The exterior metric approach to a charged axially symmetric celestial body—the fourth‐order approximate solutions of Einstein–Maxwell equations

1988 ◽  
Vol 29 (12) ◽  
pp. 2618-2621
Author(s):  
Zhou Qi‐huang
Keyword(s):  
Celestial Body ◽  
Fourth Order ◽  
Maxwell Equations ◽  
Approximate Solutions ◽  
Axially Symmetric

scholarly journals Positive Solutions and Mann Iterations of a Fourth Order Nonlinear Neutral Delay Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22
Author(s):  
Zeqing Liu ◽  
Jingjing Zhu ◽  
Jeong Sheok Ume ◽  
Shin Min Kang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Positive Solutions ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Approximate Solutions ◽  
Banach Fixed Point Theorem ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential

This paper deals with a fourth order nonlinear neutral delay differential equation. By using the Banach fixed point theorem, we establish the existence of uncountably many bounded positive solutions for the equation, construct several Mann iterative sequences with mixed errors for approximating these positive solutions, and discuss some error estimates between the approximate solutions and these positive solutions. Seven nontrivial examples are given.

A BOUNDARY APPROXIMATION METHOD FOR FOURTH ORDER PROBLEMS

1991 ◽  
Vol 01 (04) ◽  
pp. 437-445 ◽  
Author(s):  
M.I. COMODI ◽  
R. MATHON
Keyword(s):  
Boundary Conditions ◽  
Error Bound ◽  
Fourth Order ◽  
Approximation Method ◽  
Approximate Solutions ◽  
Biharmonic Problem ◽  
Fourth Order Problems ◽  
Boundary Approximation

We study approximate solutions of the biharmonic problem ∆2u=0, by a boundary approximation method for a class of given boundary conditions. We prove an O(n−r) error bound (in the space L2(Ω)) for the solution u belonging to Hr(Ω).

On the efficiency of high-order difference schemes for the Schro¨dinger equation

2021 ◽  
pp. 68-81
Author(s):  
Виктор Иванович Паасонен ◽  
Михаил Петрович Федорук
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Approximate Solutions ◽  
Correction Method ◽  
Difference Schemes ◽  
Eighth Order ◽  
Traditional Architecture ◽  
Differential Problem ◽  
Order Of Accuracy ◽  
Additional Boundary Conditions

Исследуется ряд двух- и трехслойных разностных схем, построенных на расширенных шаблонах, до восьмого порядка точности для уравнения Шрёдингера. Наряду с многоточечными схемами рассматривается метод коррекции Ричардсона в приложении к схеме четвертого порядка аппроксимации, повышающий порядок точности путем построения линейных комбинаций приближенных решений, полученных на различных вложенных сетках. Проведено сравнение методов по устойчивости, сложности реализации алгоритмов и объему вычислений, необходимых для достижения заданной точности. На основе теоретического анализа и численных экспериментов выявлены методы, наиболее эффективные для практического применения The efficiency of difference methods for solving problems of nonlinear wave optics is largely determined by the order of accuracy. Schemes up to the fourth order of accuracy have the traditional architecture of three-point stencils and standard conditions for the application of algorithms. However, a further increase in the order in the general case is associated with the need to expand the stencils using multipoint difference approximations of the derivatives. The use of such schemes forces formulating additional boundary conditions, which are not present in the differential problem, and leads to the need to invert the matrices of the strip structure, which are different from the traditional tridiagonal ones. An exception is the Richardson correction method, which is aimed at increasing the order of accuracy by constructing special linear combinations of approximate solutions obtained on various nested grids according to traditional structure schemes. This method does not require the formulation of additional boundary conditions and inversion of strip matrices. In this paper, we consider several explicit and implicit multipoint difference schemes up to the eighth order of accuracy for the Schr¨odinger equation. In addition, a simple and double Richardson correction method is also investigated in relation to the classical fourth-order scheme. A simple correction raises the order to sixth and a double correction to eighth. This large collection of schemes is theoretically compared in terms of their properties such as the order of approximation, stability, the complexity of the implementation of a numerical algorithm, and the amount of arithmetic operations required to achieve a given accuracy. The theoretical analysis is supplemented by numerical experiments on the selected test problem. The main conclusion drawn from the research results is that of all the considered schemes, the Richardson-corrected scheme is the most preferable in terms of the investigated properties

Comparison of Slip-Line Solutions With Experiment

1956 ◽  
Vol 23 (2) ◽  
pp. 225-230
Author(s):  
E. G. Thomsen
Keyword(s):  
Plane Strain ◽  
Pure Aluminum ◽  
Maximum Shear Stress ◽  
Yield Condition ◽  
Approximate Solutions ◽  
Extrusion Process ◽  
Experimental Results ◽  
Wall Friction ◽  
Axially Symmetric ◽  
Area Reduction

Abstract Plane-strain and axially symmetric approximate solutions for a perfect plastic are compared with experimental results obtained from the extrusion of a billet of commercially pure aluminum. The extrusion process was carried out at room temperature under essentially negligible external wall friction with a reduction of area of 87.8 per cent. It is found that the plane-strain solution is in good agreement with the experimental results, if a constant k = σ̄/2 = 20,000 psi (maximum shear stress for Tresca’s criterion) is substituted in the plane-strain solution. The flow stress (σ̄ ≅ 40,000 psi) was determined from a property test for the aluminum billet, at a strain equivalent to the uniform deformation in the extrusion (i.e., same area reduction without shear deformation). It is found that the Haar and von Karman yield condition is not valid for the present case, and further that the axially symmetric solution demonstrates no real advantage over the simpler plane-strain solution in predicting the stress distribution in an axially symmetric extrusion problem.

On Axially Symmetric Bending of Nearly Cylindrical Shells of Revolution

1956 ◽  
Vol 23 (1) ◽  
pp. 59-67
Author(s):  
R. A. Clark ◽  
E. Reissner
Keyword(s):  
Elastic Shell ◽  
Approximate Solutions ◽  
Rate Of Change ◽  
Axially Symmetric ◽  
Shells Of Revolution ◽  
Small Parameters ◽  
Constant Coefficients ◽  
Radial Dimension ◽  
Axial Dimension ◽  
Linear Differential

Abstract The words “nearly cylindrical” are used in this paper to describe a thin elastic shell of revolution which is such that (a) the maximum variation of the radial dimension is small compared to the average radial dimension, and (b) the rate of change of the radial dimension with respect to the axial dimension is small compared to unity. For any particular type of loading a nearly cylindrical shell may or may not exhibit a behavior similar to that of a shell which is exactly cylindrical. The purpose of this paper is to demonstrate this fact and to present a method for obtaining approximate solutions for the stresses and deflections in either event. The method involves a perturbation procedure based on the assumption that all desired quantities can be represented as expansions in powers of two small parameters. The procedure leads to a set of linear differential equations with constant coefficients, which may be solved successively.

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