Approximate solutions for a class of first order nonlinear difference equations

In this paper, a measure-theoretical approach to find the approximate solutions for a class of first order nonlinear difference equations is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a calculus of variations problem, some concepts in measure theory are used to approximate the solution. The procedure of constructing approximate solution in form of an algorithm is shown. Finally a numerical example is given.     

Bernstein polynomial basis for numerical solution of boundary value problems

The purpose of this paper is to propose a computational method for the approximate solution of linear and nonlinear two-point boundary value problems. In order to approximate the solution, the expansions in terms of the Bernstein polynomial basis have been used. The properties of the Bernstein polynomial basis and the corresponding operational matrices of integration and product are utilized to reduce the given boundary value problem to a system of algebraic equations for the unknown expansion coefficients of the solution. On this approach, the problem can be solved as a system of algebraic equations. By considering a special case of the problem, an error analysis is given for the approximate solution obtained by the present method. At last, five examples are examined in order to illustrate the efficiency of the proposed method.     

Identification of an inverse source problem for time‐fractional diffusion equation with random noise

In this paper, we consider an inverse source problem for a time fractional diffusion equation. In general, this problem is ill posed, therefore we shall construct a regularized solution using the filter regularization method in the random noise case. We will provide appropriate conditions to guarantee the convergence of the approximate solution to the exact solution. Then, we provide examples of filters in order to obtain error estimates for their approximate solutions. Finally, we present a numerical example to show efficiency of the method.     

A new approach for solving the Stokes problem

In this paper, a new approach for finding the approximate solution of the Stokes problem is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a distributed parameter control system, the theory of measure is used to approximate the velocity functions by piecewise linear functions. Then, the approximate values of pressure are obtained by a finite difference scheme.     

Numerical treatment of singularly perturbed two point boundary value problems using initial-value method

In this paper, we describe an initial-value method for linear and nonlinear singularly perturbed boundary value problems in the interval [p,q]. For linear problems, the required approximate solution is obtained by solving the reduced problem and one initial-value problems directly deduced from the given problem. For nonlinear problems the original second-order nonlinear problem is linearized by using quasilinearization method. Then this linear problem is solved as previous method. The present method has been implemented on several linear and non-linear examples which approximate the exact solution. We also present the approximate and exact solutions graphically.     

Continuity of approximate solution mappings for parametric equilibrium problems

In this paper, we obtain sufficient conditions for Hausdorff continuity and Berge continuity of an approximate solution mapping for a parametric scalar equilibrium problem. By using a scalarization method, we also discuss the Berge lower semicontinuity and Berge continuity of a approximate solution mapping for a parametric vector equilibrium problem.     

An inner approximation method incorporating a branch and bound procedure for optimization over the weakly efficient set

In this paper, we consider an optimization problem which aims to minimize a convex function over the weakly efficient set of a multiobjective programming problem. From a computational viewpoint, we may compromise our aim by getting an approximate solution of such a problem. To find an approximate solution, we propose an inner approximation method for such a problem. Furthermore, in order to enhance the efficiency of the solution method, we propose an inner approximation algorithm incorporating a branch and bound procedure.     

An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping

In this paper, we introduce and study an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping in real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping which is the unique solution of the variational inequality problem. The results presented in this paper are the supplement, extension and generalization of the previously known results in this area.     

Numerical solutions of integro-differential equations and application of a population model with an improved Legendre method

In this paper, an improved Legendre collocation method is presented for a class of integro-differential equations which involves a population model. This improvement is made by using the residual function of the operator equation. The error differential equation, gained by residual function, has been solved by the Legendre collocation method (LCM). By summing the approximate solution of the error differential equation with the approximate solution of the problem, a better approximate solution is obtained. We give the illustrative examples to demonstrate the efficiency of the method. Also we compare our results with the results of the known some methods. In addition, an application of the population model is made.     

球形扁壳超临界变形的步进求和计算

本文用步进求和法计算了球形扁壳第二类失稳问题,在球扁壳超临界变形计算上给出了优于一级近似结果,解决了该问题无法求二级近似解的困难.算例表明步进求和法收敛于二级近似解.     

