The Numerical Methods For Solving Some Delay Differential Equations

This paper deals with numerical methods and theoretical analysis for delay differential equations (DDEs). In general, the exact solutions can be given only for few DDEs. Hence, it is very important to study the numerical methods both in theory and applied aspects.The main results obtained in this paper are summarized as follows:1. The multi-pantograph delay differential equations (DDEs ) are considered. The exact solution is given in the form of series and an approximate solution is obtained by truncating the series. The error of the exact solution and an approximation is monotonically decreasing and tends to zero. If the grid points are different and dense in the interval of the solution, then approximate solution is convergent uniformly to the exact solution.2. The alternating group explicit iterative method for solving the neutral parabolic equation with initial boundary conditions is given. The second-order unconditional stable implicit difference scheme and alternating group explicit iterative method which contains parameters and is suitable for parallel computing are constructed. It is proved that the iterative method is convergent for arbitrary initial values. The numerical experiment shows that the alternating group explicit iterative method not only has higher precision, but also is convergent faster.3. The alternating direction difference method for the two-dimensional nonlinear delay parabolic differential equation is given. It is proved that the method is convergent and unconditional stability.4. The precise time-integration methods for solving the delay parabolic equations are given and numerical stability is proved. The half discrete difference scheme of forth-order is used to transform the delay partial differential equation into the delay ordinary differential equations. The higher accurate approximations are obtained when the precise time-integration method is applied to the delay differential equations.5. The classical Adomian decomposition method for the initial value problem of the multi-pantograph delay differential equation is given and its convergence is discussed. The new modified Adomian decomposition method for the initial value problem of ordinary delay differential equation is given and it is proved that it is stable and has higher accuracy.In the paper, several numerical methods based on the models of delay differential equations and partial delay differential equations are constructed. The...