非线性中立型泛函微分方程数值分析
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摘要
中立型泛函微分方程(NFDEs)常出现于生物学、物理学、控制理论及工程技术等诸多领域，由于其重要性，近四十年来，人们对这类方程的适定性及其数值方法的收敛性和稳定性进行了大量研究；但另一方面，由于其困难性，迄今国内外文献中对于其理论解和数值解的稳定性研究仍局限于线性问题和一些特殊的非线性问题．本文主要目的是试图将这项研究进一步发展到更为一般的中立型非线性问题．所获主要结果如下：
(1)提出了Banach空间中非线性NFDEs试验问题类L_(λ~*)(α，β，γ，L，τ_1，τ_2)和D_(λ~*)(α，β，γ，(?)，τ_1，τ_2)，获得了其理论解的一系列稳定性、收缩性、渐近稳定性及指数渐近稳定性结果．这些结果是本文数值稳定性分析的基础．
应当指出，2005年，李寿佛建立了Banach空间中非线性刚性Volterra泛函微分方程(VFDEs)稳定性的一般理论(参见[146])，本文的上述结果将该理论进一步发展到非线性NFDEs．在国内外其他文献中，迄今主要研究了有限维空间中的中立型延迟微分方程(NDDEs)理论解的稳定性，尚未见到关于Banach空间中一般的非线性NFDEs理论解的研究工作．而且其中大量工作是针对线性中立型问题的，例如可参见[32-65]，仅有Bellen，Guglielmi和Zennaro[93]，张诚坚[95]，Vermiglio和Torelli[97]以及王晚生和李寿佛[105，109]等人研究了一些特殊的非线性NDDEs的理论解的稳定性．
(2)研究了Banach空间中非线性NDDEs数值方法的稳定性和收敛性，给出了求解一类多延迟中立型微分方程的线性θ-方法的稳定性和渐近稳定的充分条件，获得了求解非线性变延迟中立型微分方程的一类变系数线性多步方法及若干类型的显式和对角隐式Runge-Kutta法的一系列数值稳定性结果，对于前者，同时获得了收敛性结果．
此前国内外文献中未见到有关Banach空间中非线性变延迟NDDEs数值方法的稳定性和收敛性研究工作，只有少量文献研究了有限维空间中一些特殊的非线性NDDEs的数值稳定性，例如可参见[83,93,95-109]．
(3)利用一个单边Lipschtiz条件和一些经典Lipschtiz条件，对有限维欧氏空间中求解非线性变延迟中立型微分方程的单支方法和波形松弛方法的误差进行了估计．考虑了中立项的三种不同逼近方式，证明了带线性插值的单支方法是p阶E(或EB)-收敛的当且仅当该方法A-稳定且经典相容阶为p(这里p=1，2)，同时给出了波形松弛方法的收敛性结果，部分解决了Bartoszewski和Kwapisz于2004年提到的单边Lipschitz条件不能应用于NFDEs的问题，为今后开展这方面的研究打开了突破口．数值试验结果验证了所获理论结果的正确性．
(4)获得了求解一类非线性中立型延迟积分微分方程(NDIDEs)的G(c，p)-代数稳定的单支方法和(k，l)-代数稳定的Runge-Kutta法的稳定和渐近稳定的一系列准则．同时利用单边Lipschitz条件，获得了G-稳定单支方法和代数稳定Runge-Kutta方法求解此类型的非线性NDIDEs的收敛性结果．数值试验验证了上述理论结果的正确性．
据我们所知，迄今仅有少数作者研究了线性NDIDEs的数值稳定性(参见[63-65])．2006年，余越昕和李寿佛[98]，余越昕、文立平和李寿佛[103]研究了Runge-Kutta法用于求解另两种类型非线性NDIDEs的稳定性．需要指出的是，非线性延迟积分微分方程(DIDEs)是本文研究的非线性NDIDEs的特殊情形，将本文所获数值稳定性结果应用于DIDEs，相应的结果比已有的结果更为一般和深刻(参见本文第五章第3和第4节)．
(5)获得了Hilbert空间中非线性中立型分片延迟微分方程和非线性中立型变延迟微分方程本身散逸的充分条件．对中立型分片延迟微分方程，证明了一个DJ-不可约的代数稳定的Runge-Kutta方法是(弱)E(λ)-散逸的，只要下列二条件中至少有一个成立：
1．A~(-1)存在，且|1-b~TA~(-1)e|<1；对一般的中立型有界变延迟微分方程，利用一些新的技巧，获得了DJ-不可约的代数稳定的Runge-Kutta方法的有限维散逸性和无限维散逸性结果．
在常微分方程(ODEs)和VFDEs系统本身的散逸性和数值方法的散逸性研究中，已经获得了大量重要研究成果，例如可参见[125-144]．2007年，程珍和黄乘明[145]给出了Hale型中立型延迟微分方程散逸的充分条件，并讨论了一类线性多步方法的散逸性．一般形式的非线性NDDEs本身及数值方法的散逸性研究在国内外文献中尚未见到．值得指出的是，即使对于非中立型的延迟微分方程，本文所获数值散逸性结果也比已有的同类结果更为一般和深刻(参见本文第六章第3节)．
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,τ_1,τ_2)and D_(λ~*)(α,β,γ,(?),τ_1,τ_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~(-1)e| < 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).
