Underdetermined System Of Equations Research Articles

Non-procedural large-scale modelling: Theoretical and practical significanc

One of the current major developments of computer science is a trend away from procedural programming languages, towards the creation of non-procedural languages. The paper examines the implications of this trend on large-scale mathematical modelling. In a procedural program, the basic rule is the assignment statement. In contrast, a non-procedural program consists of an underdetermined system of equations and nothing else. Consequently a non-procedural program is far more flexible and powerful than its procedural counterpart. While the latter describes only one problem the former may be used to solve every legitimate problem about the object represented by the program. The first major difficulty in designing a non- procedural modelling system is to specify the information to be derived from the non- procedural program. An interactive solution to this problem is presented,based on a purely structural analysis of the program. This solution fully exploits thesparsity of the program, can be efficiently implemented, and enables the user both to identify and to solve every legitimate problem associated with the program.

Determination of the field source distribution based on field measurements

Contents: The paper describes a method for the determination of a field source distribution in the conducting region when it is possible to measure the temperature distribution somewhere inside the conductor, or the heat flux distribution at a part of its boundary. The proposed algorithm is based on the application of a regularization method for solving an ill-conditioned, underdetermined system of equations. Numerical calculations performed in examples demonstrate all features of this method.

Asymptotic theory of turbulent shear flows

A theory is proposed in this paper to describe the behaviour of a class of turbulent shear flows as the Reynolds number approaches infinity. A detailed analysis is given for simple representative members of this class, such as fully developed channel and pipe flows and two-dimensional turbulent boundary layers. The theory considers an underdetermined system of equations and depends critically on the idea that these flows consist of two rather different types of regions. The method of matched asymptotic expansions is employed together with asymptotic hypotheses describing the order of various terms in the equations of mean motion and turbulent kinetic energy. As these hypotheses are not closure hypotheses, they do not impose any functional relationship between quantities determined by the mean velocity field and those determined by the Reynolds stress field. The theory leads to asymptotic laws corresponding to the law of the wall, the logarithmic law, the velocity defect law, and the law of the wake.