Journal of Applied Nonlinear Dynamics
Forced Oscillation of a Certain System of Fractional Partial Differential Equations
Journal of Applied Nonlinear Dynamics 10(4) (2021) 607615  DOI:10.5890/JAND.2021.12.002
R. Ramesh$^1$, P. Prakash$^2$, S. Harikrishnan$^3$
$^1$ Department of Mathematics, Muthayammal College of Engineering, Rasipuram 637408, India
$^2$ Department of Mathematics, Periyar University, Salem 636011, India
$^3$ Department of Mathematics, Sona College of Technology, Salem 636030, India
Download Full Text PDF
Abstract
In this paper, we consider a system of fractional partial differential equations subject to Dirichlet and Neumann boundary conditions. Several sufficient conditions are established for forced oscillation of solutions of such systems. Examples are given to illustrate our main results.
Acknowledgments
The authors express their sincere gratitude to the editors and anonymous referee for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results. The second author has supported by Department of Science and Technology, New Delhi, INDIA under FIST programme SR/FST/MS1115/2016.
References

[1]  Coussot, C., Kalyanam, S., Yapp, R., and Insana, M. (2009), {Fractional derivative models for ultrasonic characterization of polymer and breast tissue viscoelasticity}, IEEE Trans.Ultrasonics, Ferroelectrics, and Frequency Control, 56, 715725.


[2]  Djordjevi, V.D., Jari, J., Fabry, B., Fredberg, J.J., and Stamenovi, D. (2003), {Fractional derivatives embody essential features of cell rheological behavior}, Annals of Biomedical Engg., 31, 692699.


[3]  Yang, X.J., Feng, Y.Y., Cattani, C., and Inc, M. (2019), {Fundamental solutions of anomalous diffusion equations with the decay exponential kernel}, Math. Methods Appl. Sci., 42, 40544060.


[4]  Yang, X.J. (2019), General fractional derivatives. Theory, methods and applications. CRC Press, Boca Raton, FL.


[5]  Yang, X.J., Gao, F., Ju, Y., and Zhou, H.W. (2018), {Fundamental solutions of the general fractionalorder diffusion equations}, Math. Methods Appl. Sci., 41, 93129320.


[6]  Yang, X.J. and Machado, J.A.T. (2017), A new fractional operator of variable order: application in the description of anomalous diffusion, Phys. A, 481, 276283.


[7]  Magin, R.L. and Ovadia, M. (2008), {Modeling the cardiac tissue electrode interface using fractional calculus}, J. Vib. Control, 14, 14311442.


[8]  Abdulaziz, O., Hashim, I., and Momani, S. (2008), {Solving systems of fractional differential equations by homotopyperturbation method}, Phys. Lett. A, 372, 451459.


[9]  Deng, W. (2010), {Smoothness and stability of the solutions for nonlinear fractional differential equations}, Nonlinear Anal., 72, 17681777.


[10]  Jafari, H., Nazari, M., Baleanu, D., and Khalique, C.M. (2013), {A new approach for solving a system of fractional partial differential equations}, Comput. Math. Appl., 66, 838843.


[11]  Sun, S., Li, Q., and Li, Y. (2012), {Existence and uniqueness of solutions for a coupled system of multiterm nonlinear fractional differential equations}, Comput. Math. Appl., 64, 33103320.


[12]  Bolat, Y. (2014), {On the oscillation of fractionalorder delay differential equations with constant coefficients}, Commun. Nonlinear Sci. Numer. Simul., 19, 39883993.


[13]  Chen, D.X. (2012), {Oscillation criteria of fractional differential equations}, Adv. Difference Equ., 2012, 33.


[14]  Chen, D., Qu, P., and Lan, Y. (2013), {Forced oscillation of certain fractional differential equations}, Adv. Difference Equ., 2013, 125.


[15]  Grace, S.R., Agarwal, R.P., Wong, P.J.Y., and Zafer, A. (2012), {On the oscillation of fractional differential equations}, Fract. Calc. Appl. Anal., 15, 222231.


[16]  Wang, Y., Han, Z., Zhao, P., and Sun, S. (2014), {On the oscillation and asymptotic behavior for a kind of fractional differential equations}, Adv. Difference Equ., 2014, 50.


[17]  Yang, J., Liu, A.P., and Liu, T. (2015), {Forced oscillation of nonlinear fractional differential equations with damping term}, Adv. Difference Equ., 2015,1.


[18]  Zheng, B. (2013), {Oscillation for a class of nonlinear fractional differential equations with damping term}, J. Adv. Math. Stud., 6, 107115.


[19]  Prakash, P., Harikrishnan, S., Nieto, J.J., and Kim, J.H. (2014), {Oscillation of a time fractional partial differential equation}, Elec. J. Qual. Theory Diff. Eqns., 15, 110.


[20]  Li, W.N. (2015), {On the forced oscillation of certain fractional partial differential equations}, Appl. Math. Letters, 50, 59.


[21]  Prakash, P., Harikrishnan, S., and Benchohra, M. (2015), {Oscillation of certain nonlinear fractional partial differential equation with damping term}, Appl. Math. Letters, 43, 7279.


[22]  Li, W.N. and Sheng, W.H. (2016), {Oscillation properties for solutions of a kind of partial fractional differential equations with damping term}, J. Nonlinear Sci. Appl., 9, 16001608.


[23]  Harikrishnan, S., Prakash, P., and Nieto, J.J. (2015), {Forced oscillation of solutions of a nonlinear fractional partial differential equation}, Appl. Math. Comput., 254, 1419.


[24]  Kong, H. and Xu, R. (2017), {Forced oscillation of fractional partial differential equations with damping term}, Fract. Differ. Calc., 7, 325338.


[25]  Zhou, Y., Ahmad, B., Chen, F., and Alsaedi, A. (2017), {Oscillation for fractional partial differential equations}, Bull. Malays. Math. Sci. Soc., 42, 449465.


[26]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), {Theory and Applications of Fractional Differential Equations}, Elsevier Science B.V., Amsterdam.


[27]  Courant, R. and Hilbert, D. (1966), Methods of Mathematical Physics, Vol I, Interscience, New York.
