Banach空间中非线性泛函微分与泛函方程Runge-Kutta法的稳定性分析
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摘要
泛函微分方程(FDEs)在生物学、物理学、化学、经济学、控制理论等众多领域有广泛应用。由于其理论解很难获得，只能用数值方法进行近似计算，因而其算法理论的研究具有毋庸置疑的重要性。近几十年，众多学者对问题的适定性及数值方法的稳定性和收敛性进行了大量研究，取得了丰硕成果。这些研究大多局限于讨论有限维内积空间中的初值问题，是以内积范数和单边Lipschitz条件为基础的。然而，科学与工程技术中还存在大量刚性问题，尽管问题本身是整体良态的，但当使用内积范数时，其最小单边Lipschitz常数却不可避免地取非常巨大的正值。因此有必要突破内积范数和单边Lipschitz常数的局限，直接在Banach空间研究数值方法的理论。在此方面，文立下、王晚生等做了许多有益的工作，取得了一系列研究成果。
泛函微分与泛函方程(FDFEs)是较泛函微分方程更广泛的一类混合系统，是由泛函微分方程与泛函方程耦合而成，其理论解和数值方法的研究更具复杂性，目前仅有少量文献在内积空间基于Lipschitz条件对数值方法的稳定性进行了研究。有鉴于此，本文直接在Banach空间中对一类非线性泛函微分与泛函方程研究Runge—Kutta法的数值稳定性，获得了方法稳定的条件，数值试验也验证了所获理论的正确性。
Functional Di?erential Equations(FDEs) arise widely in physics, biology,chemistry, economics, control theory and so on. Their exact solutions are hardlyobtained. Thus, it is meaningful to investigate the theory and application ofnumerical methods for FDEs. In the last few decades, the theory of numericalmethods for FDEs have been widely discussed by many authors, and a seriesof stability and convergence results have been obtained. The existing resultsfor FDEs are mainly based on the inner product and corresponding norm inEuclidean space of finite dimension. But there exists lots of sti? problems inscience and engineering, it may happen that the one-sided Lipschitz constantis very large. Therefore, it is urgent and meaningful to break the restriction ofthe inner product and the corresponding norm to study the numerical theory inBanach space. In this field, Liping Wen and Wansheng Wang have done a lotof work and obtained a series of results.
Functional di?erential and functional equations(FDFEs) are more complexthan FDEs. They are a class of hybird problem formed by functional di?erentialequations and functional equations. In the present paper, we study the numericalstability of Runge-Kutta methods for a kind of nonlinear FDFEs in Banachspace. The stability conditions of the methods are derived. Numerical testshave given to confirm the theoretical results in the end.
泛函微分与泛函方程(FDFEs)是较泛函微分方程更广泛的一类混合系统，是由泛函微分方程与泛函方程耦合而成，其理论解和数值方法的研究更具复杂性，目前仅有少量文献在内积空间基于Lipschitz条件对数值方法的稳定性进行了研究。有鉴于此，本文直接在Banach空间中对一类非线性泛函微分与泛函方程研究Runge—Kutta法的数值稳定性，获得了方法稳定的条件，数值试验也验证了所获理论的正确性。
Functional Di?erential Equations(FDEs) arise widely in physics, biology,chemistry, economics, control theory and so on. Their exact solutions are hardlyobtained. Thus, it is meaningful to investigate the theory and application ofnumerical methods for FDEs. In the last few decades, the theory of numericalmethods for FDEs have been widely discussed by many authors, and a seriesof stability and convergence results have been obtained. The existing resultsfor FDEs are mainly based on the inner product and corresponding norm inEuclidean space of finite dimension. But there exists lots of sti? problems inscience and engineering, it may happen that the one-sided Lipschitz constantis very large. Therefore, it is urgent and meaningful to break the restriction ofthe inner product and the corresponding norm to study the numerical theory inBanach space. In this field, Liping Wen and Wansheng Wang have done a lotof work and obtained a series of results.
