Numerical Analysis Of Nonlinear Neutral Functional Differential Equations

Neutral functional differential equations (NFDEs) can be found in many scientific and technologicalfields such as biology, physics, control theory, engineering and so on. In the last four decades, basic theory of NFDEs and numerical methods for NFDEs have been widely discussed by many authors because of its importance. On the other hand, due to the difficulty of the research, so far stability analysis of the theoretical and numerical solutions are still limited to linear problems and several classes of special nonlinear problems in literature. In this dissertation, the main object is to extend the research to more general case of nonlinear NFDEs. The main results obtained in this dissertation are as follows:(1) We introduce the test problem classes L_λ^*(α,β,γ,L,Ï„₁,Ï„₂)and D_λ^*(α,β,γ,(?),Ï„₁,Ï„₂) with respect to the initial value problems of nonlinear NFDEs in Banach spaces. A series of stability, contractivity,asymptotic stability and exponential asymptotic stability results of the theoretical solutions to nonlinear NFDEs in Banach spaces are obtained. These results are the basis of numerical stability analysis in this dissertation.It should be pointed out that the aforementioned results can be regarded as an extension of the stability theory of nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces established by S. F. Li [146] in 2005. Other related results we have seen in literature are all limited to the stability analysis of the theoretical solutions to neutral delay differential equations (NDDEs) in a finite dimensional spaces. For linear NDDEs we refer to [32-65], and for nonlinear NDDEs, refer to [93, 95, 97, 105, 109].(2) We study the stability and convergence properties of numerical methods for nonlinear NDDEs in Banach spaces. Sufficient conditions for the stability and asymptotic stability of linearθ-methods for a class of NDDEs with many delays are obtained, and a series of stability results are obtained for a class of linear multistep methods with variable coefficient and for several classes of explicit and diagonal implicit Runge-Kutta methods when applied to nonlinear NDDEs with variable delay. Moreover, convergence results of a class of linear multistep methods with variable coefficient are also obtained.Up to now only a few papers in literature have researched the numerical stability of several classes of special nonlinear NDDEs in finite dimensional spaces, for example, see [83,93,95-109].(3) Using a one-sided Lipschitz condition together with some classical Lipschitz conditions, we give the error estimation of one-leg methods and waveform relaxation methods (WRM) for nonlinear NDDEs with variable delay in a finite-dimensional space. We consider three different approaches to approximating neutral term, prove that a one-leg method with linear interpolation is E (or EB)-convergent of order p if and only if it is A-stable and consistent of order p in the classical sense for ODEs, where p = 1,2. We also give the convergence results on waveform relaxation methods. Several numerical tests are given that confirm the theoretical results mentioned above. (4) A series of stability and asymptotic stability criteria of G(c, p)-algebraically stable one-leg methods and (k, l)-algebraically stable Runge-Kutta methods for a class of nonlinear neutral delay integro-differential equations (NDIDEs) are obtained. Using a one-sided Lispschitz condition, we also obtain the convergence results of G-stable one-leg methods and algebraically stable Runge-Kutta methods for the class of nonlinear NDIDEs. We have done a series of numerical experiments which confirm the theoretical results mentioned above.As far as we know, there are a few papers dealt with the linear numerical stability for NDIDEs (see [63-65]). In 2006, Y. X. Yu and S. F. Li [98], Y. X. Yu, L. P. Wen and S. F. Li [103] discussed the stability of Runge-Kutta methods for another two classes of nonlinear NDIDEs. We would like to point out that nonlinear delay integro-differential equations (DIDEs) is the special case of the nonlinear NDIDEs considered in this dissertation. Applied the results obtained in the present paper, the corresponding results are more general and deeper than the related existing results in literature (see Chapter 6, Section 3 and Section 4).(5) Some sufficient conditions for the dissipativity of the solutions to neutral differential equationswith piecewise constant delay and bounded variable delay are obtained. For neutral differential equations with piecewise constant delay, we prove that under one of the following two conditions1. A^-1 exists and |1 - b^TA^-1e| < 1;2. a_si = b_i, i = 1,2,…,s,a DJ- irreducible, algebraically stable Runge-Kutta method is (weakly) E(λ)- dissipative. By making use of some new techniques, the finite-dimensional dissipativity and the infinite-dimensional dissipativity results of DJ- irreducible and algebraically stable Runge-Kutta methods for neutral differential equations with bounded variable delay are obtained.There are a great deal of results on the dissipativity of the solutions to ordinary differential equations (ODEs) and VFDEs and on the dissipativity of numerical methods for them (for example, see [125-144]). In 2007, Z. Cheng and C. M. Huang [145] gave the sufficient conditions for the dissipativity of the solutions to NDDEs of Hale type and discussed the dissipativity of a class of linear multistep methods. It should be pointed out that even for non-neutral DDEs, the results obtained in this dissertation are more general and deeper than the related existing results in literature (see Chapter 6, Section 3).