Numerical Detections Of Blow-up Behaviors For Three Types Of Delay Differential Equations

Many real-life problems are actually influenced not only by their current situation, but also by some history information, hence, delay diff erential equations(DDEs) should be a good choice for describing such problems, and it is an important research topic of analyzing blow-up solutions of DDEs. A lot of practical problems featuring blow-up solutions, such as thermal explosion, nuclear reactor kinetics, and collapse of bridges. It is always assumed from the exiting works that the underlying solutions blow-up in finite time, however, the blow-up behaviors is strongly depend on the initial value as well as some parameters of the underlying equations. In addition, the theoretical study on blowup behaviors for differential equations is still not perfect. The conclusions obtained from these theoretical studies are always sufficient conditions which cannot satisfy the need of practical problems. Therefore, numerical detections of the blow-up behaviors for differential equations has scientific research value and practical significance.The present thesis mainly detect numerically the blow-up behaviors of three types of DDEs, namely the delay differential equations with piecewise constant arguments(EPCAs), the proportional delay ordinary differential equations(PDDEs), and the proportional delay partial differential equations(PDPDEs).First, blow-up behaviors of EPCAs are numerically detected. A framework is built to detect numerically the blow-up behaviors of piecewise smooth solutions of EPCAs, the errors for Tylor series coefficients between the true values and those calculated numerically are analyzed. The feasibility of the numerical framework is validated by comparing the numerical results with those obtained from the known theoretical analysis. The blow-up behaviors for some EPCAs are detected numerically in the present thesis, which cannot be detected theoretically from the known theory.Second, the numerical method for detecting blow-up behaviors of PDDEs is discussed. The solutions for PDDEs are extend to complex domain, the existence, uniqueness, and analyticity are proved, in addition, a theoretical method for detecting blow-up behaviors of differential equations is proposed. Based on analyticity analysis of the solutions, a feasible integral path is proposed. A series of numerical experiments illustrated that the numerical detection can be used as a supplement to theoretical study.Last, the blow-up behaviors of PDPDEs is investigated on the basis of numerical detections for PDDEs. Under the original integral path, the explicit Euler scheme of the solutions for PDPDEs is restricted by CFN condition, while the implicit Euler scheme is time consuming, therefore, the linear implicit Euler scheme is employed in this thesis. Some numerical examples are presented to illustrate the feasibility of numerical method.