Functional di?erential and functional equations(FDFEs) are more complexthan FDEs. They are a class of hybird problem formed by functional di?erentialequations and functional equations. In the present paper, we study the numericalstability of Runge-Kutta methods for a kind of nonlinear FDFEs in Banachspace. The stability conditions of the methods are derived. Numerical testshave given to confirm the theoretical results in the end.
引文
[1]Y Kuang Delay differential equations with applications in population dynam—ics[M]New"York：Academic press，1993
[2]李森林，温立志泛函微分方程[Ml_长沙：湖南科学技术出皈社，1985
[3]J Hale Theory of functional differential equations[M]New York：springer—verlag，1997
[4]郑祖庥泛函微分方程理论[Ml_合肥：安徽教育出皈社，1992
[5]秦元勋，刘水清，王联，郑祖庥带有时滞性的动力系统的运动稳定性(第二版)[M]北京：科学出皈社，1989
[6]V L Barwell Special stability problems for functional equations[J]BIT，1975，15：130—135
[7]M Z Liu，M N Spijker The stability of the 0 methods in the numerical solutionof delay differential equations[J]IMA J Numer Anal．1990．10：31—48
[8]D S Watanable，M G Roth The stability of the difference％rmulas for delaydifferential equations[J]SIAM J Numer Anal，1985，22(1)：132—145
[9]H J Tian，J X Kuang The numercial stability of linear multistep methods fordelay differential equations with equations with many delays[J]SIAM J NumerAnal，1996，33：883—889
[10]K J in’t Hout A new"interpolation proceduref for adapting Runge-Kutta meth—ods to delay differential equations[J]BIT，1992，32：634—649
[11]K J in’t Hout The stability of 0-methods for systems of for delay differentialequations[J]Appl Numer Math，1994，10：323—334
[12]K J in’t Hout Stability analysis of Runge-Kutta methods for systems of dalaydifferential equations[J]IMA J Numer Anal，1997，17：17—27
[13]T Koto A stability properties of A-stable natural Runge-Kutta methods forsystems of delay differentional equations[J]BIT，1994，34：262—267
[14]H J Tian，J X Kuang The stability of linear nmltistep methods for systemsof delay differential equations[J]Numer Math A Journal of Chniese Universi—ties(English Series)．1995．4：10—16
[15]张诚坚泛函微分方程数值解的稳定性与D一收敛性[Dl_博士学位论文，长沙：湖南大学，1998
[16]L Torelli Stability of numerical methods for delay differential equations[J]JComput Appl Math，1989，25：15—26
[17]C M Huang，H Y Fu，S F Li，G N Chen Stability analysis of Runge-Kuttamethods for nonlinear delay differential equations[J]BIT，1999，39：270—280
[18]C M Huang，H Y Fu，S F Li，G N Chen Stability and error analysis ofone—leg methods for nonlinear delay differential equations[J]J Comput ApplMath，1999，103：263—279
[19]C M Huang，S F Li，H Y Fu，G N Chen Nonlinear stability of general linearmethods for delay differential equations[J]BIT，2002，42：380—392
[20]M Zennaro Asymptotic stability analysis of Runge-Kutta methods for nonlinearsystems of delay differential equations[J]Numer Math，1997，77：549—563
[21]A Bellen Contractivity ofcontinuous Runge-Kutta methods for delay differentialequations[J]Appl Numer Math 1997，24：219—232
[22]李寿佛Banach空间中非线性刚性Volterra泛函微分方程的稳定性分析中国科学(A辑)，2005，35(3)：286—301
[23]S F Li B—theory of Runge-Kutta methods for stiff Volterra functional differentialequations[J]Science In China(Series A)，2003，5：662—674
[24]S F Li B—theory of general linear methods for Volterra functional differentialequations[J]Appl Number Math，2005，53：57—72
