Group solution of a time dependent chemical convective process

The time dependent progress of a chemical reaction over a flat vertical plate is here considered. The problem is solved using the two parameter group method which reduces the number of independent by one and leads to a set of nonlinear ordinary differential equation. The behavior of the process is numerically investigated for the chemical reaction order and different Schmidt numbers. As the problem shows a singularity at n = 1, the nonlinear system of ordinary differential equation resulting from the transformation of the problem are analytically solved through the perturbation method. The velocity and concentration of chemicals based on the analytical and numerical solutions are presented, compared and discussed.     

Numerical solution of a nonlinear reaction diffusion equation

In this paper, the authors propose a numerical method to compute the solution of a Cauchy problem with blow-up of the solution. The problem is split in two parts: a hyperbolic problem which is solved by using Hopf and Lax formula and a parabolic problem solved by a backward linearized Euler method in time and a finite element method in space. It is proved that the numerical solution blows up in a finite time as the exact solution and the support of the approximation of a self-similar solution remains bounded. The convergence of the scheme is obtained.     

Leak identification in a saturated unsteady flow via a Cauchy problem

In this initial study, we propose a numerical method for identifying multiple leak zones in a saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modeled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes because the leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we modify it and employ a previously proposed iterative regularizing method. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint to obtain a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a finite element method and the numerical results indicate that the leak zones can be identified with the proposed method.     

Numerical solution of singular perturbation problems by a terminal boundary-value technique

In this paper, we present a new approach for numerically solving linear singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in implicit form is introduced. Then, the outer region problem is solved as a two-point boundary-value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, the inner region problem is modified and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Three numerical examples have been solved to demonstrate the applicability of the method.     

Laplace方程有孔椭圆区域边值问题的解析解

本文利用相似流动替换方法 ,解决了中心有圆孔的椭园形区域上 Laplace方程第一类边值问题 ;采用分区域解法 ,给出了中心有椭园孔的椭园形区域上 Laplace方程第一类边值问题的解析通解 .这一结果在许多工程领域有重要应用 ,本文给出了油藏工程实例     

The smallest point of a polytope

This note suggests new ways for calculating the point of smallest Euclidean norm in the convex hull of a given set of points inR ⁿ . It is shown that the problem can be formulated as a linear least-square problem with nonnegative variables or as a least-distance problem. Numerical experiments illustrate that the least-square problem is solved efficiently by the active set method. The advantage of the new approach lies in the solution of large sparse problems. In this case, the new formulation permits the use of row relaxation methods. In particular, the least-distance problem can be solved by Hildreth's method.     

A Numerical Treatment of One-dimensional Models of Radiative Heat Transfer in Co-current Tube Furnaces

A model of a top-fired furnace, such as is used in chemicalindustrial plants, is described. The model gives rise to a systemof first-order differential equations, with boundary conditionsat both the top and bottom of the furnace. This also involvesa non-linear coupling of the radiation field to the energy ofthe gases flowing. Published accounts of this problem make useof shooting methods which converge extremely slowly. An alternativemethod will be described in which the radiative field equationsare solved by an algorithm of Grant and, to deal with the non-linearitiesof the system, a generalized Newton-Raphson method is employed.     

波动方程的差分反演模型

为了反演波动方程的系数函数,利用差分离散及扰动假设,推导出一个适合迭代的数值模型．解决了以往方法中正反演模型数值精度不一致问题,以及由此带来的一系列问题．经数值模拟计算说明,该方法是可行的和有效的．     

Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold

We determine the limit of the bottom of spectrum of Schrödinger operators with variable coefficients on Wiener spaces and path spaces over finite-dimensional compact Riemannian manifolds in the semi-classical limit. These are extensions of the results in [S. Aida, Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space, J. Funct. Anal. 203 (2) (2003) 401-424]. The problem on path spaces over Riemannian manifolds is considered as a problem on Wiener spaces by using Ito's map. However the coefficient operator is not a bounded linear operator and the dependence on the path is not continuous in the uniform convergence topology if the Riemannian curvature tensor on the underling manifold is not equal to 0. The difficulties are solved by using unitary transformations of the Schrödinger operators by approximate ground state functions and estimates in the rough path analysis.     

Heat transfer and Couette flow of a chemically reacting non‐linear fluid

In this paper we study the flow and heat transfer in a chemically reacting non‐linear fluid between two long horizontal parallel flat plates that are at different temperatures. The top plate is sheared, whereas the bottom plate is fixed. The fluid is modeled as a generalized power‐law fluid whose viscosity is also assumed to be a function of the concentration. The effects of radiation are neglected. The equations are made dimensionless and the boundary value problem is solved numerically; the velocity and temperature profiles are obtained for various dimensionless numbers. Published in 2009 by John Wiley & Sons, Ltd.     

Solution of a problem of nonlinear potential theory by the fixed domain method

A solution is derived for a one-dimensional boundary-value problem of nonlinear potential theory with one free end with a supplementary boundary condition. The problem is solved by a variant of the fixed domain method. In each fixed domain, the problem is reduced to the corresponding nonlinear difference problem, which is solved by Newton's method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 51–56, 1986.     

