Pantograph Integral Differential Equations Backward Euler Method Fugitive

Many problems in science and engineering are modeled by dissipative dynamical systems. These systems are characterized by possessing a bounded absorbing set which all solution obtained by arbitrary initial values enter in the finite time and thereafter remain inside. Dissipative of numerical methods for DDEs is an important issue at all times, which has been studied by many papers. When we consider the applicability of numerical methods for these systems, it is naturally hoped that numerical methods for dynamical systems can inherit dissipativity of systems.The integro-differential equations with a proportional delay arise widely in scientific fields such as biology, ecology, medicine and physics. These classes of equations play an important role in modeling diverse problems of engineering and natural science. Because it is difficult to gain the analytic solutions, the researchers make numerical analysis and numerical computation of the equations. On a number point of view , it is important that whether the numerical methods can inherit the dissipativity of the analytic solutions of the equations.This thesis is concerned with the dissipativity of several kinds of numerical methods for nonlinear integro-differential equations with a proportional delay. In chapter one, the background and present situation are introduced and summarized for the study of nonlinear delay-integro-differential equations with a proportional delay. In chapter two, we deal with the dissipativity of nonlinear delay differential equations with multi-proportional delay. It is proved that the backward Euler method inherits the dissipativity of the underlying system. In chapter three, we deal with the analytic and numerical dissipativity of integro-differential equations with a proportional delay. A sufficient condition is presented to ensure that the above nonlinear system is dissipative. It is proved the backward Euler method inherits the dissipativity of the underlying system. In chapter four, we deal with the dissipativity of nonlinear delay- integro-differential equations with multi-proportional delay, It can be regarded as an extension of that in the chapter three. A sufficient condition is pres- ented to ensure that the above nonlinear system is dissipative.