[25]A Bellen，Z Jackiewicz，M Zennaro Stability analysis of one-step methods forneutral delay differential equations[J]Numer Math，1988，52：605—619
[26]Y K Liu Runge—Kutta-collocation methods for system of functional—differentialand functional equations[J]Adv Comput Math，1999，11：315—329
[27]R Vermiglio，L Torelli A stable numerical approach for implicit non—linear neu—tral delay differential equations[J]BIT，2003，43：195—215
[28]C J Zhang Nonlinear stability of natural Runge-Kutta methods for neutraldelay differential equations[J]J Comput Math，2002，20：583—590
[29]A Bellen，N Guglielmi，M Zennaro Numerical stability of nonlinear delay differ—ential equations of neutral type[J]J Comput Appl Math，2000，125：251—263
[30]王晚生，李寿佛非线性中立型延迟微分方程稳定性分析[J]计算数学，2004，26：303—314
[31]王晚生非线性中立型泛函微分方程数值分析[Dl_博士毕业论文，湘潭：湘潭大学，2008
[32]C M Huang，Q S Chang Stability analysis of numerical methods for sys—terns of functional—differential and functional equations[J]Comput Math Appl，2003，44：717—729
[33]C M Huang，Q S Chang Stability analysis of Runge-Kutta methods for systemsof functional—differential and functional equations[J]Progr Natur Sci，2001，11：568—572
[34]S Q Gan，W M Zheng Stability of nmltistep Runge—Kutta methods for systemsof functional—differential and functional equations[J]Appl Math Letters，2004，17：585—590
[35]S Q Gan Asymptotic stability of Rosenbrock methods for systems of functionaldifferential and functional equations[J]Math Comput Modelling，2006，44：144—150
[36]S Q Gan，W M Zheng Stability of nmltistep Runge—Kutta methods for systemsof functional—differential and functional equations[J]Appl Math Letters，2004，17：585—590
[37]Y X Yu，S F Li Stability analysis of nonlinear functional differential andfunctional equations[J]Appl Math Letters，2009，22：789—791
[38]江春华泛函微分与泛函方程稳定性分析[Dl_硕士毕业论文，湘潭：湘潭大学，2010
[39]文立下抽象空间中非线性volterrra泛函微分方程的数值稳定性[D]博士毕业论文，湘潭：湘潭大学，2005
[40]李寿佛刚性微分方程算法理论[M]，长沙：湖南科学技术出版社，1997
[2]李森林，温立志泛函微分方程[Ml_长沙：湖南科学技术出皈社，1985
[3]J Hale Theory of functional differential equations[M]New York：springer—verlag，1997
[4]郑祖庥泛函微分方程理论[Ml_合肥：安徽教育出皈社，1992
[5]秦元勋，刘水清，王联，郑祖庥带有时滞性的动力系统的运动稳定性(第二版)[M]北京：科学出皈社，1989
[6]V L Barwell Special stability problems for functional equations[J]BIT，1975，15：130—135
[7]M Z Liu，M N Spijker The stability of the 0 methods in the numerical solutionof delay differential equations[J]IMA J Numer Anal．1990．10：31—48
[8]D S Watanable，M G Roth The stability of the difference％rmulas for delaydifferential equations[J]SIAM J Numer Anal，1985，22(1)：132—145
[9]H J Tian，J X Kuang The numercial stability of linear multistep methods fordelay differential equations with equations with many delays[J]SIAM J NumerAnal，1996，33：883—889
[10]K J in’t Hout A new"interpolation proceduref for adapting Runge-Kutta meth—ods to delay differential equations[J]BIT，1992，32：634—649
[11]K J in’t Hout The stability of 0-methods for systems of for delay differentialequations[J]Appl Numer Math，1994，10：323—334
[12]K J in’t Hout Stability analysis of Runge-Kutta methods for systems of dalaydifferential equations[J]IMA J Numer Anal，1997，17：17—27
[13]T Koto A stability properties of A-stable natural Runge-Kutta methods forsystems of delay differentional equations[J]BIT，1994，34：262—267
