解Riemann-Liouville分数阶导数微分方程两点边值问题(英文)  

Solving Two-Point Boundary Value Problems of Fractional Differential Equations with Riemann-Liouville Derivatives

作　　者：聂宁明[1,2] 赵艳敏[3] Salvador Jimenez 李敏[1,2] 唐贻发[1] Luis Vazquez 

机构地区：[1]中国科学院数学与系统科学研究院计算数学与科学工程计算研究所,北京100190 [2]中科院研究生院,北京100190 [3]许昌学院数学科学学院,河南许昌461000 [4]Departamento de Matematica Aplicada TTII, E.T.S.I. Telecomunicacien, Universidad Politecnica de Madrid, 28040-Madrid, Spain [5]Departamento de Matemetica Aplicada, Facultad de Informeitica, Instituto de Matemeitica Interdisciplinar (IMI), Universidad Complutense de Madrid, 28040-Madrid, Spain.

出　　处：《系统仿真学报》2010年第1期20-24,共5页Journal of System Simulation

基　　金：open project(No.47549P0)of the State Key Laboratory of Scientific and Engineering Computing,Chinese Academy of Sciences;National Natural Science Foundation of China(Grant No.10872037);Natural Science Research Project of Henan Province(Grant No.2009A110017);Ministerio de Educación y Ciencia(Spain)(grant MTM2005-05573);Sabbatical Program(SAB2006-0070)of the Spanish Ministry of Education and Science

摘　　要：研究了两类含Riemann-Liouville分数阶导数的分数阶微分方程两点边值问题。理论上,通过引入分数阶Green函数将含有Riemann-Liouville分数阶导数的两点边值问题等价转换成一个积分方程;并用Lipschitz条件和压缩映射原理给出了含有Riemann-Liouville分数阶导数的两点边值问题的解存在唯一的充分条件;数值上,设计了单打靶法,把含Riemann-Liouville分数阶导数的两点边值问题转化为含Riemann-Liouville分数阶导数的初值问题进行求解,并给出了较为精确的数值解。仿真结果表明:单打靶法是数值求解此类分数阶微分方程两点边值问题的有效工具。Two kinds of two-point boundary value problems of fractional differential equations with Riemalm-Liouville derivatives （FBVPs） were studied. Analytically, via fractional Green functions, FBVPs were transformed into equivalent integral equations, and then existence and uniqueness of the solutions were proved according to the Lipschitz conditions and the contractive mapping principle. Numerically, the single shooting methods were designed, and solving FBVPs was transformed into solving initial value problems of fractional differential equations with Riemann-Liouville derivatives （FIVPs） in order to get approximation solutions. Simulation results show that the single shooting methods are natural and efficient in numerically solving these FBVPs.

关 键 词：解的存在唯一性 分数阶微分方程 Riemann-Liouville分数阶导数 单打靶法 两点边值问题 数值仿真 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]