MONOTONE METHOD FOR MULTI-ORDER 2-SYSTEMS OF RIEMANN-LIOUVILLE FRACTIONAL DIFFERENTIAL EQUATIONS

ZACHARY DENTON

Title	MONOTONE METHOD FOR MULTI-ORDER 2-SYSTEMS OF RIEMANN-LIOUVILLE FRACTIONAL DIFFERENTIAL EQUATIONS
Publication Type	Journal Article
Year of Publication	2015
Authors	DENTON, ZACHARY
Volume	19
Issue	3
Start Page	353
Pagination	15
Date Published	2015
ISSN	1083-2564
AMS	34A08, 34A34, 34A38, 34A45
Abstract	In this paper we develop the monotone method for nonlinear multi-order 2-systems of Riemann-Liouville fractional differential equations. That is, a hybrid system of nonlinear equations of orders ${ q1 }$ and ${ q2 }$ where ${ 0 < q1, q2 < 1.}$ In the development of this method we recall any needed existence results along with any necessary changes; including results from needed linear theory. Further we prove a comparison result paramount for the discussion of fractional multi-order inequalities that utilizes lower and upper solutions of the system. The monotone method is then developed via the construction of sequences of linear systems based on the upper and lower solutions, and are used to approximate the solution of the original nonlinear multi-order system.
URL	http://www.acadsol.eu/en/articles/19/3/4.pdf
Refereed Designation	Refereed
Full Text	REFERENCES [1] M. Caputo. Linear models of dissipation whose Q is almost independent, II. Geophy. J. Roy. Astronom, 13:529–539, 1967. [2] A. Chowdhury and C. I. Christov. Memory effects for the heat conductivity of random suspensions of spheres. Proc. R. Soc. A, 466:3253–3273, 2010. [3] Z. Denton and A. S. Vatsala. Monotone iterative technique for finite systems of nonlinear Riemann-Liouville fractional differential equations. Opuscula Mathematica, 31(3):327–339, 2011. [4] K. Diethelm and A. D. Freed. On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity. In F. Keil, W. Mackens, H. Vob, and J. Werther, editors, Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, pages 217–224. Heidelberg: Springer, 1999. [5] W. G. Gl¨ockle and T. F. Nonnenmacher. A fractional calculus approach to self similar protein dynamics. Biophy. J., 68:46–53, 1995. [6] R. Hilfer, editor. Applications of Fractional Calculus in Physics. World Scientific Publishing, Germany, 2000. [7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier, North Holland, 2006. [8] V. Kiryakova. Generalized fractional calculus and applications, volume 301 of Pitman Res. Notes Math. Ser. Longman-Wiley, New York, 1994. [9] G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala. Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman Publishing Inc, 1985. [10] V. Lakshmikantham, S. Leela, and D. J. Vasundhara. Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, 2009. [11] R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher. Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phy., 103:7180–7186, 1995. [12] B. Oldham and J. Spanier. The Fractional Calculus. Academic Press, New York - London, 2002.