An approach to singular perturbation problems insoluble by asymptotic methods

Singular perturbation problems not amenable to solution by asymptotic methods require special treatment, such as the method of Carrier and Pearson. Rather than devising special methods for these problems, this paper suggests that there may be a uniform way to solve singular perturbation problems, which may or may not succumb to asymptotic methods. A potential mechanism for doing this is the author's boundary-value technique, a nonasymptotic method, which previously has only been applied to singular perturbation problems that lend themselves to asymptotic techniques. Two problems, claimed by Carrier and Pearson to be insoluble by asymptotic methods, are solved by the boundary-value method.     

Orthogonal polynomials and higher order singular Sturm-Liouville systems

In addition to the classic orthogonal polynomials which satisfy second order differential equations, there are a number of orthogonal polynomials which satisfy differential equations of orders four or six. Like the classic sets, they have distributional weight functions, are the eigenfunctions for certain self-adjoint boundary-value problems, and sometimes are involved with indefinite boundary-value problems.The purpose of this survey is to summarize the work of the last decade and to exhibit the state of the art as it now stands. Of particular interest is the development of the theory of singular Sturm-Liouville systems, which is so necessary in order to describe the boundary-value problems associated with these polynomials.     

On Spurious Solutions of Singular Perturbation Problems

A study is made of several nonlinear boundary-value problems of singular perturbation type for which a straightforward application of boundary-layer theory leads to spurious solutions. It is shown that these problems can be treated successfully by a slight modification of the method of matched asymptotic expansions. The analysis leads to several novel features which are not present in routine singular perturbation problems.     

Singuläre S-hermitesche Rand-Eigenwertprobleme

The theory of singular self-adjoint eigenvalue problems developed by Weyl [13], Stone [12], Kodaira [2] and others has been generalized by A. Schneider [8], [9], [10], [11] to real S-hermitian systems of differential equations with real boundary conditions. Here the theory of singular S-hermitian boundary-value problems for arbitrary complex systems of differential equations with complex boundary conditions is developed. Moreover the boundary conditions are allowed to depend linearly on the eigenvalue-parameter.     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

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.     

On improper integrals and differential equations in ordered Banach spaces

In this paper we study the existence of improper integrals of vector-valued mappings. The so obtained results combined with fixed point results in partially ordered functions spaces are then applied to derive existence and comparison results for least and greatest solutions of initial- and boundary-value problems in ordered Banach spaces. The considered problems can be singular, functional, nonlocal, implicit and discontinuous. Concrete examples are also solved.     

Singularly Perturbed Nonlinear Boundary-Value Problems Whose Reduced Equations Have Singular Points

Using results on implicit first-order differential equations, the existence and the asymptotic behavior of solutions of singularly perturbed second-order boundary-value problems whose reduced equations have singular points are studied. The variety of possible nonuniform behavior is investigated by means of a stability analysis of the reduced solutions and is illustrated with several model problems which are canonical in terms of the first-order singular-point theory.     

Singular points of solutions of elliptic systems in the plane

We study the structure of solutions for some important classes of singular elliptic systems in the plane. In particular, it is proved that the solutions of such systems have principally nonanalytic behavior in neighborhoods of fixed singular points. These results enable one to correctly state certain boundary-value problems and make their complete analysis.     

On Some Classes of Nonlocal Boundary-Value Problems for Singular Parabolic Equations

Mathematical Notes - We study the solvability of nonlocal boundary-value problems for singular parabolic equations of higher order in which the coefficient of the time derivative belongs to a space...     

On the existence of a solution to a singular integral equation in electromagnetic reflection

For a singular integral equation arising in a modified approach to boundary integral equations for exterior boundary-value problems from the theory of electromagnetic reflection an existence proof is given.     

B-spline finite-element method in polar coordinates

A numerical solution method for two-dimensional electromagnetic field problems is presented using the B-spline finite-element expression based on polar coordinates. The technique has two main advantages: (1) to avoid the truncation errors at some curved boundaries and (2) to improve the accuracy of singular boundary-value problems with a sharp corner. The B-spline finite-element formulation in polar coordinates is derived and its numerical applications are illustrated by an eddy current problem and several waveguide eigenvalue problems.     

Application of the boundary-value technique to singular perturbation problems at high Reynolds numbers

The boundary-value technique, advanced by Roberts for the solution of singular pertubation problems of ordinary differential equations where the small parameter multiples the highest derivative, is extended to the solution of the Navier-Stokes equation at high Reynolds numbers. Three standard flows—uniform flow past a plate, flow with a linearly adverse external velocity, and shear flow past a flat plate—have been chosen as test problems with a view to evaluating some of the features of the boundary-value technique, particularly in comparison with coefficient matching techniques as examplified by the method of matcher asymptotic expansions.     

Numerical treatment of a mathematical model arising from a model of neuronal variability

In this paper, we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability. The mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential-difference equation with small shifts. In the numerical treatment for such type of boundary-value problems, first we use Taylor approximation to tackle the terms containing small shifts which converts it to a boundary-value problem for singularly perturbed differential equation. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order. Then a parameter uniform difference scheme is constructed to solve the boundary-value problem so obtained. A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be but remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior.     

八阶奇异边值问题精确解的表达形式

该文在再生核空间W_2~9[0,1]中给出了求解八阶奇异边值问题的新算法.方程的精确解以级数形式给出.算例及数值结果验证了方法的实用性和有效性.     

Explicit forms of the Schwarz integral with applications for inverse boundary-value problems

We obtain new explicit forms of the Schwarz integral in the unit circle. By means of superpositions of logarithms we establish criteria for the appearance of singular boundary points and circles in inverse boundary-value problems.     

On the complexity of boundary integral equations with analytic coefficients with logarithmic singularities

We find the exact order of the ε-complexity of weakly singular integral equations with periodic and analytic coefficients of logarithmic singularities. This class of equations includes boundary equations for outer boundary-value problems for the two-dimensional Helmholtz equation.     

A Mixed Symmetric Problem of Electroelasticity Theory for a Layer Weakened by Through-the-Thickness Tunnel Holes

A new solution procedure for spatial mixed symmetric problems of electroelasticity theory for a layer weakened by through-the-thickness holes is proposed. The boundary-value problem is reduced to a system of one-dimensional singular integral equations consisting of 4k (k = 1, 2, ...) equations. Calculation results for characteristic stresses are presented.     

Boundary-value problem for the nonlinear Schrödinger equation

A mixed boundary-value problem for the nonlinear Schrödinger equation and its generalization is studied by the method used for the inverse scattering problem. A connection is established between conservation laws and boundary conditions in integrable boundary-value problems for higher nonlinear Schrödinger equations. It is shown that the generalized boundary-value problem requires a joint consideration of regular and singular solutions for the nonlinear Schrödinger equation with repulsion.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 151–165, 1988.     

Concentration of Mechanical Stresses near a Hole in a Piezoceramic Layer

A new procedure for solving three-dimensional mixed antisymmetric problems of elasticity theory for a layer weakened by through tunnel holes is proposed. The boundary-value problem is reduced to a system of one-dimensional singular integral equations consisting of 3k (k = 1, 2, ...) equations. The calculation results for characteristic stresses are presented.