The popularity and explosive growth of numerical analysis today are further evidence that applications are still the leading source of inspiration for mathematical creativity . Whenever new mathematical ideas are developed it is usually new applications which have pointed the way . The electronic computing machine is itself an illustration of this , are sponce to an overwhelming need for faster computation . and the appearance of such machines has made it possible to meet the demands of today s applications , in many cases , by developing more numerical methods . This is the pedigree of modern numerical analysis . It is the numerical aspect of the board field of applied analysis . It would be a mistake , however , to draw too fine a boundary between our subject and what is called pure or abstract analysis . The borderline is a fuzzy one , as borderlines usually are , and materials from both sides frequently infiltrate the other . In earlier days it was commonplace for mathematicians to be expert at both the pure and the applied . Both have long since developed to a size which makes full acquaintance with even one impossible . and reasonable competence at both an arduous objective . In spite of this the applied mathematician , icluding the numerical analyst , must try to keep a ware of what is happening across the border . The proof of the classical existence theorem , of differential equations by "applied" methods is a beautiful illustration of how applications lead eventually to abstract theory . So , our principal interest is numerical mathematics , because they are a remainder of the fuzzy borderline and of the value of infiltration in both directions . The numerical analyst is after all , an analyst . The behavior of many physical processes , particularly those in systems undergoing time- dependent change (transients) , can be described by ordinary differential equations are of great importance to engineers and scientists . Although many important differential equations can be solved by weel-know analytical techniques , a great number of physically significant differential equations can not be solved . Fortunately , the solutions of these equations can usually be enerated numerically . This project will describe the importance of these numerical methods .