Forced Oscillation of Solutions of a Fractional Neutral Partial Functional Differential Equation

Affiliation(s)

Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram, Namakkal Dt. Tamil Nadu, India.

Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram, Namakkal Dt. Tamil Nadu, India.

ABSTRACT

In this paper, we will establish the sufficient conditions for the oscillation of solutions of neutral time fractional partial differential equation of the form

for where Ω is a bounded domain in*R*^{N} with a piecewise smooth boundary is a constant, is the Riemann-Liouville fractional derivative of order a of *u* with respect to *t* and is the Laplacian operator in the Euclidean *N*-space *R*^{N} subject to the condition

In this paper, we will establish the sufficient conditions for the oscillation of solutions of neutral time fractional partial differential equation of the form

for where Ω is a bounded domain in

Cite this paper

Sadhasivam, V. and Kavitha, J. (2015) Forced Oscillation of Solutions of a Fractional Neutral Partial Functional Differential Equation.*Applied Mathematics*, **6**, 1302-1317. doi: 10.4236/am.2015.68124.

Sadhasivam, V. and Kavitha, J. (2015) Forced Oscillation of Solutions of a Fractional Neutral Partial Functional Differential Equation.

References

[1] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons, New York.

[2] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.

[3] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations, Volume 204. Elsevier Science B.V., Amsterdam.

[4] Das, S. (2008) Functional Fractional Calculus for System Identification and Controls. Springer, Berlin.

[5] Zhou, Y. (2014) Basic Theory of Fractional Differential Equations. World Scientific, Singapore.

http://dx.doi.org/10.1142/9069

[6] Wu, J. (1996) Theory of Partial Functional Differential Equations Applications. Springer, New York.

http://dx.doi.org/10.1007/978-1-4612-4050-1

[7] Thandapani, E. and Savithri, R. (2003) On Oscillation of a Neutral Partial Functional Differential Equation. Bulletin of the Institute of Mathematics - Academia Sinica, 31, 273-292.

[8] Xu, R. and Meng, F. (2013) Oscillation Criteria for Neutral Partial Functional Differential Equations. Differential Equations & Applications, 5, 69-82.

http://dx.doi.org/10.7153/dea-05-05

[9] Lakshimikantham, V. and Vasundhara Devi, J. (2008) Theory of Fractional Differential Equations in a Banach Space. European Journal of Pure and Applied Mathematics, 1, 38-45.

[10] Chen, D.X. (2012) Oscillation criteria of Fractional Differential Equations. Advances in Difference Equations, 33.

http://dx.doi.org/10.1186/1687-1847-2012-33

[11] Grace, S.R., Agarwal, R.P., Wong, P.J.Y. and Zaffer, A. (2012) On the Oscillation of Fractional Differential Equations. Fractional Calculus and Applied Analysis, 15, 222-231.

http://dx.doi.org/10.2478/s13540-012-0016-1

[12] Chen, D.X. (2013) Oscillatory Behavior of a Class of Fractional Differential Equations with Damping. U.P.B. Sci. Bull., Series A, 75, 107-118.

[13] Feng, Q.H. and Meng, F.W. (2013) Oscillation of Solutions to Nonlinear Forced Fractional Differential Equation. Electronic Journal of Differential Equations, 2013, 1-10.

[14] Zheng, B. (2013) Oscillation for a Class of Nonlinear Fractional Differential Equations with Damping Term. J. Adv. Math. Stud, 6, 107-115.

[15] Han, Z.L. Zhao, Y.G., Sun, Y. and Zhang, C. (2013) Oscillation for a Class of Fractional Differential Equations. Discrete Dynamics in Nature and Society, 2013, 6 p.

[16] 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, 1-10.

http://dx.doi.org/10.14232/ejqtde.2014.1.15

[17] Li, W.N. (2015) Forced Oscillation Criteria for a Class of Fractional Partial Differential Equations with Damping Term. Mathematical Problems in Engineering, 2015, Article ID: 410904, 6 p.

[1] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons, New York.

[2] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.

[3] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations, Volume 204. Elsevier Science B.V., Amsterdam.

[4] Das, S. (2008) Functional Fractional Calculus for System Identification and Controls. Springer, Berlin.

[5] Zhou, Y. (2014) Basic Theory of Fractional Differential Equations. World Scientific, Singapore.

http://dx.doi.org/10.1142/9069

[6] Wu, J. (1996) Theory of Partial Functional Differential Equations Applications. Springer, New York.

http://dx.doi.org/10.1007/978-1-4612-4050-1

[7] Thandapani, E. and Savithri, R. (2003) On Oscillation of a Neutral Partial Functional Differential Equation. Bulletin of the Institute of Mathematics - Academia Sinica, 31, 273-292.

[8] Xu, R. and Meng, F. (2013) Oscillation Criteria for Neutral Partial Functional Differential Equations. Differential Equations & Applications, 5, 69-82.

http://dx.doi.org/10.7153/dea-05-05

[9] Lakshimikantham, V. and Vasundhara Devi, J. (2008) Theory of Fractional Differential Equations in a Banach Space. European Journal of Pure and Applied Mathematics, 1, 38-45.

[10] Chen, D.X. (2012) Oscillation criteria of Fractional Differential Equations. Advances in Difference Equations, 33.

http://dx.doi.org/10.1186/1687-1847-2012-33

[11] Grace, S.R., Agarwal, R.P., Wong, P.J.Y. and Zaffer, A. (2012) On the Oscillation of Fractional Differential Equations. Fractional Calculus and Applied Analysis, 15, 222-231.

http://dx.doi.org/10.2478/s13540-012-0016-1

[12] Chen, D.X. (2013) Oscillatory Behavior of a Class of Fractional Differential Equations with Damping. U.P.B. Sci. Bull., Series A, 75, 107-118.

[13] Feng, Q.H. and Meng, F.W. (2013) Oscillation of Solutions to Nonlinear Forced Fractional Differential Equation. Electronic Journal of Differential Equations, 2013, 1-10.

[14] Zheng, B. (2013) Oscillation for a Class of Nonlinear Fractional Differential Equations with Damping Term. J. Adv. Math. Stud, 6, 107-115.

[15] Han, Z.L. Zhao, Y.G., Sun, Y. and Zhang, C. (2013) Oscillation for a Class of Fractional Differential Equations. Discrete Dynamics in Nature and Society, 2013, 6 p.

[16] 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, 1-10.

http://dx.doi.org/10.14232/ejqtde.2014.1.15

[17] Li, W.N. (2015) Forced Oscillation Criteria for a Class of Fractional Partial Differential Equations with Damping Term. Mathematical Problems in Engineering, 2015, Article ID: 410904, 6 p.