(1)提出了Banach空间中非线性NFDEs试验问题类L_(λ~*)(α，β，γ，L，τ_1，τ_2)和D_(λ~*)(α，β，γ，(?)，τ_1，τ_2)，获得了其理论解的一系列稳定性、收缩性、渐近稳定性及指数渐近稳定性结果．这些结果是本文数值稳定性分析的基础．
应当指出，2005年，李寿佛建立了Banach空间中非线性刚性Volterra泛函微分方程(VFDEs)稳定性的一般理论(参见[146])，本文的上述结果将该理论进一步发展到非线性NFDEs．在国内外其他文献中，迄今主要研究了有限维空间中的中立型延迟微分方程(NDDEs)理论解的稳定性，尚未见到关于Banach空间中一般的非线性NFDEs理论解的研究工作．而且其中大量工作是针对线性中立型问题的，例如可参见[32-65]，仅有Bellen，Guglielmi和Zennaro[93]，张诚坚[95]，Vermiglio和Torelli[97]以及王晚生和李寿佛[105，109]等人研究了一些特殊的非线性NDDEs的理论解的稳定性．
(2)研究了Banach空间中非线性NDDEs数值方法的稳定性和收敛性，给出了求解一类多延迟中立型微分方程的线性θ-方法的稳定性和渐近稳定的充分条件，获得了求解非线性变延迟中立型微分方程的一类变系数线性多步方法及若干类型的显式和对角隐式Runge-Kutta法的一系列数值稳定性结果，对于前者，同时获得了收敛性结果．
此前国内外文献中未见到有关Banach空间中非线性变延迟NDDEs数值方法的稳定性和收敛性研究工作，只有少量文献研究了有限维空间中一些特殊的非线性NDDEs的数值稳定性，例如可参见[83,93,95-109]．
(3)利用一个单边Lipschtiz条件和一些经典Lipschtiz条件，对有限维欧氏空间中求解非线性变延迟中立型微分方程的单支方法和波形松弛方法的误差进行了估计．考虑了中立项的三种不同逼近方式，证明了带线性插值的单支方法是p阶E(或EB)-收敛的当且仅当该方法A-稳定且经典相容阶为p(这里p=1，2)，同时给出了波形松弛方法的收敛性结果，部分解决了Bartoszewski和Kwapisz于2004年提到的单边Lipschitz条件不能应用于NFDEs的问题，为今后开展这方面的研究打开了突破口．数值试验结果验证了所获理论结果的正确性．
(4)获得了求解一类非线性中立型延迟积分微分方程(NDIDEs)的G(c，p)-代数稳定的单支方法和(k，l)-代数稳定的Runge-Kutta法的稳定和渐近稳定的一系列准则．同时利用单边Lipschitz条件，获得了G-稳定单支方法和代数稳定Runge-Kutta方法求解此类型的非线性NDIDEs的收敛性结果．数值试验验证了上述理论结果的正确性．
据我们所知，迄今仅有少数作者研究了线性NDIDEs的数值稳定性(参见[63-65])．2006年，余越昕和李寿佛[98]，余越昕、文立平和李寿佛[103]研究了Runge-Kutta法用于求解另两种类型非线性NDIDEs的稳定性．需要指出的是，非线性延迟积分微分方程(DIDEs)是本文研究的非线性NDIDEs的特殊情形，将本文所获数值稳定性结果应用于DIDEs，相应的结果比已有的结果更为一般和深刻(参见本文第五章第3和第4节)．
(5)获得了Hilbert空间中非线性中立型分片延迟微分方程和非线性中立型变延迟微分方程本身散逸的充分条件．对中立型分片延迟微分方程，证明了一个DJ-不可约的代数稳定的Runge-Kutta方法是(弱)E(λ)-散逸的，只要下列二条件中至少有一个成立：
1．A~(-1)存在，且|1-b~TA~(-1)e|<1；对一般的中立型有界变延迟微分方程，利用一些新的技巧，获得了DJ-不可约的代数稳定的Runge-Kutta方法的有限维散逸性和无限维散逸性结果．
在常微分方程(ODEs)和VFDEs系统本身的散逸性和数值方法的散逸性研究中，已经获得了大量重要研究成果，例如可参见[125-144]．2007年，程珍和黄乘明[145]给出了Hale型中立型延迟微分方程散逸的充分条件，并讨论了一类线性多步方法的散逸性．一般形式的非线性NDDEs本身及数值方法的散逸性研究在国内外文献中尚未见到．值得指出的是，即使对于非中立型的延迟微分方程，本文所获数值散逸性结果也比已有的同类结果更为一般和深刻(参见本文第六章第3节)．
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,τ_1,τ_2)and D_(λ~*)(α,β,γ,(?),τ_1,τ_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~(-1)e| < 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).
引文
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