Numerical solutions of delay differential equations and applications

Numerical Solutions of Delay Differential Equations and ApplicationsüKorhan GUNELSUMMARYThe mathematical modelling using delay differential equations takes place, widely,in the application area of medical science, population dynamics, biology, economy,control systems, physics and more fields of science and engineering. The modelingusing delay differential equations (DDEs) have been started in the late of 1940?swith the problem of distribution of prime numbers introduced by Lord Cherwelland the modelling, made by Minorsky, to ensure the equilibration of ships inthe II. World War. The modelling of Lord Cherwell, drew Wright?s attention.Wright said that, ?Lord Cherwell drew my attention to an equation, which hehad encountered in his application of probability methods to the problem of thedistribution of primes. My thanks are due to him for thus introducing me to aninteresting problem.? Thus, the studies about delay differential equations theoryhas been originated.The general theory of DDEs is widely developed by Wright, Bellmann,Cooke, Hale, Driver, El?sgol?ts, Norkin and Kuang. The numerical methods ofDDEs are based on ordinary differential equations. However, the main difficultyin the adopting the methods for DDEs occurs because of the delay term. Inthe literature two different approaches exist to solve delay differential equations,numerically. The first is the method of steps introduced by Bellmann. Secondly,the approximate value of the delay term is calculated with an interpolation. Al-though the stability analysis of the DDEs have been widely studied, this area isuncovered entirely.In the thesis, the numerical methods for solving DDEs and the numericalstability analysis have been examined.