A NUMERICAL METHOD FOR FRACTIONAL INTEGRAL WITH APPLICATIONS

Applied Mathematics and Mechanics (English Edition) ›› 2003, Vol. 24 ›› Issue (4): 373-384.

• 论文 • 下一篇

A NUMERICAL METHOD FOR FRACTIONAL INTEGRAL WITH APPLICATIONS

朱正佑^1,2, 李根国⁴, 程昌钧^1,3

1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P. R. China;
2. Department of Mathematics, Shanghai University, Shanghai 200436, P. R. China;
3. Department of Mechanics, Shanghai University, Shanghai 200436, P. R. China;
4. Shanghai Supercomputer Center, Shanghai 201203, P. R. China

收稿日期:2001-10-30 修回日期:2003-01-06 出版日期:2003-04-18 发布日期:2003-04-18
基金资助:
the National Natural Science Foundation of China (60273048):the Science Foundation of Shanghai Municipal Commission of Education (99A01):the Science Foundation of Shanghai Municipal Commission of Sicences and Technology (98JC14032)

A NUMERICAL METHOD FOR FRACTIONAL INTEGRAL WITH APPLICATIONS

ZHU Zheng-you^1,2, LI Gen-guo⁴, CHENG Chang-jun^1,3

1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P. R. China;
2. Department of Mathematics, Shanghai University, Shanghai 200436, P. R. China;
3. Department of Mechanics, Shanghai University, Shanghai 200436, P. R. China;
4. Shanghai Supercomputer Center, Shanghai 201203, P. R. China

Received:2001-10-30 Revised:2003-01-06 Online:2003-04-18 Published:2003-04-18
Supported by:
the National Natural Science Foundation of China (60273048):the Science Foundation of Shanghai Municipal Commission of Education (99A01):the Science Foundation of Shanghai Municipal Commission of Sicences and Technology (98JC14032)

摘要/Abstract

摘要： A new numerical method for the fractional integral that only stores part history data is presented, and its discretization error is estimated. The method can be used to solve the integro-differential equation including fractional integral or fractional derivative in a long history. The difficulty of storing all history data is overcome and the error can be controlled. As application,motion equations governing the dynamical behavior of a viscoelastic Timoshenko beam with fractional derivative constitutive relation are given. The dynamical response of the beam subjected to a periodic excitation is studied by using the separation variables method. Then the new numerical method is used to solve a class of weakly singular Volterra integro-differential equations which are applied to describe the dynamical behavior of viscoelastic beams with fractional derivative constitutive relations. The analytical and unmerical results are compared. It is found that they are very close.

Abstract: A new numerical method for the fractional integral that only stores part history data is presented, and its discretization error is estimated. The method can be used to solve the integro-differential equation including fractional integral or fractional derivative in a long history. The difficulty of storing all history data is overcome and the error can be controlled. As application,motion equations governing the dynamical behavior of a viscoelastic Timoshenko beam with fractional derivative constitutive relation are given. The dynamical response of the beam subjected to a periodic excitation is studied by using the separation variables method. Then the new numerical method is used to solve a class of weakly singular Volterra integro-differential equations which are applied to describe the dynamical behavior of viscoelastic beams with fractional derivative constitutive relations. The analytical and unmerical results are compared. It is found that they are very close.

中图分类号:

朱正佑;李根国;程昌钧. A NUMERICAL METHOD FOR FRACTIONAL INTEGRAL WITH APPLICATIONS[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(4): 373-384.

ZHU Zheng-you;LI Gen-guo;CHENG Chang-jun. A NUMERICAL METHOD FOR FRACTIONAL INTEGRAL WITH APPLICATIONS[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(4): 373-384.

参考文献

[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] CHENG Chang-jun, ZHANG Neng-hui. Chaos and hyperchaos motion of viscoelastic rectangular plates under a transverse periodic load [J]. Acta Mechanica Sinica, 1998, 30(6):690-699. (in Chinese).
[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] CHEN Li-qun, CHENG Chang-jun. Dynamical behavior of nonlinear viscoelastic beams [J]. Applied Mathematics and Mechanics (English Edition ), 2000, 21(9):995-1001..
[15] Atkinson K E. An Introduction to Numerical Analysis[M]. London:John Wiley & Sons, 1978, 120-128.
[16] Timoshenko S, Gere J. Mechanics of Materials [M]. Nostrand Reinhold Company, 1972..
[17] Makris N. Three-dimensional constitutive viscoelastic law with fractional order time derivatives[J]. J Rheology, 1997, 41(5):1007-1020.
[18] LIU Yan-zhu, CHEN Wen-liang, CHEN Li-qun. Mechanics of Vibrations [M]. Beijing:Advanced Educational Press, 1998, 143-147. (in Chinese).
[19] YANG Ting-qing. Viscoelastic Mechanics [M]. Wuhan:Huazhong University of Science and Technology Press, 1990, 55-102. (in Chinese).

A NUMERICAL METHOD FOR FRACTIONAL INTEGRAL WITH APPLICATIONS

A NUMERICAL METHOD FOR FRACTIONAL INTEGRAL WITH APPLICATIONS

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

编辑推荐

Metrics

本文评价

[1]	朱正佑;李根国;程昌钧. QUASI-STATIC AND DYNAMICAL ANALYSIS FOR VISCOELASTIC TIMOSHENKO BEAM WITH FRACTIONAL DERIVATIVE CONSTITUTIVE RELATION[J]. Applied Mathematics and Mechanics (English Edition), 2002, 23(1): 1-12.
[2]	根国;朱正佑;程昌钧. DYNAMICAL STABILITY OF VISCOELASTIC COLUMN WITH FRACTIONAL DERIVATIVE CONSTITUTIVE RELATION[J]. Applied Mathematics and Mechanics (English Edition), 2001, 22(3): 294-303.