Document Type: Research Paper
Authors
- Devaraj Vivek ^{1}
- Omid Baghani ^{} ^{2}
- Kuppusamy Kanagarajan ^{1}
^{1} Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore-641020, India.
^{2} Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran.
Abstract
We develop the theory of hybrid fractional differential equations with the complex order $\theta\in \mathbb{C}$, $\theta=m+i\alpha$, $0<m\leq 1$, $\alpha\in \mathbb{R}$, in Caputo sense. Using Dhage's type fixed point theorem for the product of abstract nonlinear operators in Banach algebra; one of the operators is $\mathfrak{D}$- Lipschitzian and the other one is completely continuous, we prove the existence of mild solutions of initial value problems for hybrid fractional differential equations. Finally, an application to solve one-variable linear fractional Schr\"odinger equation with complex order is given.
Keywords
- Hybrid fractional differential equations
- Initial value problem
- Complex order
- Dhage's fixed point theorems
- Existence of mild solution
Main Subjects
[1] B. Ahmad and S.K. Ntouyas, An existence theorem for fractional hybrid differential inclusions of hadamard type with Dirichlet boundary conditions, Abstr. Appl. Anal., (2014), Article ID 705809, 7 pages.
[2] B. Ahmad and S.K. Ntouyas, Initial value problems for hybrid Hadamard fractional differential equations, Electron. J. Diff. Eq., 161 (2014), pp. 1-8.
[3] R. Andriambololona, R. Tokiniaina, and H. Rakotoson, Definitions of complex order integrals and complex order derivatives using operator approach, Int. J. Latest Res. Sci. Tech., 1 (2012), pp. 317-323.
[4] T.M. Atanackovic, S. Konjik, S. Pilipovic, and D. Zorica, Complex order fractional derivatives in viscoelasticity, Mech. Time-Depend. Mater., 1 (2016), pp. 1-21.
[5] O. Baghani, On fractional Langevin equation involving two fractional orders, Commun. Nonlinear
Sci. Numer. Simulat., 42 (2017), pp. 675-681.
[6] B.C. Dhage, On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett., 18 (2005), pp. 273-280.
[7] B.C. Dhage, On some variants of Schauder's fixed point principle and applications to nonlinear integral equations, J. Math. Phys. Sci., 25 (1988), pp. 603-611.
[8] B.C. Dhage and V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear Anal. Hybrid Syst., 4 (2010), pp. 414-424.
[9] P. Gorka, H. Prado, and J. Trujillo, The time fractional Schr"odinger equation on Hilbert space, Integr. Equ. Oper. Theory, 88 (2017), pp. 1-14.
[10] M.A.E. Herzallah and D. Baleanu, On fractional order hybrid differential equations, Abst. Appl. Anal., 2014, Article ID 389386, 7 pages.
[11] R. Hilfer, Application of Fractional Calculus in Physics, World Scientific, Singapore, 1999.
[12] A.A. Kilbas, H.M. Srivasta, and J.J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier B. V., Netherlands, 2016.
[13] N. Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Anal., 70 (2009), pp. 2521-2529.
[14] E.R. Love, Fractional derivatives of imaginary order, J. London Math. Soc., 2 (1971), pp. 241-259.
[15] A. Neamaty, M. Yadollahzadeh, and R. Darzi, On fractional differential equation with complex order, Progr. Fract. Differ. Appl., 1 (2015), pp. 223-227.
[16] C.M.A. Pinto and J.A.T. Machado, Complex order Van der Pol oscillator, Nonlinear Dyn., 65 (2011), pp. 247-254.
[17] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[18] B. Ross and F. Northover, A use for a derivative of complex order in the fractional calculus, Int. J. Pure Appl. Math., 9 (1978), pp. 400-406.
[19] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Philadelphia, 1993.
[20] D. Vivek, K. Kanagarajan, and S. Harikrishnan, Dynamics and stability results for integro-differential equations with complex order, Discontinuity, Nonlinearity and Complexity, In Press, 2018.
[21] D. Vivek, K. Kanagarajan, and S. Harikrishnan, Dynamics and stability results for pantograph equations with complex order, Journal of Applied Nonlinear Dynamics, 7 (2018), pp. 179-187.
[22] Y. Zhao, S. Sun, Z. Han, and Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl., 62 (2011), pp. 1312-1324.