A study on nabla discrete fractional operator in mass - spring - damper system

The fractional calculus that is one of the new trends in science and engineering is concept of derivative and integral with arbitrary order. And, discrete fractional calculus (DFC) has an important place in fractional calculus which studied for the last 300 years. In present paper, we solved the equations of motion in mass-spring-damper system by using nabla () discrete fractional operator. And, we also introduced some instructive examples.

Keywords:

Fractional calculus, discrete fractional calculus nabla discrete fractional operator,

PDF

___

K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press Cambridge, MA, USA (1974).
J.B. Diaz and T.J. Osler, Differences of fractional order, American Mathematical Society, 28, (1974), 185-202.
K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation, 1st ed., Wiley, NJ, USA, (1993).
I. Podlubny, Matrix approach to discrete fractional calculus, Fract Calc Appl Anal., 3 (4), (2000), 359-386.
F.M. Atici and P.W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory. Differ. Equ., 3, (2009), 1-12.
F.M. Atici and N. Acar, Exponential functions of discrete fractional calculus, Appl. Anal. Discrete. Math., 7, (2013), 343-353.
R. Yilmazer, M. Inc, F. Tchier and D. Baleanu, Particular solutions of the confluent hypergeometric differential equation by using the nabla fractional calculus operator, Entropy, 18 (2), (2016), 49.
F.M. Atici and M. Uyanik, Analysis of discrete fractional operators, Appl. Anal. Discrete Math., 9 (1), (2015), 139-149.
J. Baoguo, L. Erbe and A. Peterson, Convexity for nabla and delta fractional differences, Journal of Difference Equations and Applications, 21 (4), (2015), 360-373.
D. Mozyrska and M. Wyrwas, The L-transform method and delta type fractional difference operators, Discrete Dynamics in Nature and Society, 2015, (2015), 12 pages.
G.-C. Wu and D. Baleanu, New applications of the variational iteration method-from differential Equations to q-fractional difference equations, Advances in Difference Equations, 2013 (21), (2013), 16 pages.
M.D. Ortigueira, F.J.V. Coito and J.J. Trujillo, A new look into the discrete-time fractional calculus: Derivatives and exponentials, fractional differentiation and its applications, 6 (1), (2013), 629-634.
J. Jonnalagadda, Solutions of perturbed linear nabla fractional difference equations, Differ. Equ. Dyn. Syst., 22 (3), (2014) 281-292.
F. Chen, X. Luo and Y. Zhou, Existence results for nonlinear fractional difference equation, Advances in Difference Equations, 2011, (2011), 12 pages.
T. Abdeljawad and D. Baleanu, Fractional differences and integration by parts, Journal of Computational Analysis and Applications, 13 (3), (2011), 574-582.
T. Miyakoda, On an almost free damping vibration equation using N-fractional calculus, Journal of Computational and Applied Mathematics, 144, (2002), 233-240.
A. Palfalvi, Efficient solution of a vibration equation involving fractional derivatives, International Journal of Non-Linear Mechanics, 45, (2010), 169-175.
L.-L. Liu and J.-S. Duan, A detailed analysis for the fundamental solution of fractional vibration equation, Open Mathematics, 13 (1), (2015), 2391-5455.
E.S. Panakhov and M. Sat, Inverse problem for the interior spectral data of the equation of hydrogen atom, Ukrainian Mathematical Journal, 64 (11), (2013), 1716-1726.
M. Sat and E.S. Panakhov, A uniqueness theorem for Bessel operator from interior spectral data, Abstr Appl Anal., 2013, (2013), 6 pages.
M. Sat and E.S. Panakhov, Spectral problem for diffusion operator, Applicable Analysis, 93 (6), (2014), 1178-1186.
R. Yilmazer and O. Ozturk, Explicit solutions of singular differential equation by means of fractional calculus operators, Abstr Appl Anal., 2013, (2013), 6 pages.
R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed., Addison-Wesley, Reading, MA, USA, (1994).
G. Boros and V. Moll, Irresistible Integrals: Symbols, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, (2004).
W.G. Kelley and A.C. Peterson, Difference Equations: An Introduction with Applications. Academic Press, Cambridge, MA, USA, (2001).
J.J. Mohan and G.V.S.R. Deekshitulu, Solutions of fractional difference equations using S-transforms, Malaya J Math., 3 (1), (2013), 7-13.