Implicit-Explicit And Linearized Methods For Several Classes Of Nonlinear Functional Differential Equations

Functional differential equations(FDEs)have a wide applications in many scientific areas,such as physics,chemistry,biology,power and circuit analysis,neural network,medicine,economy and so forth.Compared to differential equations without delay,FDEs can more accurately describe the development trend of objective things.Generally,to obtain the exact solutions of this type of equations are difficult.Hence,it's significant to construct efficient methods to obtain the numerical solutions.Up to now,many references and monographs are related to the numerical computation and analysis for FDEs,where linear multistep methods,one-leg methods,Runge-Kutta methods,general linear methods,boundary value methods are included.However,when applying the above methods to solve nonlinear stiff FDEs,a large computational cost is always requested.In view of this,to improve the computational efficiency of the schemes,the main purpose of the present paper is to extend implicit-explicit and linearized methods to solve several classes of FDEs,such as stiff neutral equations,nonlinear delay Sobolev equations and Allen-Cahn equations.The framework of the present paper is organized as follows.In Chapter 1,we briefly introduce the historical background and the recent research status of FDEs.Then,we give a brief review to implicit-explicit and linearized methods.In the end,the main work of this thesis is presented.In Chapter 2,we construct a class of extended implicit-explicit one-leg methods for solving nonlinear stiff neutral equations.It's proved under some suitable conditions that the methods are convergent and stable.Moreover,several numerical examples are given to confirm the computational effectiveness,theoretical accuracy and the advantage in computational efficiency of the methods.In Chapter 3,a class of extended implicit-explicit Runge-Kutta methods for solving semi-linear stiff neutral equations are considered.Under some suitable conditions,we present an error analysis for the methods,and with several numerical examples,we further illustrate the theoretical results and computational effectiveness of the methods.Finally,a numerical comparison with fully implicit Runge-Kutta methods suggests that the derived methods have higher computational efficiency.In Chapter 4,we study the extended implicit-explicit general linear methods for solving semi-linear stiff neutral equations.Under some suitable conditions,we derive the convergence and stability of the methods.Moreover,with several numerical examples,the theoretical results and computational effectiveness of the methods are verified.Finally,a numerical comparison with fully implicit general linear methods shows that the methods have great advantages in computational efficiency.In Chapter 5,we focus on the numerical computation and analysis for nonlinear delay Sobolev equations.For solving 1D and 2D problems,we construct linearized compact difference methods,respectively.Then we analyse the solvability and convergence of the methods and it's shown that the methods are convergent of order two in time and order four in space.Moreover,in order to improve the accuracy in time,we consider the Richardson extrapolation technique and it's proved that the derived algorithms have fourth order accuracy in both time and space.Finally,several numerical examples are given to testify the theoretical results and computational effectiveness of the methods.In Chapter 6,combining the second order central difference with Crank-Nicolson scheme and Newton linearized technique,we present a novel two-level linearized difference method for solving Allen-Cahn equations.It's proved under some suitable conditions that the scheme can preserve the discrete maximum principle and energy stability.Moreover,we give an error estimate for the scheme and it's shown that the scheme has second order accuracy in both time and space.Finally,with several numerical examples,we confirm the derived theoretical results and computational effectiveness of the methods.In last chapter,a summary of the whole thesis is given and some future work is listed.