This book serves as a concise textbook for students in an advanced undergraduate or first-year
graduate course in various disciplines such as applied mathematics control and engineering
who want to understand the modern standard of numerical methods of ordinary and delay
differential equations. Experts in the same fields can also learn about the recent developments
in numerical analysis of such differential systems. Ordinary differential equations (ODEs)
provide a strong mathematical tool to express a wide variety of phenomena in science and
engineering. Along with its own significance one of the powerful directions toward which ODEs
extend is to incorporate an unknown function with delayed argument. This is called delay
differential equations (DDEs) which often appear in mathematical modelling of biology
demography epidemiology and control theory. In some cases the solution of a differential
equation can be obtained by algebraic combinations of known mathematical functions. In many
practical cases however such a solution is quite difficult or unavailable and numerical
approximations are called for. Modern development of computers accelerates the situation and
moreover launches more possibilities of numerical means. Henceforth the knowledge and
expertise of the numerical solution of differential equations becomes a requirement in broad
areas of science and engineering. One might think that a well-organized software package such
as MATLAB serves much the same solution. In a sense this is true but it must be kept in mind
that blind employment of software packages misleads the user. The gist of numerical solution of
differential equations still must be learned. The present book is intended to provide the
essence of numerical solutions of ordinary differential equations as well as of delay
differential equations. Particularly the authors noted that there are still few concise
textbooks of delay differential equations and then they set about filling the gap through
descriptions as transparent as possible. Major algorithms of numerical solution are clearly
described in this book. The stability of solutions of ODEs and DDEs is crucial as well. The
book introduces the asymptotic stability of analytical and numerical solutions and provides a
practical way to analyze their stability by employing a theory of complex functions.
