Paper title: Oscillation Theorems for Fractional Order Neutral Differential Equations
Published in: Issue 2, (Vol. 10) / 2016Download
Publishing date: 2016-10-20
Pages: 46-51
Author(s): GANESAN Vellaiyappaudaiyar, KUMAR Sathish M.
Abstract. The purpose of this paper is to study the oscillation of the fractional order neutral differential equation ๐‘ซ๐’•๐œถ[๐’“(๐’•)[๐‘ซ๐’•๐œถ(๐’™(๐’•)+๐’‘(๐’•)๐’™(๐‰(๐’•)))]๐œธ]+๐’’(๐’•)๐’™๐œธ๐œธ(๐ˆ(๐’•))=๐ŸŽ, where ๐‘ซ๐’•๐œถ(โ‹…) is a modified Riemann-Liouville derivative. The obtained results are based on the new comparison theorems, which enable us to reduce the oscillatory problem of ๐Ÿ๐œถ-order fractional differential equation to the oscillation of the first order equation. The results are easily verified.
Keywords: Oscillation; Comparison Theorem; Fractional Differential Equation; Modified Riemann-Liouville Derivative

1. Bagley, R.L. and Torvik, P.J., 1983. A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology (1978-present), 27(3), pp.201-210.

2. Mainardi, F., 1997. Fractional calculus: Some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, 291-348.

3. Mandelbrot, B., 1967. Some noises with 1/f spectrum, a bridge between direct current and white noise. IEEE transactions on Information Theory, 13(2), pp.289-298.

4. Rossikhin, Y.A. and Shitikova, M.V., 1997. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Applied Mechanics Reviews, 50(1), pp.15-67.

5. Baillie, R.T., 1996. Long memory processes and fractional integration in econometrics. Journal of econometrics, 73(1), pp.5-59.

6. Magin, R.L., 2004. Fractional calculus in bioengineering, part 1. Critical Reviewsโ„ข in Biomedical Engineering, 32(1) pp.1-377.

7. Miller, K.S. and Ross, B., 1993. An introduction to the fractional calculus and fractional differential equations.

8. Podlubny, I., 1999. Fractional Differential Equations, Academic Press, San Diego.

9. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., 2006. Preface. North-Holland mathematics studies, 204, pp.vii-x.

10. Lakshmikantham, V., Leela, S., Vaaundhara Devi, J., 2009. Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers.

11. Baleanu, D., Machado, J.A.T. and Luo, A.C. eds., 2011. Fractional dynamics and control. Springer Science & Business Media.

12. Saadatmandi, A. and Dehghan, M., 2010. A new operational matrix for solving fractional - order differential equations. Computers & mathematics with applications, 59(3), pp.1326-1336.

13. Zhou, Y., Jiao, F. and Li, J., 2009. Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Analysis: Theory, Methods & Applications, 71(7), pp.2724-2733.

14. Galeone, L. and Garrappa, R., 2009. Explicit methods for fractional differential equations and their stability properties. Journal of Computational and Applied Mathematics, 228(2), pp.548-560.

15. Deng, W., 2010. Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications, 72(3), pp.1768-1777.

16. Edwards, J.T., Ford, N.J. and Simpson, A.C., 2002. The numerical solution of linear multi-term fractional differential equations: systems of equations. Journal of Computational and Applied Mathematics, 148(2), pp.401-418.

17. Muslim, M., 2009. Existence and approximation of solutions to fractional differential equations. Mathematical and Computer Modelling, 49(5), pp.1164-1172.

18. Jiang, J. and Li, X., 2003. Oscillation of second order nonlinear neutral differential equations. Applied Mathematics and Computation, 135(2), pp.531-540.

19. Prakash, P., Harikrishnan, S., Nieto, J.J. and Kim, J.H., 2014. Oscillation of a time fractional partial differential equation. Electronic Journal of Qualitative Theory of Differential Equations, 15, pp.1-10.

20. Parhi, N., 2011. Oscillation and non-oscillation of solutions of second order difference equations involving generalized difference. Applied Mathematics and Computation, 218(2), pp.458-468.

21. Erbe, L., Hassan, T.S. and Peterson, A., 2010. Oscillation of third order nonlinear functional dynamic equations on time scales. Differential Equations and Dynamical Systems, 18(1-2), pp.199-227.

22. Qi, C. and Cheng, J., 2013. Interval oscillation criteria for a class of fractional differential equations with damping term. Mathematical Problems in Engineering, pp.1-8.

23. Baculรญkovรก, B. and Dลพurina, J., 2011. Oscillation theorems for second order neutral differential equations. Computers & Mathematics with Applications, 61(1), pp.94-99.

24. Baculรญkovรก, B. and Dลพurina, J., 2011. Oscillation theorems for second-order nonlinear neutral differential equations. Computers & Mathematics with Applications, 62(12), pp.4472-4478.

25. Chen, D.X., 2012. Oscillation criteria of fractional differential equations, Adv. Difference Equ. No. 33, 10.

26. Feng, Q. and Meng, F., 2013. Oscillation of solutions to nonlinear forced fractional differential equations. Electronic Journal of Differential Equations, 2013(169), pp.1-10.

27. Grace, S., Agarwal, R., Wong, P. and Zafer, A., 2012. On the oscillation of fractional differential equations. Fractional Calculus and Applied Analysis, 15(2), pp.222-231.

28. Liu, T., Zheng, B. and Meng, F., 2013. Oscillation on a class of differential equations of fractional order. Mathematical Problems in Engineering, 2013, pp 1-13.

29. Qin, H. and Zheng, B., 2013. Oscillation of a class of fractional differential equations with damping term. The Scientific World Journal , 1-9.

30. Zheng, B., 2013. Oscillation for a class of nonlinear fractional differential equations with damping term. Journal of Advanced Mathematical Studies, 6(1), pp.107-115.

31. Ladde, G.S., Lakshmikantham, V. and Zhang, B.G., 1987. Oscillation theory of differential equations with deviating arguments (Vol. 110). Marcel Dekker Inc.

32. Philos, C.G., 1981. On the existence of nonoscillatory solutions tending to zero at โˆž for differential equations with positive delays. Archiv der Mathematik, 36(1), pp.168-178.

33. Wang, Y.Z., Han, Z.L., Zhao, P. and Sun, S.R., 2015. Oscillation theorems for fractional neutral differential equations. Hacettepe Journal of Mathematics and Statistics, 44(6), pp.1477-1488.

34. Jumarie, G., 2006. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications, 51(9), pp.1367-1376.

Back to the journal content
Creative Commons License
This article is licensed under a
Creative Commons Attribution-ShareAlike 4.0 International License.
Home | Editorial Board | Author info | Archive | Contact
Copyright JACSM 2007-2020