A Numerical Method for Fractional Integral With Applications
-
摘要: 提出了一种只需要存储部分历史数据的分数积分的数值计算方法,并给出了误差估计。这种方法可对包含分数积分和分数导数的积分-微分方程进行较长时间的数值计算,克服了存储全部历史数据的困难,并能对计算误差进行控制。作为应用,给出了具有分数导数型本构关系的粘弹性Timoshenko梁的动力学行为研究的控制方程,利用分离变量法讨论梁在简谐激励作用下的动力响应,然后用新提出的数值方法对控制方程进行数值计算,数值计算结果和理论结果进行了比较,它们比较吻合。
-
关键词:
- 分数微积分 /
- 数值计算方法 /
- 分数导数型本构关系 /
- 弱奇异性Volterra积分-微分方程
Abstract: A new numerical method for the liactional integral that only stores part history data is preseated, and its discretization error is estimated.The method can be used to solve the integno-diffemntial equation including fiactional integral or fractional derivative in a long history.The difficulty of storing all history data is overcoane and the error can be controlled. As application, motion equations goverring the dynandcal behavior of a viscoelastic Timoshenko beam with fractional derivative constitutiverelation are gniven.The dynamical response of the beam subjected to a periodic excitation is studied by using the separation variables metiwd. Then the new numerical method is used to solve a class of wealdy singular Voltena integro-differential equations which are applied to descaibe the dynamical behavior of viscoelastic beams with fractional derivative constitutive relations. The analytical and unmeiical results are compared.It is foiurd that they are very close. -
[1] Ross B.A Brief History and Exposition of the Fundamental Theory of Fractional Calculus[M].Lecture Notes in Math,Vol 457,New York:Springer-Verlag,1975,40-130. [2] Samko S G,Kilbas A A,Marichev O L.Fractional Integrals and Derivatives:Theory and Application[M].New York:Gordon and Breach Science Publishers,1993,24-56,120-140. [3] Gemant A.On fractional differences[J].Phil Mag,1938,25(1):92-96. [4] Delbosco D,Rodino L.Existence and uniqueness for a nonlinear fractional differential equation[J].J Math Anal Appl,1996,204(4):609-625. [5] Koeller R C.Applications of the fractional calculus to the theory of viscoelasticity[J].J Appl Mech,1984,51(2):294-298. [6] Bagley R L,Torvik P J.On the fractional calculus model of viscoelasticity behavior[J].J Rheology,1986,30(1):133-155. [7] Rossikhin Y A,Shitikova M V.Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solid[J].Appl Mech Rev,1997,50(1):15-67. [8] Enelund M,Mahler L,Runesson K,et al.Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws[J].Int J Solids Strut,1999,36(18):2417-2442. [9] Enelund M,Olsson P.Damping described by fading memory-analysis and application to fractional derivative models[J].Int J Solids Strut,1999,36(5):939-970. [10] Argyris J.Chaotic vibrations of a nonlinear viscoelastic beam[J].Chaos Solitons Fractals,1996,7(1):151-163. [11] 程昌钧,张能辉.横向周期载荷作用的粘弹性矩形板的混沌和超混沌运动[J].力学学报,1998,30(6):690-699. [12] Akoz Y,Kadioglu F.The mixed finite element method for the quasi-static and dynamic analysis of viscoelastic Timoshenko beams[J].Int J Numer Mech Engng,1999,44(5):1909-1932. [13] Suire G,Cederbaum G.Periodic and chaotic behavior of viscoelastic nonlinear (elastica) bars under harmonic excitations[J].Int J Mech Sci,1995,37(2):753-772. [14] 陈立群,程昌钧.非线性粘弹性梁的动力学行为[J].应用数学和力学,2000,21(9):897-902. [15] Atkinson K E.An Introduction to Numerical Analysis[M].London:John Wiley & Sons,1978,120-128. [16] Timoshenko S,Gere J.材料力学[M].胡大礼译.北京:科学出版社,1978,230-236. [17] Makris N.Three-dimensional constitutive viscoelastic law with fractional order time derivatives[J].J Rheology,1997,41(5):1007-1020. [18] 刘延柱,陈文良,陈立群.振动力学[M].北京:高等教育出版社,1998,143-147. [19] 杨挺青.粘弹性力学[M].武汉:华中理工大学出版社,1990,55-102.
点击查看大图
计量
- 文章访问数: 2590
- HTML全文浏览量: 99
- PDF下载量: 894
- 被引次数: 0