An evolutionary and constructive approach to a crew scheduling problem in underground passenger transport

Operations management of subway systems is associated with combinatorial optimization problems (i.e. crew and train scheduling and rostering) which belong to the np-hard class of problems. Therefore, they are generally solved heuristically in real situations. This paper considers the problem of duty generation, i.e. identifying an optimal trips set that the conductors should complete in one workday. With regard to operational and labor conditions, the problem is to use the lowest possible number of conductors and minimize total idle time between trips. The problem is modeled and solved using a constructive hybrid approach, which has the advantage of visualizing a solution construction similar to the manual approach typically used. Our approach takes advantage of the benefits offered by evolutionary methods, which store a candidate solutions population in each stage, thus controlling the combinatorial explosion of possible solutions. The results thus obtained for problems similar to those that are solved manually in the Santiago Metro System were compared with two alternative approaches, based on tabu search and a greedy method. The hybrid method produced solutions with the minimum number of duties in six of the ten problems solved. However, the tabu search method provided better results in terms of idle time than either the hybrid method or the greedy method.     

Electromagnetic Scattering by a Chiral Grating in a Homogeneous Chiral Environment and its Finite Element Method with Perfectly Matched Absorbing Layers

1 Introduction The phenomenon of optical activity in special materials has been known since the beginning of the last century. Whereas optical activity has been considered in optics and in quantum mechanics for many years, its analysis within the framewor…     

Boundary Identification for a Blast Furnace

In this paper,the authors discuss an inverse boundary problem for the axi- symmetric steady-state heat equation,which arises in monitoring the boundary corrosion for the blast-furnace.Measure temperature at some locations are used to identify the shape of the corrosion boundary. The numerical inversion is complicated and consuming since the wear-line varies during the process and the boundary in the heat problem is not fixed.The authors suggest a method that the unknown boundary can be represented by a given curve plus a small perturbation,then the equation can be solved with fixed boundary,and a lot of computing time will be saved. A method is given to solve the inverse problem by minimizing the sum of the squared residual at the measuring locations,in which the direct problems are solved by axi- symmetric fundamental solution method. The numerical results are in good agreement with test model data as well as industrial data,even in severe corrosion case.     

Three-dimensional kirsch problem for a transtropic plate

Qualitative and quantitative study of the concentration of stresses in the Kirsch problem for isotropic plates is carried out in the three-dimensional formulation using the superposition method /1/ and the method of homogeneous solutions /2/. An asymptotic method of solving the Kirsch problem for transtropic plates is given in /3/. Below the problem in question is solved for the transtropic bodies of finite dimensions.     

A Contact Problem for a Half-Space and a Rigid Base with an Axially Symmetric Recess

An axially symmetric problem of frictionless contact interaction of an elastic half-space and a rigid base that has a small surface recess is considered. A corresponding problem of elasticity theory is formulated. The problem is solved by the method of double integral equations. The contact pressure, the surface shape of the half-space after compression, and the size of the gap are found in closed form.     

Numerical analysis for the scattering by obstacles in a homogeneous chiral environment

The scattering of time-harmonic electromagnetic waves propagating in a homogeneous chiral environment by obstacles is studied. The problem is simplified to a two-dimensional scattering problem, and the existence and the uniqueness of solutions are discussed by a variational approach. The diffraction problem is solved by a finite element method with perfectly matched absorbing layers. Our computational experiments indicate that the method is efficient.     

Third-order accurate finite-volume method on a triangular grid

A third-order accurate finite-volume method on triangular grids is proposed for the numerical solution of conservation law systems. The method is described in detail as applied to the advection equation. The accuracy of its numerical solutions obtained on grids of various degrees of detail is compared. The viscous flow over a plate and the unsteady flow around a cylinder are solved. For comparison purposes, the latter problem is also solved using third- and fifth-order accurate compact approximations.     

A numerical method for reconstructing the coefficient in a wave equation

We present a numerical method for reconstructing the coefficient in a wave equation from a single measurement of partial Dirichlet boundary data. The original inverse problem is converted to a nonlinear integral differential equation, which is solved by an iterative method. At each iteration, one linear second‐order elliptic problem is solved to update the reconstruction of the coefficient, then the reconstructed coefficient is used to solve the forward problem to obtain the new data for the next iteration. The initial guess of the iterative method is provided by an approximate model. This model extends the approximate globally convergent method proposed by Beilina and Klibanov, which has been well developed for the determination of the coefficient in a special wave equation. Numerical experiments are presented to demonstrate the stability and robustness of the proposed method with noisy data.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 289–307, 2015     

Cubic spline method for a class of nonlinear singularly-perturbed boundary-value problems

In this paper, we present a numerical method for solving a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. The original second-order problem is reduced to an asymptotically equivalent first-order problem and is solved by a numerical method using a fourth-order cubic spline in the inner region. The method has been analyzed for convergence and is shown to yield anO(h ⁴) approximation to the solution. Some test examples have been solved to demonstrate the efficiency of the method.The authors thank the referee for his helpful comments.