THERMAL SCIENCE

International Scientific Journal

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Resat Yilmazer

,

Okkes Ozturk

ON NABLA DISCRETE FRACTIONAL CALCULUS OPERATOR FOR A MODIFIED BESSEL EQUATION

ABSTRACT
In thermal sciences, it is possible to encounter topics such as Bessel beams, Bessel functions or Bessel equations. In this work, we also present new discrete fractional solutions of the modified Bessel differential equation by means of the nabla-discrete fractional calculus operator. We consider homogeneous and non-homogeneous modified Bessel differential equation. So, we acquire four new solutions of these equations in the discrete fractional forms via a newly developed method
KEYWORDS
discrete fractional calculus, modified Bessel equation, nabla operator
PAPER SUBMITTED: 2017-06-14
PAPER REVISED: 2017-11-10
PAPER ACCEPTED: 2017-11-20
PUBLISHED ONLINE: 2018-01-07
DOI REFERENCE: https://doi.org/10.2298/TSCI170614287Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2018, VOLUME 22, ISSUE Supplement 1, PAGES [S203 - S209]
REFERENCES
  1. Oldham, K.B., Spanier, J., The Fractional Calculus, Academic Press, New York, USA, 1974.
  2. Podlubny, I., Fractional Differential Equations, in: Mathematics in Science and Engineering, Academic Press, San Diego, USA, 1999.
  3. Ozturk, O., Yilmazer, R., Solutions of the Radial Component of the Fractional Schrödinger Equation using N-fractional Calculus Operator, Differ. Equ. Dyn. Syst., (2016), pp.1-9, doi: 10.1007/s12591-016- 0308-8.
  4. Tchier, F., Inc, M., Korpinar, Z., Baleanu, D., Solutions of the Time Fractional Reaction-diffusion Equations with Residual Power Series Method, Adv. Mech. Eng., 8 (2016), 10, pp.1-10.
  5. Ozturk, O., A Study on Nabla Discrete Fractional Operator in Mass-Spring-Damper System, New Trends Math. Sci., 4 (2016), 4, pp.137-144.
  6. Atici, F.M., Eloe, P.W., A Transform Method in Discrete Fractional Calculus, Int. J. Difference Equations, 2 (2007), 2, pp.165-176.
  7. Atici, F.M., Eloe, P.W., Initial Value Problems in Discrete Fractional Calculus, Proc. American Math. Soc., 137 (2009), 3, pp.981-989.
  8. Atici, F.M., Eloe, P.W., Discrete Fractional Calculus with the Nabla Operator, Electronic J. Qualitative Theory Dif. Eqs. Spec. Ed. I, 3 (2009) 1-12.
  9. Atici, F.M., Sengul, S., Modeling with Fractional Difference Equations, J. Math. Anal. Applic., 369 (2010), pp.1-9.
  10. Bastos, N.R.O., Torres, D.F.M., Combined Delta-nabla Sum Operator in Discrete Fractional Calculus, Commun. Frac. Calculus, 1 (2010), pp.41-47.
  11. Holm, M., Sum and Difference Compositions in Discrete Fractional Calculus, CUBO A Math. J., 13 (2011), 3, pp.153-184.
  12. Anastassiou, G.A., Right Nabla Discrete Fractional Calculus, Int. J. Difference Equations, 6 (2011), 2, pp.91-104.
  13. Mohan, J.J., Solutions of Perturbed Nonlinear Nabla Fractional Difference Equations, Novi Sad J. Math., 43 (2013), 2, pp.125-138.
  14. Han, Z.L., Pan,Y.Y., Yang, D.W., The Existence and Nonexistence of Positive Solutions to a Discrete Fractional Boundary Value Problems with a Parameter, App. Math. Lett., 36 (2014), pp.1-6.
  15. Okrasi´nski, W., Płociniczak, Ł., A Note on Fractional Bessel Equation and Its Asymptotics, Fract. Calc. Appl. Anal., 16 (2013), 3, pp.559-572.
  16. Robin,W., Ladder-operator Factorization and the Bessel Differential Equations, Math. Forum, 9 (2014), 2, pp.71-80.
  17. Lin, S.-D., Ling,W.-C., Nishimoto, K., Srivastava, H.M., A Simple Fractional-calculus Approach to the Solutions of the Bessel Difeerential Equation of General Order and Some of Its Applications, Comput. Math. Appl., 49 (2005), pp.1487-1498.
  18. Chadan, K., Sabatier, P., Inverse Problems in Quantum Scattering Theory, Springer, New York, USA, 1977.
  19. Doan, D.H., Iida, R., Kim, B., Satoh, I., Fushinobu, K., Bessel Beam Laser-scribing of Thin Film Silicon Solar Cells by ns Pulsed Laser, J. Therm. Sci. Tech-Jpn, 11 (2016), pp.1-6.
  20. Castañeda, J.A., Pérez-Pascual, R., Jáuregui, R., Chaotic Dynamics of Dilute Thermal Atom Clouds on Stationary Optical Bessel Beams, J. Phys. B: At. Mol. Opt., 46 (2013), pp.1-10.
  21. Zhang, Q., Cheng, X.M., Chen, H.W., He, B., Ren, Z.Y., Zhang, Y., Bai, J.T., Diffraction-free, Selfreconstructing Bessel Beam Generation using Thermal Nonlinear Optical Effect, Appl. Phys. Lett., 111 (2017), pp.1-5.
  22. Acton, Q.A., Issues in Mechanical Engineering: 2013 Edition, Scholarly Editions, Atlanta, USA, 2013.
  23. Jones, J.C., The Principles of Thermal Sciences and Their Application to Engineering, CRC Press, Boca Raton, USA, 2000.
  24. Stauffer, F., Bayer, P., Blum, P., Giraldo, N.M., Kinzelbach,W., Thermal Use of Shallow Groundwater, CRC Press, Boca Raton, USA, 2013.
  25. Swain, J., Gaurav, K., Singh, D., Sen, P.K., Bohidar, S.K., A Comparative Study on Heat Transfer in Straight Triangular Fin and Porous Pin Fin under Natural Convection, Int. J. Innov. Sci. Res., 11 (2014), 2, pp.611-619.
  26. Goodrich, C.S., Continuity of Solutions to Discrete Fractional Initial Value Problems, Comp. Math. Applic., 59 (2010), pp.3489-3499.
  27. Goodrich, C.S., Existence and Uniqueness of Solutions to a Fractional Difference Equation with Nonlocal Conditions, Comp. Math. Applic., 61 (2011), pp.191-202.
  28. Yilmazer, R., N-fractional Calculus Operator N Method to a Modified Hydrogen Atom Equation, Math. Commun., 15 (2010), pp.489-501.
  29. Kelley, W.G., Peterson, A.C., Difference Equations: An Introduction with Applications, Academic Press, San Diego, USA, 2001.
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On nabla discrete fractional calculus operator for a modified Bessel equation

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence