Stability Analysis Of Two Kinds Of Exponential Methods For Semi-linear Differential Equations

Delay differential equations(DDEs) have been widely used in many fields such as economy, biology, physics, automation etc. But because of complexity of DDEs, it is usually difficult to obtain the analytic expression of solutions of the systems. So many people dedicate themselves to the research of the numerical methods of DDEs. Recently, there are many results of the stability of numerical methods for DDEs. DDEs can describe the rules in these fields more accurately.The thesis is organized as follow.Firstly, we present many applications of DDEs and the research of both analytic solution and numerical solution theory of DDEs, along with exponential integrators for the recent years.Secondly, we discuss the stability of exponential Runge-Kutta methods for ODEs. Then, we define the concept of B-stability and exponential algebraic stability. Finally, we prove that if an exponential Runge-Kutta method is exponential algebraic stable, it is B-stable.Thirdly, we discuss the stability of exponential Runge-Kutta methods for DDEs. Then, we define the concept of R-stability. Finally, we prove that if an exponential Runge-Kutta method is exponential algebraic stable, it is R-stable. Finally, we discuss the stability of exponential Rosenbrock methods for DDEs. Then we obtain the scheme of exponential Rosenbrock methods. Finally, we discuss the theory of the stability bounds of discrete evolution operation.The end part concludes studies above,and shows the directions of researching in future.