Asymptotic solution of one-dimensional spectral boundary-value problems with rapidly varying coefficients: The case of multiple spectra

We suggest a method for the construction of complete asymptotic expansions for eigenvalues and eigen-functions of spectral boundary-value problems for differential equations with rapidly varying coefficients in the case of multiple spectra of the averaged problem. The effect of splitting of multiple eigen-values is illustrated by an example of a special fourth-order problem. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 10, pp. 1399–1408, October, 1998.     

An approximate solution of linear multicriteria control problems through the multicriteria simplex method

This paper deals with multicriteria, linear, continuous optimal control problems through the application of the multicriteria simplex method. Since the system is linear, the state variables at a given time can be expressed in terms of definite integrals. Upon using standard numerical integration formulas, the definite integrals are approximately expressed by means of weighted sums of the integrands. Then, after introducing some suitable auxiliary variables, approximate linear multicriteria programming problems are formulated. Illustrative numerical examples are provided to indicate the efficiency of the proposed method.The author is indebted to Professor Y. Sawaragi of Kyoto University for his constant encouragement. The author also wishes to thank T. Suwa for his cooperation in preparing the programming for the digital computer.     

An approximation method in a boundary-value problem for a linear differential equation with polynomial coefficients

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 2, pp. 248–253, March–April, 1988.     

Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides

In the spirit ofWilkinson's [1, 2] backward error analysis, conditions are established under which a given approximate solution of a system ofn linear equations withn unknowns is the exact solution of a modified system whose coefficients and right-hand sides are within a given neighborhood of those of the original system.     

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.     

Spline functions as approximate solutions of boundary-value problems

In this paper, an algorithm for finding a spline function that will satisfy a given linear differential equation at a discrete set of points and a given set of boundary conditions is described.     

An effective boundary-integral approach for the mixed boundary-value problems of linear elastostatics

A recent BIE formulation is applied to the mixed problem of two-dimensional elastostatics. The discretized form of the equations is presented in detail for easy numerical implementation. Some numerical examples are presented from which it is seen that this method is a significant improvement over a previous indirect approach.