Generalized Polynomial Chaos For Random Delay Differential Equations

Random delay differential equations are used widely to model the undetermined phe-nomena from physics, biology, aerospace, material science and other fields. It is very dif-ficult to solve such equations theoretically. Thus, it becomes an significant issue to study such equations numerically.This thesis is devoted to the investigation of the generalized polynomial chaos (gPC) method which is currently prevailing at home and abroad. Firstly, we further study the gPC method for random ordinary differential equation(RODE). Then, we extend the gPC method to solve several kinds of significant random delay systems. The main work of the thesis is as follows:In Chapter2, we further study the gPC method for RODE. The error estimation of the method is derived. Also, with both the theoretical and the numerical results, it is confirmed that the error due to the finite dimensional noise assumption is important for the global gPC error, which is a key feature of gPC. What’s more, we investigate the tolerance of the error due to the finite dimensional noise assumption to correlation length and variance. In addition, we provide the first clear demonstration of the contribution of variance to "the curse of dimensionality".In Chapter3, the gPC method is extended to solve nonlinear random delay differential equation(NRDDE). The error estimation of the method is derived, which shows that the global gPC error is controlled by the number of random variables used in the gPC method and the highest order of the gPC basis as well as the discretization error. With several numerical examples, the theoretical results are further illustrated, and it is verified that the gPC method is effective for such NRDDE.Chapter4deals with gPC for nonlinear random pantograph delay differential equation. We introduce the gPC procedures for such kind of equation, and give an estimation of the global gPC error. Some numerical results are reported showing the computational efficiency of the gPC method and confirming the significant impact of the error due to the finite-dimensional noise assumption on the gPC performance.Chapter5is concerned with the numerical solution of nonlinear random delay differ- ential equation with piecewise continuous argument. A gPC method is used to compute the statistical characteristics of the solution, and Runge-Kutta(RK) methods are presented to solve the deterministic delay differential equations with piecewise continuous argument (EPCA) resulted from the gPC method. The error estimation of the gPC method is derived. Also, the convergence property of the RK methods applied to the deterministic nonlinear EPCA is analyzed, and it is proved that the RK methods here conserve their original orders for ODEs. Numerical experiments are performed to support our theoretical analysis and test the efficiency of the gPC method.