# Fractional Calculus Publications

1. Narahari Achar, John W. Hanneken, Thomas Enck, Ted Clarke, “Dynamics of the Fractional Oscillator”, Physica A 297 (2001) 361-367. http://dx.doi.org/10.1016/S0378-4371(01)00200-X

2. B. N. Narahari Achar, John W. Hanneken, and Ted Clarke, “Response Characteristics of a Fractional Oscillator”, Physica A 309 (2002) 275-288. http://dx.doi.org/10.1016/S0378-4371(02)00609-X

3. B.N. Narahari Achar, John W. Hanneken, “Surprising Non-Scaling Behavior in Fractional Diffusion”, Physica A 322 (2003) 195-200 and 326 (2003) 609-610. http://dx.doi.org/10.1016/S0378-4371(02)01911-8 and http://dx.doi.org/10.1016/S0378-4371(03)00390-X

4. John W. Hanneken, B. N. Narahari Achar, David M. Vaught, Kristine L. Harrington, “A Random Walk Simulation of Fractional Diffusion”, Journal of Molecular Liquids 114 (2004) 153-157 http://dx.doi.org/10.1016/j.molliq.2004.02.013

5. B.N. Narahari Achar, John W. Hanneken, “Fractional Radial Diffusion in a Cylinder”, Journal of Molecular Liquids 114 (2004) 147-151 http://dx.doi.org/10.1016/j.molliq.2004.02.012

6. Ted Clarke, B. N. Narahari Achar, John W. Hanneken, “Mittag-Leffler Functions and Transmission Lines”, Journal of Molecular Liquids 114 (2004) 159-163 http://dx.doi.org/10.1016/j.molliq.2004.02.014

7. B. N. Narahari Achar, John W. Hanneken, T. Clarke, “Damping characteristics of a fractional oscillator”, Physica A 339 (2004) 311-319 http://dx.doi.org/10.1016/j.physa.2004.03.030

8. John W. Hanneken, David M. Vaught and B. N. Narahari Achar, “The Number of Real Zeros of the Single Parameter Mittag-Leffler Function for Parameter Values Between 1 and 2”, Proceedings of IDETC/CIE 2005: 2005 ASME International Design Engineering Technical Conferences & Computer and Information in Engineering Conference held September 24-28, 2005 in Long Beach, California. DETC2005-84172 pp.1-5 ISBN: 0-7918-3766-1

9. B.N. Narahari Achar and John W. Hanneken, “Damping in a Fractional Relaxor-Oscillator Driven by a Harmonic Force”, Proceedings of IDETC/CIE 2005: 2005 ASME International Design Engineering Technical Conferences & Computer and Information in Engineering Conference held September 24-28, 2005 in Long Beach, California. DETC2005-84345 pp.1-7 ISBN: 0-7918-3766-1

10. John W. Hanneken, David M. Vaught and B. N. Narahari Achar, “Enumeration of the Real Zeros of the Mittag-Leffler Function Ea(z), 1 < a < 2 “, in Sabatier, J., Agrawal, O. P.,Tenreiro Machado, J. A. (Eds), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007 pp. 15-26. ISBN: 978-1-4020-6041-0

11. B. N. Narahari Achar and John W. Hanneken, “Dynamic Response of the Fractional Relaxor-Oscillator to a Harmonic Driving Force”, in Sabatier, J., Agrawal, O. P.,Tenreiro Machado, J. A. (Eds), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007 pp. 243-256. ISBN: 978-1-4020-6041-0

12. B. N. Narahari Achar, Tanya Prozny, John W. Hanneken, “Linear Chain of Coupled Fractional Oscillators; Response Dynamics and its Continuum Limit”, Proceedings of the ASME 2007 International Design Engineering Technical Conference & Computers and Information in Engineering Conference held September 4-7, 2007 in Las Vegas, Nevada DETC2007-35403 pp. 1-7

13. B. N. Narahari Achar and John W. Hanneken, “Microscopic formulation of fractional calculus theory of viscoelasticity based on lattice dynamics”, Proceedings of the Third IFAC Workshop on Factional Differentiation and Its Applications, Cankaya University, Ankara, Turkey, November 5-7, 2008.

