existence result for a free boundary shallow water model using a Lagrangian scheme

We propose a free boundary shallow water model for which we give an existence theorem. The proof uses an original Lagrangian discrete scheme in order to build a sequence of approximate solutions. The properties of this scheme allow to treat the difficulties linked to the boundary motion. These approximate solutions verify some compactness results which allow us to pass to the limit in the discrete problem.     

Analysis of a meshless method for the time fractional diffusion-wave equation

In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is \(\mathcal {O}(\tau ^{3-\alpha })\). Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when \(\beta \rightarrow +\infty \) solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is \(\mathcal {O}(h)\). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.     

Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip

This paper addresses the finite element method for the two-dimensional time-dependent Schrödinger equation on an infinite strip by using artificial boundary conditions. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying the Crank-Nicolson scheme in time and a bilinear or quadratic finite element approximation in space. This scheme, by a rigorous analysis, has been proved to be unconditionally stable and convergent, and its convergence order has also been obtained. Finally, two numerical examples are given to verify the accuracy of the scheme.     

Stability and monotonicity of a conservative difference scheme for a multidimensional nonlinear scalar conservation law

We prove the existence, uniqueness, and monotonicity of the solution of an upwind conservative explicit difference scheme approximating an initial-boundary value problem for a many-dimensional nonlinear scalar conservation law with a quadratic nonlinearity under some specific conditions imposed only on the input data of the problem. We show that the resulting solution is not necessarily stable. Under some additional conditions on the input data, which provide the absence of shock waves, we prove the stability of the unique solution of the difference scheme for any finite time.     

Polynomial Time Approximation Scheme for Two Parallel Machines Scheduling with a Common Due Date to Maximize Early Work

We study the scheduling problem with a common due date on two parallel identical machines and the total early work criterion. The problem is known to be NP-hard. We prove a few dominance properties of optimal solutions of this problem. Their proposal was inspired by the results of some auxiliary computational experiments. Test were performed with the dynamic programming algorithm and list algorithms. Then, we propose the polynomial time approximation scheme, based on structuring problem input. Moreover, we discuss the relationship between the early work criterion and the related late work criterion. We compare the computational complexity and approximability of scheduling problems with both mentioned objective functions.     

Prolate spheroidal wave functions on a disc—Integration and approximation of two-dimensional bandlimited functions

We consider the problem of integrating and approximating 2D bandlimited functions restricted to a disc by using 2D prolate spheroidal wave functions (PSWFs). We derive a numerical scheme for the evaluation of the 2D PSWFs on a disc, which is the basis for the numerical implementation of the presented quadrature and approximation schemes. Next, we derive a quadrature formula for bandlimited functions restricted to a disc and give a bound on the integration error. We apply this quadrature to derive an approximation scheme for such functions. We prove a bound on the approximation error and present numerical results that demonstrate the effectiveness of the quadrature and approximation schemes.     

对一群签名方案的分析与改进

通过对夏祥盛等人的动态门限群签名方案的研究,指出该方案的若干不足,其中最主要的不足是通过伪造和不可追踪性,并对该方案进行了改进．与现有群签名方案不同,新方案中用户的秘密数由用户自己选取,从而避免了双线性对的计算,大大提高了效率．分析说明改进的群签名方案几乎克服了原方案的所有缺点．     

Parallel decomposition methods for linearly constrained problems subject to simple bound with application to the SVMs training

We consider the convex quadratic linearly constrained problem with bounded variables and with huge and dense Hessian matrix that arises in many applications such as the training problem of bias support vector machines. We propose a decomposition algorithmic scheme suitable to parallel implementations and we prove global convergence under suitable conditions. Focusing on support vector machines training, we outline how these assumptions can be satisfied in practice and we suggest various specific implementations. Extensions of the theoretical results to general linearly constrained problem are provided. We included numerical results on support vector machines with the aim of showing the viability and the effectiveness of the proposed scheme.     

Numerical analysis and simulations of a quasistatic frictional contact problem with damage in viscoelasticity

We consider a quasistatic problem which models the bilateral contact between a viscoelastic body and a foundation, taking into account the damage and the friction. The damage which results from tension or compression is then involved in the constitutive law and it is modelled using a nonlinear parabolic inclusion. The variational problem is formulated as a coupled system of evolutionary equations for which we state the existence of a unique solution. Then, we introduce a fully discrete scheme using the finite element method to approximate the spatial variable and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity hypotheses, the convergence of the numerical scheme obtained. Finally, a numerical algorithm and results are presented for some two-dimensional examples.     

A Hybrid Viscosity Approximation Method for a Common Solution of a General System of Variational Inequalities,an Equilibrium Problem,and Fixed Point Problems

In this paper, we introduce a new iterative method based on the hybrid viscosity approximation method for finding a common element of the set of solutions of a general system of variational inequalities, an equilibrium problem, and the set of common fixed points of a countable family of nonexpansive mappings in a Hilbert space. We prove a strong convergence theorem of the proposed iterative scheme under some suitable conditions on the parameters. Furthermore, we apply our main result for W-mappings. Finally, we give two numerical results to show the consistency and accuracy of the scheme.