[14]H J Tian，J X Kuang The stability of linear nmltistep methods for systemsof delay differential equations[J]Numer Math A Journal of Chniese Universi—ties(English Series)．1995．4：10—16
[15]张诚坚泛函微分方程数值解的稳定性与D一收敛性[Dl_博士学位论文，长沙：湖南大学，1998
[16]L Torelli Stability of numerical methods for delay differential equations[J]JComput Appl Math，1989，25：15—26
[17]C M Huang，H Y Fu，S F Li，G N Chen Stability analysis of Runge-Kuttamethods for nonlinear delay differential equations[J]BIT，1999，39：270—280
[18]C M Huang，H Y Fu，S F Li，G N Chen Stability and error analysis ofone—leg methods for nonlinear delay differential equations[J]J Comput ApplMath，1999，103：263—279
[19]C M Huang，S F Li，H Y Fu，G N Chen Nonlinear stability of general linearmethods for delay differential equations[J]BIT，2002，42：380—392
[20]M Zennaro Asymptotic stability analysis of Runge-Kutta methods for nonlinearsystems of delay differential equations[J]Numer Math，1997，77：549—563
[21]A Bellen Contractivity ofcontinuous Runge-Kutta methods for delay differentialequations[J]Appl Numer Math 1997，24：219—232
[22]李寿佛Banach空间中非线性刚性Volterra泛函微分方程的稳定性分析中国科学(A辑)，2005，35(3)：286—301
[23]S F Li B—theory of Runge-Kutta methods for stiff Volterra functional differentialequations[J]Science In China(Series A)，2003，5：662—674
[24]S F Li B—theory of general linear methods for Volterra functional differentialequations[J]Appl Number Math，2005，53：57—72
[25]A Bellen，Z Jackiewicz，M Zennaro Stability analysis of one-step methods forneutral delay differential equations[J]Numer Math，1988，52：605—619
[26]Y K Liu Runge—Kutta-collocation methods for system of functional—differentialand functional equations[J]Adv Comput Math，1999，11：315—329
[27]R Vermiglio，L Torelli A stable numerical approach for implicit non—linear neu—tral delay differential equations[J]BIT，2003，43：195—215
[28]C J Zhang Nonlinear stability of natural Runge-Kutta methods for neutraldelay differential equations[J]J Comput Math，2002，20：583—590
[29]A Bellen，N Guglielmi，M Zennaro Numerical stability of nonlinear delay differ—ential equations of neutral type[J]J Comput Appl Math，2000，125：251—263
[30]王晚生，李寿佛非线性中立型延迟微分方程稳定性分析[J]计算数学，2004，26：303—314
[31]王晚生非线性中立型泛函微分方程数值分析[Dl_博士毕业论文，湘潭：湘潭大学，2008
[32]C M Huang，Q S Chang Stability analysis of numerical methods for sys—terns of functional—differential and functional equations[J]Comput Math Appl，2003，44：717—729
[33]C M Huang，Q S Chang Stability analysis of Runge-Kutta methods for systemsof functional—differential and functional equations[J]Progr Natur Sci，2001，11：568—572
[34]S Q Gan，W M Zheng Stability of nmltistep Runge—Kutta methods for systemsof functional—differential and functional equations[J]Appl Math Letters，2004，17：585—590
[35]S Q Gan Asymptotic stability of Rosenbrock methods for systems of functionaldifferential and functional equations[J]Math Comput Modelling，2006，44：144—150
[36]S Q Gan，W M Zheng Stability of nmltistep Runge—Kutta methods for systemsof functional—differential and functional equations[J]Appl Math Letters，2004，17：585—590
[37]Y X Yu，S F Li Stability analysis of nonlinear functional differential andfunctional equations[J]Appl Math Letters，2009，22：789—791
[38]江春华泛函微分与泛函方程稳定性分析[Dl_硕士毕业论文，湘潭：湘潭大学，2010
[39]文立下抽象空间中非线性volterrra泛函微分方程的数值稳定性[D]博士毕业论文，湘潭：湘潭大学，2005
[40]李寿佛刚性微分方程算法理论[M]，长沙：湖南科学技术出版社，1997