14. John W. Hanneken, B. N. Narahari Achar, Raymond Puzio, and David Vaught, “Properties of the Mittag-Leffler function for negative alpha”, Proceedings of the Third IFAC Workshop on Factional Differentiation and Its Applications, Cankaya University, Ankara, Turkey, November 5-7, 2008.

15. John W. Hanneken, B. N. Narahari Achar, David M. Vaught, James M. Andrews and Avery T. Carr, “Sums of the zeros of the single parameter Mittag-Leffler function with the parameter equal to one-half”, Proceedings of the ASME 2009 Design Engineering Technical Conferences & Computers and Information in Engineering Conference, SanDiego, CA, August 30 – September 2, 2009, DETC2009-86220 pp. 1-7. ISBN 978-0-7918-3856-3

16. B. N. Narahari Achar and John W. Hanneken, “Response dynamics in the continuum limit of the lattice dynamical theory of viscoelasticity (fractional calculus approach)”, ”, Proceedings of the ASME 2009 Design Engineering Technical Conferences & Computers and Information in Engineering Conference, SanDiego, CA, August 30 – September 2, 2009, DETC2009-86218 pp. 1-7. ISBN 978-0-7918-3856-3

17. John W. Hanneken, B. N. Narahari Achar, Raymond Puzio, and David Vaught, “Properties of the Mittag-Leffler function for negative alpha”, Physica Scripta T136 (2009) 014037 (5pp)

18. B. N. Narahari Achar and John W. Hanneken, “Microscopic formation of fractional calculus theory of viscoelasticity based on lattice dynamics”, Physica Scripta T136 (2009) 014011 (7pp)

19. B. N. Narahari Achar, John W. Hanneken, Nicholas Becker and Matthew Morgan, “Intrinsic dissipation in fractional order dynamics:, Proceedings of the International Workshop on Fractional Differentiation and its Applications (FDA’10) held at University of Extremadura, Badajoz, Spain, October 18-20, 2010.

20. B. N. Narahari Achar and John W. Hanneken, “Fractional order dynamics under gravity: Falling Body and Kepler problems”, Proceedings of the International Workshop on Fractional Differentiation and its Applications (FDA’10) held at University of Extremadura, Badajoz, Spain, October 18-20, 2010.

21. John W. Hanneken, B. N. Narahari Achar, David M. Vaught, Stephan T. Spencer and Trenton R. Ensley , “An alpha-beta phase diagram representation of the zeros and properties of the Mittag-Leffler function”, Proceedings of the International Workshop on Fractional Differentiation and its Applications (FDA’10) held at University of Extremadura, Badajoz, Spain, October 18-20, 2010.

22. John W. Hanneken, James M. Andrews, David M. Vaught, B. N. Narahari Achar and Raymon Puzio, “Mittag-Leffler functions: Their infinite product representations and the sums of their zeros”, Proceedings of the International Workshop on Fractional Differentiation and its Applications (FDA’10) held at University of Extremadura, Badajoz, Spain, October 18-20, 2010.

23. B. N. Narahari Achar and John W. Hanneken, “Microscopic Formulation of Fractional Theory of Viscoelasticity”, in Viscoelasticity – From Theory to Biological Applications, by Juan de Vicente (Ed), Intech Publishing, NY, 2012 ISBN: 978-953-51-0841-2 http://dx.doi.org/10.5772/51493

24. B. N. Narahari Achar, Bradley T. Yale, and John W. Hanneken, “Time Fractional Schrodinger Equation Revisited”, Advances in Mathematical Physics 2013 (2013), 290216 (11 pages) http://dx.doi.org/10.1155/2013/290216

25. John W. Hanneken, B. N. Narahari Achar and David M. Vaught, “An Alpha-Beta Phase Diagram Representation of the Zeros and Properties of the Mittag-Leffler Function”, Advances in Mathematical Physics 2013 (2013), 421685 (13 pages) http://dx.doi.org/10.1155/2013/421685

26. John W. Hanneken and B. N. Narahari Achar, "Finite Series Representation of the Inverse Mittag-Leffler Function", Mathematical Problems in Engineering 2014 (2014), 252393 (18 pages) http://dx.doi.org/10.1155/2014/252393