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions.     

Initial-Value Technique for Singularly-Perturbed Turning-Point Problems Exhibiting Twin Boundary Layers

The initial-value technique that was originally developed for solving singularly-perturbed nonturning-point problems (Ref. 1) is used here to solve singularly-perturbed turning-point problems exhibiting twin boundary layers. In this method, the required approximate solution is obtained by combining solutions of the reduced problem, an initial-value problem, and a terminal-value problem. Error estimates for approximate solutions are obtained. The implementation of the method on parallel architectures is discussed. Numerical examples are presented to illustrate the present technique.     

On a differential-algebraic problem

The method of quasilinearization is a procedure for obtaining approximate solutions of differential equations. In this paper, this technique is applied to a differential-algebraic problem. Under some natural assumptions, monotone sequences converge quadratically to a unique solution of our problem.     

A NUMERICAL PERTURBATION EXPANSION METHOD FOR THE SOLUTION OF A LIENARD-TYPE INITIAL-VALUE PROBLEM WITH PERIODIC DECAYING FORCING TERM

A numerical perturbation expansion method is developed, analysed and implemented for the numerical solution of a second-order initial-value problem. The differential equation in this problem exhibits cubic damping, a cubic restoring force and a decaying forcing-term which is periodic with constant frequency. The method is compared with the numerical method by Twizell [1]. In fact, the later is first perturbation approximate solution in the present paper.     

一类球型区域上变系数反向热传导问题

考虑了一类球型区域上变系数反向热传导问题.这个问题是不适定的,即问题的解（若存在）并不连续依赖于测量数据.构造了投影迭代正则化方法,得到了该反问题的正则近似解,同时给出了在先验和后验参数选取规则下精确解与正则近似解之间的收敛性误差估计.最后,通过数值结果验证了该方法的有效性.     

ASYMPTOTIC METHOD OF TRAVELLING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR REACTION DIFFUSION EQUATION

In this article the travelling wave solution for a class of nonlinear reaction diffusion problem;are considered.Using the homotopic method and the theory of travelling wave transform,the approximate solution for the corresponding problem is obtained.     

Bundle-based decomposition for large-scale convex optimization: Error estimate and application to block-angular linear programs

Robinson has proposed the bundle-based decomposition algorithm to solve a class of structured large-scale convex optimization problems. In this method, the original problem is transformed (by dualization) to an unconstrained nonsmooth concave optimization problem which is in turn solved by using a modified bundle method. In this paper, we give a posteriori error estimates on the approximate primal optimal solution and on the duality gap. We describe implementation and present computational experience with a special case of this class of problems, namely, block-angular linear programming problems. We observe that the method is efficient in obtaining the approximate optimal solution and compares favorably with MINOS and an advanced implementation of the Dantzig—Wolfe decomposition method.     

A fundamental solution method for three-dimensional viscous flow problems with obstacles in a periodic array

In this paper, we propose a fundamental solution method for three-dimensional viscous flow problems with obstacles in a periodic array. Our problem is mathematically a boundary value problem of the Stokes equation with periodic boundary conditions, to which it is difficult to give a good approximation by the ordinary fundamental solution method. Our method gives an approximate solution by a linear combination of the periodic fundamental solutions. In addition, we can compute the drag forces on the obstacles by using the data obtained in our method. Numerical examples for the problems of flows past spheres show the effectiveness of our method.     

Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes’ equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by Maz’ya (DFG-Kolloquium des DFG-Forschungsschwerpunktes Randelementmethoden, 1991) and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes’ equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström’s method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

LEAST-SQUARES MIXED FINITE ELEMENT METHOD FOR SADDLE-POINT PROBLEM

1. IntroductionThere are many work to investigate the stability of the mired finite element methodfor the saddle-point problems, i.e., to construct the finite element spaces, such that theso-called discrete BB-codition is satisfied (c.f. [1],[21,[7],[81 and the references therein).To circumvent the discrete BB-conditon, recently there has been an increased interest inuse of least-squares approach for the solution of the mixed finite element approximationof the saddel-point problem (c.f.[3]--[…