The author is greatly indebted to Professor Dr Abbas I Abdel Karim, Professor of Mathematics and Head of Department, University College of Women, Ain Shams University for suggesting the topic of this thesis, for his helpful guidance and kind advice throughout the supervision of this research work. Sincere thanks are expressed to Prof Dr M S El Mohandis, Professor of Mathematics, University College of Women, for his kind help and encouragement.

This thesis contains five main chapters. The first chapter is considered as fundamental for solving sets of linear simultaneous equations. The most important problem of numerical algebra is the development of algorithms for the solution. Essentially there are two reasons why this is such a basic problem. First of all, today nearly every linear problem in applied mathematics - as a boundary value problem for a linear ordinary or partial differential equation is reduced by appropriate techniques to a system of linear equations, especially when electronic computers are to be employed for its solution. Secondly, nonlinear problems can in most cases be solved by approximating them by linear problems. There are several different methods for the solution of sets of equations, the method used depending on the calculating aids available, the type of equations to be solved and the accuracy required in the solution. In general, there are two type of numerical techniques for solving simultaneous linear equations: direct methods, which are finite, and indirect methods which are infinite. The two methods are useful, both have advantages and limitations.

By the numerical treatment of differential equations, the question of stability is still important. It is advantageous to state alternative methods for investigating the stability of multi-step formulae. Among these methods are: determination of the characteristic roots, graphical method, Hermitian form obtained from the characteristic equation or from the multi-step formula, application of Newton's theorem and using Sturm's theorem. The stability of an illustrative formula is discussed by using each of these methods.

There is much current interest in the problem of devising numerical methods of stiff ODEs. Stiffly stable variable step methods of order 3 and 4 are established. Some stability properties of multistep formulas will be discussed, some theorems and lemmas are proved ...

The author has plagiarized part of a paper that had already appeared in 'J. Math. Biol.' 24 (1986), 583-601. One of the conditions of submission of a paper for publication is that authors declare explicitly that their work is original and has not appeared in a publication elsewhere. Re-use of any data should be appropriately cited. As such this article represents a severe abuse of the scientific publishing system. The scientific community takes a very strong view on this matter and we apologize to readers of the journal that this was not detected during the submission process.