A Numerical Method for Fractional Integral With Applications

ZHU Zheng-you; LI Gen-guo; CHENG Chang-jun

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Article Navigation > Applied Mathematics and Mechanics > 2003 > 24(4): 331-341

ZHU Zheng-you, LI Gen-guo, CHENG Chang-jun. A Numerical Method for Fractional Integral With Applications[J]. Applied Mathematics and Mechanics, 2003, 24(4): 331-341.

Citation:

ZHU Zheng-you, LI Gen-guo, CHENG Chang-jun. A Numerical Method for Fractional Integral With Applications[J]. Applied Mathematics and Mechanics, 2003, 24(4): 331-341.

Citation:

ZHU Zheng-you, LI Gen-guo, CHENG Chang-jun. A Numerical Method for Fractional Integral With Applications[J]. Applied Mathematics and Mechanics, 2003, 24(4): 331-341.

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A Numerical Method for Fractional Integral With Applications

1.
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P. R. China;
2.
Department of Mathematics, Shanghai University, Shanghai 200072, P. R. China;
3.
Department of Mechanics, Shanghai University, Shanghai 200072, P.R. China;
4.
Shanghai Supercomputer Center, Shanghai 201203, P. R. China

Received Date: 2001-10-30
Rev Recd Date: 2003-01-06
Publish Date: 2003-04-15

Abstract

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.

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References(19)

References

[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.