关于耦合型对流-扩散方程组的奇异摄动边值问题

本文考虑下述耦合型对流-扩散方程组的奇异摄动边值问题:本文提出两种方法:一种是初值化解法,用这种方法,原始问题转化成一系列没有扰动的一阶微分方程或方程组的初值问题,从而得到一个渐近展开式;第二种是边值化解法,用这种方法,原始问题转化成一组没有边界层现象的边值问题,从而可以得到精确解和使用经典的数值方法去得到具有关于摄动参数ε一致的高精度数值解.     

An <Emphasis Type="BoldItalic">O</Emphasis>(<Emphasis Type="BoldItalic">h</Emphasis><Superscript><Emphasis Type="Bold">6</Emphasis></Superscript>) numerical solution of general nonlinear fifth-order two point boundary value problems

A sixth-order numerical scheme is developed for general nonlinear fifth-order two point boundary-value problems. The standard sextic spline for the solution of fifth order two point boundary-value problems gives only O(h ²) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. O(h ⁶) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior of the orders of convergence.     

Numerical solution of the boundary-value problem for a nonlinear diffusion equation in image processing

Diffusion filtering methods involve solving an initial boundary-value problem for the diffusion equation in which the initial condition is specified by a function representing the filtered image. The output of this filter is the solution u(x, y, t) of the initial boundary-value problem at some fixed time t = T. In a previous study we have proposed a new version of the diffusion filtering method that ensures improved noise removal due to inclusion of a dependence of the diffusion coefficient on local image intensity. The present study analyzes the resulting finite-difference method for the initial boundary-value problem, examines its numerical implementation, and analyzes its efficiency on prototype and real images. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 35–43, 2006.     

Multipoint solution of two-point boundary-value problems

The purpose of this paper is to report on the application of multipoint methods to the solution of two-point boundary-value problems with special reference to the continuation technique of Roberts and Shipman. The power of the multipoint approach to solve sensitive two-point boundary-value problems with linear and nonlinear ordinary differential equations is exhibited. Practical numerical experience with the method is given.Since employment of the multipoint method requires some judgment on the part of the user, several important questions are raised and resolved. These include the questions of how many multipoints to select, where to specify the multipoints in the interval, and how to assign initial values to the multipoints.Three sensitive numerical examples, which cannot be solved by conventional shooting methods, are solved by the multipoint method and continuation. The examples include (1) a system of two linear, ordinary differential equations with a boundary condition at infinity, (2) a system of five nonlinear ordinary differential equations, and (3) a system of four linear ordinary equations, which isstiff.The principal results are that multipoint methods applied to two-point boundary-value problems (a) permit continuation to be used over a larger interval than the two-point boundary-value technique, (b) permit continuation to be made with larger interval extensions, (c) converge in fewer iterations than the two-point boundary-value methods, and (d) solve problems that two-point boundary-value methods cannot solve.     

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.     

Solution of boundary-value problems for elliptic equations in the space of distributions

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order2m. The representation of a distribution from (C ^∞ (S))’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.     

Matrix-differential methods of solving boundary-value problems for the nonlinear one-dimensional heat-conduction equation

Matrix numerical differentiation algorithms are applied to construct numerical-analytical methods for approximate solution of boundary-value problems for the nonlinear one-dimensional equation of heat conduction. The problems are reduced to a system of differential equations for the values of the sought approximate solution in the interior grid nodes and also to numerical formulas for the solution values at the boundary nodes. A numerical experiment is conducted. The error relative to grid spacing is established.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 37–43, 1986.     

Numerical analysis of the dispersion of electroelastic waves in a cylinder

The wave problem of electroelastic waves in a cylinder is reduced to a spectral problem for a system of eight ordinary differential equations. We give an algorithm for numerical solution based on reducing the original boundary-value problem to four Cauchy problems and determining the roots of the dispersion equation by the method of bisection. One figure. Bibliogrpahy: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 149–153.     

A refined problem on the strain of orthotropic noncircular cylindrical shells

Based on the refined Timoshenko theory, a semi-analytic method for solving problems of statics of orthotropic noncircular cylindrical shells is developed. The essence of this method consists in the spline-approximation of a solution in one coordinate direction and utilization of the collocation method and numerical solution of a high-order one-dimensional boundary-value problem by the discrete orthogonalization method in the second direction. The state of stress and strain of an open elliptic cylindrical shell under external load is investigated in the case where three contours rest upon supports and the fourth contour is rigidly fixed. Bibliography: 4 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 67–70.     

Exact three-point difference scheme for a nonlinear boundary-value problem on the semiaxis

For the numerical solution of boundary-value problems on the semiaxis for second-order nonlinear ordinary differential equations, an exact three-point difference scheme is constructed and substantiated. Under the conditions of existence and uniqueness of solution of a boundary-value problem, we prove the existence and uniqueness of solution of the exact three-point difference scheme and convergence of the method of successive approximations for its solution.     

