On Finite Part Integrals and Hadamard-Type Fractional Derivatives

[+] Author and Article Information
Li Ma

School of Mathematics,
Hefei University of Technology,
Hefei 230009, China
e-mail: mali20062787@163.com

Changpin Li

Department of Mathematics,
Shanghai University,
Shanghai 200444, China
e-mail: lcp@shu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 18, 2017; final manuscript received August 29, 2017; published online July 26, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(9), 090905 (Jul 26, 2018) (7 pages) Paper No: CND-17-1317; doi: 10.1115/1.4037930 History: Received July 18, 2017; Revised August 29, 2017

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Kilbas, A. A. , Srivastava, H. M. , and Trujillo, J. J. , 2006, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, The Netherlands.
Li, C. P. , and Ma, L. , 2016, “ Lyapunov-Schmidt Reduction for Fractional Differential Systems,” ASME J. Comput. Nonlinear Dyn., 11(5), p. 051022. [CrossRef]
Li, C. P. , and Zeng, F. H. , 2015, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, Boca Raton, FL.
Machado, J. A. T. , Silva, M. F. , Barbosa, R. S. , Jesus, I. S. , Reis, C. M. , Marcos, M. G. , and Galhano, A. F. , 2010, “ Some Applications of Fractional Calculus in Engineering,” Math. Probl. Eng., 2010, p. 639801.
Ma, L. , and Li, C. P. , 2016, “ Center Manifold of Fractional Dynamical System,” ASME J. Comput. Nonlinear Dyn., 11(2), p. 021010. [CrossRef]
Li, C. P. , Yi, Q. , and Kurths, J. , 2017, “ Fractional Convection,” ASME J. Comput. Nonlinear Dyn., epub.
Pinto, C. M. A. , and Machado, J. A. T. , 2013, “ Fractional Model for Malaria Transmission Under Control Strategies,” Comput. Math. Appl., 66(5), pp. 908–916. [CrossRef]
Pinto, C. M. A. , 2017, “ Persistence of Low Levels of Plasma Viremia and of the Latent Reservoir in Patients Under ART: A Fractional-Order Approach,” Commun. Nonlinear Sci. Numer. Simul., 43, pp. 251–260. [CrossRef]
Pinto, C. M. A. , and Carvalho, A. R. M. , 2017, “ The Role of Synaptic Transmission in a HIV Model With Memory,” Appl. Math. Comput., 292, pp. 76–95.
Jarad, F. , Abdeljawad, T. , and Baleanu, D. , 2017, “ On the Generalized Fractional Derivatives and Their Caputo Modification,” J. Nonlinear Sci. Appl., 10(5), pp. 2607–2619. [CrossRef]
Baleanu, D. , Wu, G. C. , and Zeng, S. D. , 2017, “ Chaos Analysis and Asymptotic Stability of Generalized Caputo Fractional Differential Equations,” Chaos Solitons Fractals, 102, pp. 99–105. [CrossRef]
Kilbas, A. A. , 2001, “ Hadamard-Type Fractional Calculus,” J. Korean Math. Soc., 38(6), pp. 1191–1204. http://citeseerx.ist.psu.edu/viewdoc/download?doi=
Hadamard, J. , 1892, “ Essai Sur l’étude des Fonctions Données Par Leur Développment de Taylor,” J. Math. Pures Appl., 8(4), pp. 101–186.
Gong, Z. Q. , Qian, D. L. , Li, C. P. , and Guo, P. , 2012, “ On the Hadamard Type Fractional Differential System,” Fractional Dynamics and Control, D. Baleanu , J. Machado, and A. Luo, eds., Springer, New York, pp. 159–171. [CrossRef]
Pooseh, S. , Almeida, R. , and Torres, D. F. M. , 2012, “ Expansion Formulas in Terms of Integer-Order Derivatives for the Hadamard Fractional Integral and Derivative,” Numer. Funct. Anal. Optim., 33(3), pp. 301–319. [CrossRef]
Jarad, F. , Abdeljawad, T. , and Baleanu, D. , 2012, “ Caputo-Type Modification of the Hadamard Fractional Derivatives,” Adv. Differ. Equations, 2012(1), pp. 1–8. [CrossRef]
Ma, L. , and Li, C. P. , 2017, “ On Hadamard Fractional Calculus,” Fractals, 25(3), p. 1750033. [CrossRef]
Hadamard, J. , 1923, Lectures on Cauchy's Problem in Partial Differential Equations, Yale University Press, New Haven, CT.
Ioakimidis, N. I. , 1982, “ Application of Finite-Part Integrals to the Singular Integral Equations of Crack Problems in Plane and Three-Dimensional Elasticity,” Acta Mech., 45(1), pp. 31–47. [CrossRef]
Elliott, D. , 1993, “ An Asymptotic Analysis of Two Algorithms for Certain Hadamard Finite-Part Integrals,” IMA J. Numer. Anal., 13(3), pp. 3273–3286. [CrossRef]
Samko, S. G. , Kilbas, A. A. , and Marichev, O. I. , 1993, Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Philadelphia, PA.
Monegato, G. , 2009, “ Definitions, Properties and Applications of Finite-Part Integrals,” J. Comput. Appl. Math., 229(2), pp. 425–439. [CrossRef]
Kutt, H. R. , 1975, “ The Numerical Evaluation of Principal Value Integrals by Finite-Part Integration,” Numer. Math., 24(3), pp. 205–210. [CrossRef]
Paget, D. F. , 1981, “ The Numerical Evaluation of Hadamard Finite-Part Integrals,” Numer. Math., 36(4), pp. 447–453. [CrossRef]
George, T. , and George, D. , 1990, “ Gauss Quadrature Rules for Finite Part Integrals,” Int. J. Numer. Methods Eng., 30(1), pp. 13–26. [CrossRef]
Diethelm, K. , 1997, “ Generalized Compound Quadrature Formulae for Finite-Part Integrals,” IMA J. Numer. Anal., 17(3), pp. 479–493. [CrossRef]
Sun, W. W. , and Wu, J. M. , 2008, “ Interpolatory Quadrature Rules for Hadamard Finite-Part Integrals and Their Superconvergence,” IMA J. Numer. Anal., 28(3), pp. 580–597. [CrossRef]
Wu, J. M. , Dai, Z. H. , and Zhang, X. P. , 2010, “ The Superconvergence of the Composite Midpoint Rule for the Finite-Part Integral,” J. Comput. Appl. Math., 233(8), pp. 1954–1968. [CrossRef]
Zeng, F. H. , Mao, Z. P. , and Karniadakis, G. E. , 2017, “ A Generalized Spectral Collocation Method With Tunable Accuracy for Fractional Differential Equations With End-Point Singularities,” SIAM J. Sci. Comput., 39(1), pp. A360–A383. [CrossRef]
Butzer, P. L. , Kilbas, A. A. , and Trujillo, J. J. , 2002, “ Fractional Calculus in the Mellin Setting and Hadamard-Type Fractional Integrals,” J. Math. Anal. Appl., 269(1), pp. 1–27. [CrossRef]
Varberg, D. E. , 1965, “ On Absolutely Continuous Functions,” Am. Math. Mon., 72(8), pp. 831–841. [CrossRef]
Cannarsa, P. , and D'Aprile, T. , 2015, Introduction to Measure Theory and Functional Analysis, Springer International Publishing, Cham, Switzerland. [CrossRef]




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In