Forced oscillation of conformable fractional partial delay differential equations with impulses

Saker, S. H. ; Logaarasi, K. ; Sadhasivam, V.

Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 74 (2020) no. 2.

Résumé

In this paper, we establish some interval oscillation criteria for impulsive conformable fractional partial delay differential equations with a forced term. The main results will be obtained by employing Riccati technique. Our results extend and improve some results reported in the literature for the classical differential equations without impulses. An example is provided to illustrate the relevance of the new theorems.

Détail

Keywords: Oscillation, fractional differential equations, impulse

Bibliographie

[1] Atangana, A., Baleanu, D., Alsaedi, A., New properties of conformable derivative, Open Math. 13 (2015), 889–898.

[2] Bainov, D. D., Simenov, P.S., Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Harlow, 1993.

[3] Chatzarakis, G. E., Logaarasi, K., Raja, T., Sadhasivam, V., Interval Oscillation Criteria for Impulsive Conformable Partial Differential Equations, Appl. Anal. Discrete Math. 13 (2019), 325–345.

[4] Chatzarakis, G. E., Logaarasi, K., Raja, T., Sadhasivam, V., Interval oscillation criteria for conformable fractional differential equations with impulses, Applied Mathematics E-Notes, 19 (2019), 354–369.

[5] Chen, D. X., Oscillation criteria of fractional differential equations, Adv. Diff. Equ. 2012 (2012), 1–10.

[6] Delbosco, D., Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (2) (1996), 609–625.

[7] El-Sayed, A. M. A., Fractional-order diffusion-wave equation, Internat. J. Theoret. Phys. 35 (2) (1996), 311–322.

[8] V. Gafiychuk, B. Datsko and V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J. Comput. Appl. Math. 220 (1–2) (2008), 215–225.

[9] Grace, S. R., Agarwal, R. P.,Wong, P. J. Y., Zafer, A., On the oscillation of fractional differential equations, Frac. Calc. Appl. Anal. 15 (2012), 222–231.

[10] He, J., Some applications of nonlinear fractional differential equations and their approximations, Bulletin of Science and Technology 15 (1999), 86–90.

[11] Kalaimani, T., Raja, T., Sadhasivam, V., Saker, S.H., Oscillation of impulsive neutral partial differential equations with distributed deviating arguments, Bul. Math. Soc. Sci. Math. Roumanie (N. S.) 61(109) (2018), 51–68.

[12] Khalil, R. R., Horani, M. Al., Yousef, A., Sababheh, M., A new definition of fractional derivative, J. Com. Appl. Math. 264 (2014), 65–70.

[13] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[14] Kubiaczyk, I., Saker, S. H., Oscillation of parabolic delay differential equations with positive and negative coefficients, Comment. Math. 42 (2002), 221–236.

[15] Kubiaczyk, I., Saker, S. H., Oscillation of delay parabolic differential equations with several coefficients, J. Comp. Appl. Math. 147 (2) (2002), 263–275.

[16] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S., Theory of Impulsive Differential Equations, World Scientific Publishers, Singapore, 1989.

[17] Li, Q. L., Cheung, W. S., Interval oscillation criteria for second-order forced delay differential equations under impulsive effects, Electron. J. Qual. Theor. Diff. Eq. 2013 (43) (2013), 1–11.

[18] Metzler, R., Schick, W., Kilian, H. G, Nonnenmacher, T. F., Relaxation in filled polymers: a fractional calculus approach, Journal of Chemical Physics 103 (1995), 7180–7186.

[19] Muthulakshmi, V., Thandapani, E., Interval criteria for oscillation of second-order impulsive differential equation with mixed nonlinearities, ,Electron. J. Differ. Eq., 2011(40) (2011), 1–14.

[20] Ozbekler, A., Zafer, A., Oscillation of solutions of second order mixed nonlinear differential equations under impulsive perturbations, Comput. Math. Appl. 61 (2011), 933–940.

[21] Philos, Ch. G., Oscillation theorems for linear differential equations of second order, Arch. Math. 53 (1989), 482–492.

[22] Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.

[23] Ryabov, Y. E., Puzenko, A., Damped oscillations in view of the fractional oscillator equation, Physics Review B 66 (2002), 184–201.

[24] Sadhasivam, V., Logaarasi ,K., Raja, T., Interval oscillation criteria for impulsive partial differential equations, Int. J. Math. and Appl. 6(1-B) (2018), 229–242.

[25] Sadhasivam, V., Raja, T., Logaarasi, K., On the interval oscillation of impulsive partial differential equations with damping term, Int. J. Engg. Sci. Math. 6 (2017), 328–340.

[26] Saker, S. H., Oscillation of hyperbolic nonlinear differential equations with deviating arguments, Publ. Math. Debr. 62 (2003), 165–185.

[27] Samko, S. G., Kilbas, A. A., Maritchev, O. I., Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Sci. Publ., Amsterdam, 1993.

[28] Thandapani, E., Manju, E., Pinelas, S., Interval oscillation criteria for second order forced impulsive delay differential equations with damping term, Springer Plus 5 (2016), 1–16.

[29] Vladimirov, V. S., Equations of Mathematical Physics. Translated from the Russian by Eugene Yankovsky, Nauka, Moscow, 1964.

[30] Wu, J. H., Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996.

[31] Xiaoliang, Z., Zhonghai, G., Wu-Sheng, W., Interval oscillation criteria for superhalf-linear impulsive differential equations with delay, J. Appl. Math. (2012), art. id. 285051, 22 pp.

[32] Yoshida, N., Oscillation Theory of Partial Differential Equations, World Scientific, Singapore, 2008.