Enhanced RFB method

The residual-free bubble method (RFB) is a parameter-free stable finite element method that has been successfully applied to a wide range of boundary-value problems exhibiting multiple-scale behaviour. If some local features of the solution are known a priori, the approximation properties of the RFB finite element space can be improved by enriching it on selected edges of the partition by edge-bubbles that are supported on pairs of neighbouring elements. Motivated by this idea, we define and analyse an enhanced residual-free bubble method for the solution of convection-dominated convection-diffusion problems in 2-D. Our a priori analysis highlights the limitations of the RFB method and the improved global approximation properties of the new method. The theoretical results are supported by detailed numerical experiments.     

A Steffensen's type method in Banach spaces with applications on boundary-value problems

In this paper, a modified Steffensen's type iterative scheme for the numerical solution of a system of nonlinear equations is studied. Two convergence theorems are presented. The numerical solution of boundary-value problems by the multiple shooting method using the proposed iterative scheme is analyzed.     

An approach to numerical solution of multipoint boundary-value problems

An approach is proposed to solving multipoint boundary-value problems for linear differential equation of w-th order, based on reduction to two-point boundary-value problems. The two-point problems are solved by the stable discrete orthogonalization method. Some numerical examples are considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 36–45, 1986.     

Axisymmetric deformation of bimetallic shells of revolution in the supercritical region

Geometrically nonlinear relationships of the theory of thin layered shells are applied to analyze axisymmetric strain of bimetallic shells of revolution in a temperature field. One-dimensional nonlinear boundary-value problems are solved by a combination of the linearization method and the discrete orthogonalization method. A numerical approach is proposed to solve the boundary-value problems in the supercritical strain region.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 52–56, 1990.     

A boundary-value problem for the polarized-radiation transfer equation with Fresnel interface conditions for a layered medium

A boundary-value problem for the polarized-radiation transfer equation for a layered medium with Fresnel matching conditions at the boundaries of the medium partition is examined. The theorems of solvability of the boundary-value problem are proved, and the continuity properties for its solution are examined. A numerical algorithm based on the Monte Carlo method for solving the boundary-value problem is proposed and proved.     

Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method

This paper considers two-point boundary-value problems using the differential transformation method. An iterative procedure is proposed for both the linear and nonlinear cases. Using the proposed approach, an analytic solution of the two-point boundary-value problem, represented by an mth-order Taylor series expansion, can be obtained throughout the prescribed range.     

Numerical Analysis of Boundary-Value Problems for Singularly-Perturbed Differential-Difference Equations with Small Shifts of Mixed Type

In this paper, we use a numerical method to solve boundary-value problems for a singularly-perturbed differential-difference equation of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift. Similar boundary-value problems are associated with expected first exit time problems of the membrane potential in models for the neuron. The stability and convergence analysis of the method is given. The effect of a small shift on the boundary-layer solution is shown via numerical experiments. The numerical results for several test examples demonstrate the efficiency of the method.     

Numerical solution of unstable two-point boundary-value problems by quasilinearization and orthonormalization

This paper reports on a method of numerical solution of sensitive nonlinear two-point boundary-value problems. The method consists of a modification of the continuation technique in quasilinearization obtained by combination with an orthogonalization procedure for linear boundary-value problems.This work was supported by CNR, Rome, Italy, within the framework of GNAFA.     

Solution differentiability for parametric nonlinear control problems with control-state constraints

This paper considers parametric nonlinear control problems subject to mixed control-state constraints. The data perturbations are modeled by a parameterp of a Banach space. Using recent second-order sufficient conditions (SSC), it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof blends numerical shooting techniques for solving the associated boundary-value problem with theoretical methods for obtaining SSC. In a first step, a differentiable family of extremals for the underlying parameteric boundary-value problem is constructed by assuming the regularity of the shooting matrix. Optimality of this family of extremals can be established in a second step when SSC are imposed. This is achieved by building a bridge between the variational system corresponding to the boundary-value problem, solutions of the associated Riccati ODE, and SSC.Solution differentiability provides a theoretical basis for performing a numerical sensitivity analysis of first order. Two numerical examples are worked out in detail that aim at reducing the considerable deficit of numerical examples in this area of research.This paper is dedicated to Professor J. Stoer on the occasion of his 60th birthday.The authors are indebted to K. Malanowski for helpful discussions.     

Numerical Method for the Solution of Linear Boundary-Value Problems for Integrodifferential Equations Based on Spline Approximations

Ukrainian Mathematical Journal - We propose a numerical method for the solution of linear boundary-value problems for systems of integrodifferential equations. The method is based on the...